Fractional Partial Differential Equation

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 483791 19 pageshttpdxdoiorg1011552013483791

Research ArticleFractional Partial Differential Equation Fractional TotalVariation and Fractional Steepest Descent Approach-BasedMultiscale Denoising Model for Texture Image

Yi-Fei Pu12 Ji-Liu Zhou1 Patrick Siarry3 Ni Zhang4 and Yi-Guang Liu1

1 School of Computer Science and Technology Sichuan University Chengdu 610065 China2 State Key Laboratory of Networking and Switching Technology Beijing University of Posts and TelecommunicationsBeijing 100876 China

3Universite de Paris 12 (LiSSi EA 3956) 61 avenue du General de Gaulle 94010 Creteil Cedex France4 Library of Sichuan University Chengdu 610065 China

Correspondence should be addressed to Yi-Fei Pu puyifei 007163com

Received 14 June 2013 Accepted 11 August 2013

Academic Editor Juan J Trujillo

Copyright copy 2013 Yi-Fei Pu et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The traditional integer-order partial differential equation-based image denoising approaches often blur the edge and complextexture detail thus their denoising effects for texture image are not very good To solve the problem a fractional partial differentialequation-based denoisingmodel for texture image is proposed which applies a novel mathematical methodmdashfractional calculus toimage processing from the view of system evolutionWe know fromprevious studies that fractional-order calculus has some uniqueproperties comparing to integer-order differential calculus that it can nonlinearly enhance complex texture detail during the digitalimage processingThegoal of the proposedmodel is to overcome the problemsmentioned above by using the properties of fractionaldifferential calculus It extended traditional integer-order equation to a fractional order and proposed the fractional Greenrsquos formulaand the fractional Euler-Lagrange formula for two-dimensional image processing and then a fractional partial differential equationbased denoising model was proposedThe experimental results prove that the abilities of the proposed denoisingmodel to preservethe high-frequency edge and complex texture information are obviously superior to those of traditional integral based algorithmsespecially for texture detail rich images

1 Introduction

Fractional calculus has been an important branch of math-ematical analysis over the last 300 years [1ndash4] however itis still little known by many mathematicians and physicalscientists in both the domestic and overseas engineeringfields Fractional calculus of theHausdorffmeasure is notwellestablished after more than 90 years studies [5 6] whereasfractional calculus in the Euclidean measure seems morecompleted So Euclidean measure is commonly required inmathematics [5 6] In general fractional calculus in theEuclidean measure extends the integer step to a fractionalstep Random variable of physical process in the Euclideanmeasure can be deemed to be the displacement of particlesby random movement thus fractional calculus can be used

for the analysis and processing of the physical states andprocesses in principle [7ndash15] Fractional calculus has oneobvious feature that is that most fractional calculus is basedon a power function and the rest is based on the addition orproduction of a certain function and a power function [1ndash6]It is possible that this feature indicates some changing lawof nature Scientific research has proved that the fractional-order or dimensional mathematical approach provides thebest description for many natural phenomena [16ndash19] Frac-tional calculus in the Euclidean measure has been usedin many fields including diffusion process viscoelasticitytheory and random fractal dynamics Methods to applyfractional calculus to modern signal analysis and processing[18ndash30] especially to digital image processing [31ndash38] are anemerging branch to study which has been seldom explored

2 Abstract and Applied Analysis

Integer-order partial differential equation-based imageprocessing is an important branch in the field of image pro-cessing By exploring the essence of image and image process-ing people tend to reconstruct the traditional image process-ing approaches through strictly mathematical theories andit will be a great challenge to practical-oriented traditionalimage processing Image denoising is a significant researchbranch of integer-order partial differential equation-basedimage processing with two kinds of denoising approach thenonlinear diffusion-based method and the minimum energynorm-based variational method [39ndash42] They have twocorresponding models which are the anisotropic diffusionproposed by Perona and Malik [43] (Perona-Malik or PM)and the total variation model proposed by Rudin et al [44](Rudin-Osher-Fatemi or ROF) The PM model simulatesthe denoising process by a thermal energy diffusion processand the denoising result is the balanced state of thermaldiffusion while the ROF model describes the same thermalenergy by a total variation In further study some researchershave applied the PM model and the ROF model to colorimages [45 46] discussed the selection of the parameters forthe models [47ndash51] and found the optimal stopping pointin iteration process [52 53] Rudin and his team proposeda variable time step method to solve the Euler-Lagrangeequation [44] Vogel and Oman proposed improving thestability of ROF model by a fixed point iteration approach[54] Darbon and Sigelle decomposed the original problemsinto independent optimal Markov random fields by usinglevel set methods and obtained globally optimal solution byreconstruction [55ndash57] Wohlberg and Rodriguez proposedto solve the total variation by using an iterative weightednorm to improve the computing efficiency [58] MeanwhileCatte et al proposed to perform a Gaussian smoothingprocess in the initial stage to improve the suitability of thePM model [59] However PM model and ROF model havesome obvious defects in image denoising that is they caneasily lose the contrast and texture details and can producestaircase effects [39 60 61] Some improved models havebeen proposed to solve these problems To maintain thecontrast and texture details some scholars have proposedto replace the 119871

2 norm with the 1198711 norm [62ndash65] while

Osher et al proposed an iterative regularizationmethod [66]Gilboa et al proposed a denoising method using a numericaladaptive fidelity term that can change with the space [67]Esedoglu andOsher proposed to decompose images using theanisotropic ROFmodel and retaining certain edge directionalinformation [68] To remove the staircase effects Blomgren etal proposed to extend the total variation denoising model bychanging it with gradients [69 70] Some scholars introducedhigh-order derivative to energy norm [71ndash76] Lysaker etal integrated high-order deductive to original ROF model[77 78] while other scholars proposed a two-stage denoisingalgorithm which smoothes the corresponding vector fieldfirst and then fits it by using the curve surface [79 80]The above methods have provided some improvements inmaintaining contrast and texture details and removing thestaircase effect but they still have some drawbacks Firstthe improved algorithms have greatly increased calculationcomplexity for real-time processing and excessive storage and

computational requirements will lead them to be impracticalSecond the above algorithms are essentially integer-orderdifferential based algorithm and thus they may cause theedge field to be somewhat fuzzy and the texture-preservationeffect to be less effective than expected

We therefore propose to introduce a new mathematicalmethodmdashfractional calculus to the denoising field for textureimage and implementing a fractional partial differentialequation to solve the above problems by the integer-orderpartial differential equation-based denoising algorithms [2333ndash38] Guidotti and Lambers [81] and Bai and Feng [82]have pushed the classic anisotropic diffusion model to thefractional field extended gradient operator of the energynorm from first-order to fractional-order and numericallyimplemented the fractional developmental equation in thefrequency domain which has some effects on image denois-ing However the algorithm still has certain drawbacks Firstit simply took the gradient operator of the energy normfrom first order to fractional order and still cannot essentiallysolve the problem of how to nonlinearly maintain the texturedetails via the anisotropic diffusion Therefore the textureinformation is not retained well after denoising Second thealgorithm does not include the effects of the fractional powerof the energy norm and the fractional extreme value onnonlinearly maintaining texture details Third the methoddoes not deduce the corresponding fractional Euler-Lagrangeformula according to fractional calculus features and directlyreplace it according to the complex conjugate transposefeatures of the Hilbert adjoint operator It greatly increasedthe complex of the numerical implementation of the frac-tional developmental equation in frequency field Finally thetransition function of fractional calculus in Fourier transformdomain is (119894120596)V Its form looks simple but the Fourierrsquosinverse transform of (119894120596)V belongs to the first kind of Eulerintegral which is difficult to calculate theoretically Thealgorithm simply converted the first-order difference into thefractional-order difference in the frequency domain form andreplaced the fractional differential operator which has notsolved the computing problem of the Euler integral of the firstkind

The properties of fractional differential are as follows [2324 38] First the fractional differential of a constant is non-zero whereas it must be zero under integer-order differentialFractional calculus varies from a maximum at a singularleaping point to zero in the smooth areas where the signal isunchanged or not changed greatly note that by default anyinteger-order differential in a smooth area is approximatedto zero which is the remarkable difference between thefractional differential and integer-order differential Secondthe fractional differential at the starting point of a gradientof a signal phase or slope is nonzero which nonlinearlyenhances the singularity of high-frequency componentsWith the increasing fractional order the strengthening of thesingularity of high-frequency components is also greater Forexample when 0 lt V lt 1 the strengthening is less thanwhen V = 1 The integral differential is a special case of thefractional calculus Finally the fractional calculus along theslope is neither zero nor constant but is a nonlinear curvewhile integer-order differential along slope is the constant

Abstract and Applied Analysis 3

From this discussion we can see that fractional calculus cannonlinearly enhance the complex texture details during thedigital image processing Fractional calculus can nonlinearlymaintain the low-frequency contour features in smootharea to the furthest degree nonlinearly enhance the high-frequency edge and texture details in those areas wheregray level changes frequently and nonlinearly enhance high-frequency texture details in those areas where gray level doesnot change obviously [23 33ndash38]

A fractional partial differential equation-based denoisingalgorithm is proposed The experimental results prove thatit can not only preserve the low-frequency contour featurein the smooth area but also nonlinearly maintain the high-frequency edge and texture details both in the areas wheregray level did not change obviously or change frequently Asfor texture-rich images the abilities for preserving the high-frequency edge and complex texture details of the proposedfractional based denoising model are obviously superior tothe traditional integral based algorithms The outline of thepaper is as follows First it introduces three common-useddefinitions of fractional calculus that is Grumwald-LetnikovRiemann-Liouville and Caputo which are the premise of thefractional developmental equation-based model Second weobtain fractional Greenrsquos formula for two-dimensional imageby extending the classical integer-order to a fractional-orderand also fractional Euler-Lagrange formula On the basis afractional partial differential equation is proposed Finallywe show the denoising capabilities of the proposed modelby comparing with Gaussian denoising fourth-order TVdenoising bilateral filtering denoising contourlet denoisingwavelet denoising nonlocal means noise filtering (NLMF)denoising and fractional-order anisotropic diffusion denois-ing

2 Related Work

The common-used definitions of fractional calculus inthe Euclidean measure are Grumwald-Letnikov definitionRiemann-Liouville definition and Caputo definition [1ndash6]

Grumwald-Letnikov defined that V-order differential ofsignal 119904(119909) can be expressed by

119863V119866-119871119904 (119909) =

119889V

[119889 (119909 minus 119886)]V 119904 (119909)

10038161003816100381610038161003816100381610038161003816119866-119871

= lim119873rarrinfin

((119909 minus 119886) 119873)

minusV

Γ (minusV)

times

119873minus1

sum

119896=0

Γ (119896 minus V)

Γ (119896 + 1)119904 (119909 minus 119896 (

119909 minus 119886

119873))

(1)

where the duration of 119904(119909) is [119886 119909] and V is any real num-ber (fraction included) 119863V

119866-119871 denotes Grumwald-Letnikovdefined fractional-order differential operator and Γ isGamma function Equation (1) shows that Grumwald-Letnikov definition in the Euclidean measure extends thestep from integer to fractional and thus it extends the order

from integer differential to fractional differential Grumwald-Letnikov defined fractional calculus is easily calculatedwhich only relates to the discrete sampling value of 119904(119909minus119896((119909minus119886)119873)) that correlates to 119904(119909) and irrelates to the derivative orthe integral value

Riemann-Liouville defined the V-order integral when V lt0 is shown as

119863V119877-119871119904 (119909) =

119889V

[119889 (119909 minus 119886)]V 119904 (119909)

10038161003816100381610038161003816100381610038161003816119877-119871

=1

Γ (minusV)int

119909

119886

(119909 minus 120578)minusVminus1

119904 (120578) 119889120578

=minus1

Γ (minusV)int

119909

119886

119904 (120578) 119889(119909 minus 120578)minusV V ≺ 0

(2)

where 119863V119877-119871 represents the Riemann-Liouville defined frac-

tional differential operator As for V-order differential whenV ge 0 119899 satisfies 119899 minus 1 lt V le 119899 Riemann-Liouville definedV-order differential can be given by

119863V119877-119871119904 (119909) =

119889V

[119889 (119909 minus 119886)]V 119904 (119909)

10038161003816100381610038161003816100381610038161003816119877-119871

=119889

119899

119889119909119899

119889Vminus119899

[119889(119909 minus 119886]Vminus119899

119904 (119909)

10038161003816100381610038161003816100381610038161003816119877-119871

=

119899minus1

sum

119896=0

(119909 minus 119886)119896minusV119904(119896)(119886)

Γ (119896 minus V + 1)

+1

Γ (119899 minus V)int

119909

119886

119904(119899)(120578)

(119909 minus 120578)Vminus119899+1

119889120578 0 le V ≺ 119899

(3)

Fourier transform of the 119904(119909) is expressed as

FT [119863V119904 (119909)] = (119894120596)

VFT [119904 (119909)] minus119899minus1

sum

119896=0

(119894120596)119896 119889

Vminus1minus119896

119889119909Vminus1minus119896119904 (0)

(4)

where 119894 denotes imaginary unit and 120596 represents digitalfrequency If 119904(119909) is causal signal (4) can be simplified to read

FT [119863V119904 (119909)] = (119894120596)

VFT [119904 (119909)] (5)

3 Theoretical Analysis for Fractional PartialDifferential Equation Fractional TotalVariation and Fractional Steepest DescentApproach Based Multiscale DenoisingModel for Texture Image

31 The Fractional Green Formula for Two-DimensionalImage The premise of implementing Euler-Lagrange for-mula is to obtain the proper Greenrsquos formula [83] Wetherefore extend the order ofGreenrsquos formula from the integerto a fractional first in order to implement fractional Euler-Lagrange formula of two-dimensional image

Consider Ω to be simply connected plane region takingthe piecewise smooth curve 119862 as a boundary then the differ-integrable functions119875(119909 119910) and119876(119909 119910) [1ndash6] are continuous


d

c

a b

C1

C2

y = 1206011(x)

y = 1206012(x)

Ω

y

x

x = 1205951(y)

x = 1205952(y)A

B

0

Figure 1 Simply connected spaceΩ and its smooth boundary curve119862

in Ω and 119862 and the fractional continuous partial derivativesfor 119909 and 119910 exist If we consider 1198631 to represent the first-order differential operator then 119863

V represents the V-orderfractional differential operator 1198681 = 119863

minus1 denotes the first-order integral operator and 119868

V= 119863

minusV represents V-orderfractional integral operator when V gt 0 Note that (119868V

119909119868V119910)Ω

represents the V-order integral operator of curve surface inthe Ω plane 119868V

119862(1198601198621119861)

is the V-order integral operator in the

1198601198621119861 section of curve 119862 along the direction of

997888997888997888997888rarr

1198601198621119861 119868V

119862minus

is V-order fractional integral operator in the closed curve 119862along counter-clockwise direction Consider that boundary119862 is circled by two curves 119910 = 120601

1(119909) 119910 = 120601

2(119909) 119886 le 119909 le 119887 or

119909 = 1205951(119910) 119909 = 120595

2(119910) 119888 le 119910 le 119889 as shown in Figure 1

As for differintegrable function 119875(119909 119910) [1ndash6] when 119875 minus

119863minusV1119863

V1119875 = 0 it follows that 119863

V1119863

V2119875 = 119863

V1+V2119875 minus

119863V1+V2(119875 minus 119863

minusV1119863

V1119875) Thus when 119868

V2

119909119868V2

119910119863

V1

119910119875(119909 119910) =

119868V2

119909119863

V1minusV2

119910119875(119909 119910) minus 119863

V1minusV2

119910[119875(119909 119910)minus119863

minusV1

119910119863

V1

119910119875(119909 119910)] it has

(119868V2

119909119868V2

119910)Ω119863

V1

119910119875 (119909 119910)

=119887

119886119868V2

119909

1206012(119909)

1206011(119909)

119868V2

119910119863

V1

119910119875 (119909 119910)

=119887

119886119868V2

119909119863

V1minusV2

119910119875 (119909 119910)

minus119863V1minusV2

119910[119875(119909 119910) minus 119863

minusV1

119910119863

V1

119910119875(119909 119910)]

10038161003816100381610038161003816

1206012(119909)

1206011(119909)

= minus 119868V2

119909

119862(1198611198622119860)

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

minus 119868V2

119909

119862(1198601198621119861)

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

= minus119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

(6)

Similarly it has

(119868V2

119909119868V2

119910)Ω119863

V1

119909119876 (119909 119910)

= 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

(7)

Fractional Greenrsquos formula of two-dimensional image can beexpressed by

(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

+ 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

(8)

When119863V1 and119863minusV

1 are reciprocal that is 120593 minus119863minusV1119863

V1120593 = 0

it follows that119863V1119863

V2120593 = 119863

V1+V2120593Then (8) can be simplified

to read

(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910) + 119868

V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)

(9)

We know from (9) the following First when V1= V

2= V

(9) can be simplied as (119868V119909119868V119910)Ω(119863

V119909119876(119909 119910) minus 119863

V119910119875(119909 119910)) =

(119868V119909)119862minus119875(119909 119910) + (119868

V119910)119862minus119876(119909 119910) which is the expression of

fractional Greenrsquos formula in reference [84] Second whenV1

= V2

= 1 it follows that (1198681

1199091198681

119910)Ω(119863

1

119909119876(119909 119910) minus

1198631

119910119875(119909 119910)) = (119868

1

119909)119862minus119875(119909 119910) + (119868

1

119910)119862minus119876(119909 119910) The classical

integer-order Greenrsquos formula is the special case of fractionalGreen formula

32 The Fractional Euler-Lagrange Formula for Two-Dimen-sional Image To implement a fractional partial differentialequation-based denoising model we must obtain the frac-tional Euler-Lagrange formula first and thus we furtherlydeduce the fractional Euler-Lagrange formula for two-dimensional image based on the above fractional Greenrsquosformula

Consider the differintegrable numerical function in two-dimensional space to be 119906(119909 119910) and the differintegrablevector function to be (119909 119910) = 119894120593

119909+ 119895120593

119910[1ndash6] the V-order

fractional differential operator is 119863V= 119894(120597

V120597119909

V) + 119895(120597

V120597119910

V) = 119894119863

V119909+ 119895119863

V119910

= (119863V119909 119863

V119910) Here 119863V is a type of

linear operator When V = 0 then 1198630 represents an equalityoperator which is neither differential nor integral where 119894and 119895 respectively represent the unit vectors in the 119909- and 119910-directions In general the two-dimensional image regionΩ isa rectangular simply connected space and thus the piecewise


y

x

C3(x0 y1) C2(x1 y1)

C1(x1 y0)C0(x0 y0)

Ω

0

Figure 2 The two-dimensional simply connected space Ω and itspiecewise smooth boundary curve 119862

smooth boundary 119862 is also a rectangular closed curve asshown in Figure 2

Referring to (2) it follows that 119868V119909119904(119909 119910) = (1

Γ(V)) int119909

119886119909

(119909 minus 120578)Vminus1119904(120578 119910)119889120578 119868V

119910119904(119909 119910) = (1Γ(V)) int

119910

119886119910

(119910 minus

120577)Vminus1119904(119909 120577)119889120577 and 119868

V119909119868V119910119904(119909 119910) = (1Γ

2(V)) int

119909

119886119909

int119910

119886119910

(119909 minus

120578)Vminus1(119910 minus 120577)

Vminus1119904 (120578 120577)119889120578 119889120577 From the fractional Green

formula (8) and Figure 2 we can derive

(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

+ 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

=1199091

1199090

119868V2

119909119863

V1minusV2

119910119875 minus 119863

V1minusV2

119910[119875 minus 119863

minusV1

119910119863

V1

119910119875]

+1199090

1199091

119868V2

119909119863

V1minusV2

119910119875 minus 119863

V1minusV2

119910[119875 minus 119863

minusV1

119910119863

V1

119910119875]

+1199101

1199100

119868V2

119910119863

V1minusV2

119909119876 minus 119863

V1minusV2

119909[119876 minus 119863

minusV1

119909119863

V1

119909119876]

+1199100

1199101

119868V2

119910119863

V1minusV2

119909119876 minus 119863

V1minusV2

119909[119876 minus 119863

minusV1

119909119863

V1

119909119876] equiv 0

(10)

Since suminfin

119898=0sum

119898

119899=0equiv sum

infin

119899=0sum

infin

119898=119899and ( V

119903+119899 ) (119903+119899

119899) equiv (

V119899 ) (

Vminus119899

119903)

we have

119863V119909minus119886

(119891119892) =

infin

sum

119899=0

[(V119899) (119863

Vminus119899

119909minus119886119891)119863

119899

119909minus119886119892] (11)

Referring to the homogeneous properties of fractional cal-culus and (10) we can derive that (119868V2

119909119868V2

119910)Ω[119863

V1

119909(119906120593

119909) +

119863V1

119910(119906120593

119910)] = (119868

V2

119909119868V2

119910)Ω[119863

V1

119909(119906120593

119909) minus 119863

V1

119910(minus119906120593

119910)] = 0 Then we

have

(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙

= (119868V2

119909119868V2

119910)Ω[(119863

V1

119909119906) 120593

119909+ (119863

V1

119910119906) 120593

119910]

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [(119863

V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910]

(12)

where the sign ∙ denotes the inner product Similar to thedefinition of fractional-order divergence operator divV =

119863V∙ = 119863

V119909120593

119909+ 119863

V119910120593

119910 we consider the V-order fractional

differential operator to be 119875V= sum

infin

119899=1(V119899 ) [119894((119863

Vminus119899

119909119906)119906)119863

119899

119909+

119895((119863Vminus119899

119910119906)119906)119863

119899

119910] and the V-order fractional divergence oper-

ator to be div119875V = 119875

V∙ = 119875

V119909120593

119909+ 119875

V119910120593

119910 where both div119875V

and 119875Vare the linear operators In light of Hilbert adjoint

operator theory [85] and (12) we can derive

(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙ = ⟨119863

V1119906 ⟩

V2

= ⟨119906 (119863V1)

lowast

⟩V2

= (119868V2

119909119868V2

119910)Ω119906 ((119863

V1)

lowast

)

= minus(119868V2

119909119868V2

119910)Ω119906 (119875

V1 ∙ )

= minus(119868V1

119909119868V2

119910)Ω119906 (div119875V

1 )

(13)

where ⟨ ⟩V2 denotes the integral form of the V

2-order

fractional inner product (119863V)lowast is V-order fractional Hilbert

adjoint operator of119863V Then it follows that

(119863V)lowast

= minusdiv119875V (14)

where (119863V)lowast is a fractional Hilbert adjoint operator When

V1= V

2= 1 (12) becomes

(1198681

1199091198681

119910)Ω119863

1119906 ∙ = ⟨119863

1119906 ⟩

1

= ⟨119906 (1198631)lowast

⟩

1

= (1198681

1199091198681

119910)Ω119906 ((119863

1)lowast

) = minus1198681

1199091198681

119910

Ω

119906 (div1)

(15)

where ⟨ ⟩1 denotes the integral form of the first-order

inner product div1 = 119863

1∙ = 119863

1

119909120593

119909+ 119863

1

119910120593

119910represents

the first-order divergence operator and (1198631)lowast represents

the first-order Hilbert adjoint operator of 1198631 As for digitalimage we find that

(1198631)lowast

= minusdiv1 (16)


Equations (14) and (16) have shown that the first-orderHilbert adjoint operator is the special case of that of fractionalorder When (119868V2

119909119868V2

119910)Ω119863

V1119906 ∙ = 0 (14) becomes

(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [(119863

V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910]

= 0

(17)Since the 119909 line and 119910 line meet at right angle it has(119863

V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910= (119863

V1minus119899

119909119906119863

V1minus119899

119910119906) ∙ (119863

119899

119909120593

119909

119863119899

119910120593

119910) As for any two-dimensional function 119906 the

corresponding 119863V1minus119899

119909119906 and 119863

V1minus119899

119910119906 are randomly chosen

According to the fundamental lemmaof calculus of variations[83] we know that (119863119899

119909120593

119909 119863

119899

119910120593

119910) = (0 0) is required

to make (17) established Since 119899 is the positiveinteger belonging to 1 rarr infin we only need (119863

1

119909120593

119909

1198631

119910120593

119910) = (0 0) to make (119863

119899

119909120593

119909 119863

119899

119910120593

119910) = (0 0) estab-

lished Equation (17) can be rewritten asminus(119868

V2

119909119868V2

119910)Ω(V1

1) [(119863

V1minus1

119909119906)119863

1

119909120593

119909+ (119863

V1minus1

119910119906)119863

1

119910120593

119910] =

minus(119868V2

119909119868V2

119910)Ω(V1

1) (119863

V1minus1

119909119906119863

V1minus1

119910119906) ∙ (119863

1

119909120593

119909 119863

1

119910120593

119910) = 0

When and only when the below equation is satisfied (17) isestablished

minus(V1

1) [119863

1

119909120593

119909+ 119863

1

119910120593

119910] =

Γ (1 minus V1)

Γ (minusV1)[119863

1

119909120593

119909+ 119863

1

119910120593

119910] = 0

(18)Equation (18) is the fractional Euler-Lagrange formula whichcorresponds to (119868V2

119909119868V2

119910)Ω119863

V1119906 ∙ = 0

Also if one considersΦ1(119863

V1119906) to be the numerical func-

tion of vector function 119863V1119906 and Φ

2() to be the numerical

function of differintegrable vector function (119909 119910) = 119894120593119909+

119895120593119910[1ndash6] similarly when (119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙ =

0 (14) can be written as

(119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙

= (119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)119863

V1119906 ∙ (Φ

2() )

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [ (Φ

1(119863

V1119906)119863

V1minus119899

119909119906)

times 119863119899

119909(Φ

2() 120593

119909)

+ (Φ1(119863

V1119906)119863

V1minus119899

119910119906)

times119863119899

119910(Φ

2() 120593

119910)] = 0

(19)

Equation (19) is established when and only when

minus (V1

1) [119863

1

119909(Φ

2() 120593

119909) + 119863

1

119910(Φ

2() 120593

119910)]

=Γ (1 minus V

1)

Γ (minusV1)[119863

1

119909(Φ

2() 120593

119909)

+1198631

119910(Φ

2() 120593

119910)] = 0

(20)

Equation (20) is the fractional Euler-Lagrange formula cor-responding to (119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙ = 0

Since it has 119863V119909minus119886

[0] equiv 0 no matter what V is we knowthat the fractional Euler-Lagrange formulas (18) and (20) areirrelevant to the integral order V

2of fractional surface integral

(119868V2

119909119868V2

119910)Ω We therefore only adopt the first-order surface

integral (119868V119909119868V119910)Ω

instead of that of fractional order whenwe discuss the energy norm of fractional partial differentialequation-based model for texture denoising below

33 The Fractional Partial Differential Equation-BasedDenoising Approach for Texture Image Based on thefractional Euler-Lagrange formula for two-dimensionalimage we can implement a fractional partial differentialequation-based denoising model for texture image

119904(119909 119910) represents the gray value of the pixel (119909 119910) whereΩ sub 119877

2 is the image region that is (119909 119910) isin Ω Consider119904(119909 119910) to be the noised image and 119904

0(119909 119910) to represent the

desired clean image Since the noise can be converted toadditive noise by log processing when it is multiplicativenoise and to additive noise by frequency transform and logprocessing when it is convolutive noise we assume that119899(119909 119910) is additive noise that is 119904(119909 119910) = 119904

0(119909 119910) + 119899(119909 119910)

without loss of generality Consider 119899(119909 119910) to represent theadditive noise that is 119904(119909 119910) = 119904

0(119909 119910) + 119899(119909 119910) we

adopted the fractional extreme to form the energy normSimilar to the fractional 120575-cover in the Hausdorff measure[96 97] we consider the fractional total variation of image

119904 to be |119863V1119904|

V2= (radic(119863

V1

119909 119904)2

+ (119863V1

119910 119904)2

)

V2

where V2is any

real number including fractional number and |119863V1119904|

V2 is

the hypercube measure We assume the fractional variation-based fractional total variation as

119864FTV (119904) = (1198681

1199091198681

119910)Ω[119891 (

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

)] = ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

) 119889119909 119889119910

(21)

Consider 119904 to be the V3-order extremal surface of 119864FTV

the test function is the admitting curve surface close to theextremal surface that is 120585(119909 119910) isin 119862infin

0(Ω) we then correlate

andmerge 119904 and 120585 by 119904+(120573minus1)120585When120573 = 1 it is the V3-order

extremal surface 119904 Assume that Ψ1(120573) = 119864FTV[119904 + (120573 minus 1)120585]

and Ψ2(120573) = ∬

Ω120582[119904 + (120573 minus 1)120585 minus 119904

0]119904

0119889119909 119889119910 where Ψ

2(120573)

is the cross energy of the noise and clean signal s0 that is

[119904+(120573minus1)120585minus1199040] and it also themeasurement of the similarity

between [119904 + (120573 minus 1)120585 minus 1199040] and 119904

0 We therefore can explain

the anisotropic diffusion as energy dissipation process forsolving the V

3-order minimum of fractional energy norm

119864FTV that is the process for solving minimum of Ψ2(120573) is

to obtain the minimum similarity between the noise and theclean signal Here Ψ

2(120573) plays the role of nonlinear fidelity

during denoising and 120582 is regularized parameter Fractionaltotal variation-based fractional energy norm in surface family119904 + (120573 minus 1)120585 can be expressed by

Ψ (120573) = Ψ1(120573) + Ψ

2(120573)

= (1198681

1199091198681

119910)Ω[119891 (

1003816100381610038161003816119863V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+ 120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0]


= ∬Ω

[119891 (1003816100381610038161003816119863

V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0] 119889119909 119889119910

(22)

As for Ψ1(120573) if V

3-order fractional derivative of

119891(|119863V1119904 + (120573 minus 1)119863

V1120585|

V2) exists Ψ

1(120573) has the V

3-order

fractional minimum or stationary point when 120573 = 1Referring to the linear properties of fractional differentialoperator we have

119863V3

120573Ψ

1(120573)

100381610038161003816100381610038161003816120573=1

=120597V3

120597120573V3∬

Ω

119891 (10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= 0

(23)

where it has 2= ||

2= [radic()

2]

2

= ∙ as for the vector =119863

V1119904+(120573minus1)119863

V1120585 and the signal ∙ denotes the inner product

Unlike the traditional first-order variation (23) is the V3-

order fractional extreme of Ψ1(120573) which aims to nonlinearly

preserve the complex texture details as much as possiblewhen denoising by using the special properties of fractionalcalculus that it can nonlinearly maintain the low-frequencycontour feature in the smooth area to the furthest degree andnonlinearly enhance the high-frequency edge informationin those areas where gray level changes frequently and alsononlinearly enhance the high-frequency texture details inthose areas where gray level does not change obviously [33ndash38]

Provided that V is a fractional number when 119899 gt V it has(V119899 ) = (minus1)

119899Γ(119899 minus V)Γ(minusV)Γ(119899 + 1) = 0 Referring to (11) and

Faa deBruno formula [95] we canderive the rule of fractionalcalculus of composite function as

119863V120573minus119886

119891 [119892 (120573)] =(120573 minus 119886)

minusV

Γ (1 minus V)119891

+

infin

sum

119899=1

(V119899)

(120573 minus 119886)119899minusV

Γ (119899 minus V + 1)119899

times

119899

sum

119898=1

119863119898

119892119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573minus119886119892

119896

]

]

119875119896

(24)

where 119892(120573) = ||V2 and 119886 is the constant 119899 = 0 is separated

from summation item From (24) we know that the fractionalderivative of composite function is the summation of infiniteitems Here 119875

119896satisfies

119899

sum

119896=1

119896119875119896= 119899

119899

sum

119896=1

119875119896= 119898 (25)

The third signal sum in (25) denotes the summation ofprod

119899

119896=1(1119875

119896)[119863

119896

120573minus119886119892119896]

119875119896

|119898=1rarr119899

of the combination of

119875119896|119898=1rarr119899

that satisfied (25) Recalling (23) (24) and theproperty of Gamma function we can derive

∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120573minusV3119891 (

10038161003816100381610038161003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(V3

119899)

120573119899minusV3

Γ (119899 minus V3+ 1)

119899

times

119899

sum

119898=1

119863119898

| |V2119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)

119896

]

]

119875119896

119889119909 119889119910

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)

Γ (minusV3) Γ (119899 minus V

3+ 1)

times

119899

sum

119898=1

119863119898

| |V2119891

100381610038161003816100381610038161003816120573=1

sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

times 119889119909119889119910 = 0

(26)

Without loss of the generality we consider 119891(120578) = 120578 forsimple calculation it then has1198631

120578119891(120578) = 1 and119863119898

120578119891(120578)

119898ge2

= 0thus (26) can be reduced as

∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

119898=1

119889119909 119889119910 = 0

(27)

We know from (25) when 119898 = 1 119875119896satisfies 119875

119899= 1 and

1198751= 119875

2= sdot sdot sdot = 119875

119899minus1= 0 then it can be furtherly reduced as

∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119899119889119909 119889119910 = 0

(28)

When 119899 takes odd number (119899 = 2119896 + 1 119896 = 0 1 2 3 )and even number (119899 = 2119896 119896 = 1 2 3 ) respectively theexpressions of119863119899

120573(||

V2)|

120573=1are also different

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896+1

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus119899minus1

times1003816100381610038161003816119863

V11205851003816100381610038161003816

119899minus1

(119863V1120585) ∙ (119863

V1119904)

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus1198991003816100381610038161003816119863

V11205851003816100381610038161003816

119899

(29)


Put (29) into (28) then it becomes

∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus2119896minus21003816100381610038161003816119863

V11205851003816100381610038161003816

2119896

Γ (minusV3) (2119896)

(119863V1119904)

∙ 119863V1 [

Γ (2119896 minus V3)

Γ (2119896 minus V3+ 1)

119904 minus(V

2minus 2119896) Γ (2119896 minus V

3+ 1)

(2119896 + 1) Γ (2119896 minus V3+ 2)

120585]

times 119889119909 119889119910 = 0

(30)

We consider that prod119899

120591=1(V

2minus 120591 + 1)

119899=0

= 1 Since the corre-sponding 119863V

1120585 is randomly chosen for any two-dimensionalnumerical function 120585119863V

1[(Γ(2119896minusV3)Γ(2119896minusV

3+1))119904 minus ((V

2minus

2119896)Γ(2119896minusV3+1)(2119896+1)Γ(2119896minusV

3+2)120585)] is random Referring

to (20) the corresponding fractional Euler-Lagrange formulaof (30) is

Γ (1 minus V1)

Γ (minusV1) Γ (minusV

3)

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816

V2minus2119896minus2

119863V1

119909119904) + 119863

1

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)] = 0

(31)

where it has prod119899

120591=1(V

2minus 120591 + 1)

119899=0

= 1 It is the correspondingfractional Euler-Lagrange formula of (23)

As for Ψ2(120573) it has

119863V3

120573Ψ

2(120573)

100381610038161003816100381610038161003816120573=1

= 119863V3

120573(119868

1

1199091198681

119910)Ω120582 [119904 + (120573 minus 1) 120585 minus 119904

0] 119904

0

100381610038161003816100381610038161003816120573=1

= ∬Ω

1205821199040

Γ (1 minus V3) Γ (2 minus V

3)

times [Γ (2 minus V3) (119904 minus 119904

0minus 120585) + Γ (1 minus V

3) 120585] 119889119909 119889119910 = 0

(32)

Since the test function 120585 is randomly chosen Γ(2minusV3)(119904minus119904

0minus

120585) + Γ(1 minus V3)120585 is also random According to the fundamental

lemma of calculus of variations [83] we know that to make(32) established it must have

1205821199040


3)= 0 (33)

Equations (23) and (32) respectively are the V3-order mini-

mal value ofΨ1(120573) andΨ

1(120573) Thus when we take V = V

3= 1

2 and 3 the V3-order minimal value of (22) can be expressed

by

120597V3119904

120597119905V3=

minusΓ (1 minus V1)


3)

times

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

minus120582119904

0


3)

(34)


120591=1(V

2minus 120591 + 1)

119899=0

= 1 120597V3119904120597119905V3 is cal-culated by the approach of fractional difference We mustcompute 120582(t) If image noise 119899(119909 119910) is the white noise it has∬

Ω119899(119909 119910)119889119909 119889119910 = ∬

Ω(119904minus119904

0)119889119909 119889119910 = 0When 120597V3119904120597119905V3 = 0

it converges to a stable stateThen wemerelymultiply (119904minus1199040)2

at both sides of (34) and integrate by parts overΩ and the leftside of (34) vanishes

120582 (119905) =minusΓ (1 minus V

1) Γ (1 minus V

3) Γ (2 minus V

3)

1205902Γ (minusV1) Γ (minusV

3) 119904

0

times∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

times (119904 minus 1199040)2

119889119909 119889119910

(35)

Here we simply note the fractional partial differentialequation-based denoising model as FDM (a fractional devel-opmental mathematics-based approach for texture imagedenoising) When numerical iterating we need to performlow-pass filtering to completely remove the faint noise in verylow-frequency and direct current We know from (34) and(35) that FDM enhances the nonlinear regulation effects oforder V

2by continually multiplying functionprod2119896

120591=1(V

2minus120591+1)

and power V2minus 2119896 minus 2 of |119863V

1119904| and enhance the nonlinearregulation effects of order V

3by increasing Γ(minusV

3) in the

denominator Also we know from (34) that FDM is the tradi-tional potential equation or elliptic equation when V

3= 0 the

traditional heat conduction equation or parabolic equationwhen V

3= 1 and the traditional wave equation or hyperbolic

equation when V3= 2 FDM is the continuous interpolation

of traditional potential equation and heat conduction equa-tion when 0 lt V

3lt 1 and the continuous interpolation

of traditional heat conduction equation and wave equationwhen 1 lt V

3lt 2 FDM has pushed the traditional integer-

order partial differential equation-based image processingapproach from the anisotropic diffusion of integer-order heatconduction equation to that of fractional partial differentialequation in the mathematical and physical sense

4 Experiments and Theoretical Analyzing

41 Numerical Implementation of the Fractional PartialDifferential Equation-Based Denoising Model for TextureImage We know from (34) and (35) that we should obtain


Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1

Cs119899middot middot middot

middot middot middot


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(a)

Cs119899minus2Cs119899minus1

Cs119896Cs1

Cs0Csminus1

Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middot middot middot middot middot middot0 0 0 0 0 0 0

0 0 0 0 0 0 0

(b)

Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1

Cs119899




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(c)


Cs119896Cs1Cs0

Csminus1Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(d)

Csminus1middot middot middot

middot middot middot middot middot middot





0

0

0

0 0

0

00

0 0

0

0

0

0Cs0

Csk

Csn

Cs1 middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

(e)

Csnmiddot middot middot


middot middot middotmiddot middot middot




0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middot middot middot 0

(f)

Csn middot middot middot


middotmiddotmiddot





0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middotmiddotmiddot



middotmiddotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middot middot middot0

(g)

Csminus1 middot middot middot

middotmiddotmiddot

middotmiddotmiddot






0

0

0

00

0

0 0

0 0

0

0

0

0 Cs0

Csk

Csn

Cs1middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Cs119899minus2

Cs119899minus1

(h)

Figure 3 Fractional differential masks119863V respectively on the eight directions (a) Fractional differential operator on 119909-coordinate negativedirection noted as 119863V

119909minus (b) Fractional differential operator on 119910-coordinate negative direction noted as 119863V

119910minus (c) Fractional differential

operator on 119909-coordinate positive direction noted as 119863V119909+ (d) Fractional differential operator on 119910-coordinate positive direction noted

as 119863V119910+ (e) Fractional differential operator on left downward diagonal noted as 119863V

ldd (f) Fractional differential operator on right upwarddiagonal noted as 119863V

rud (g) Fractional differential operator on left upward diagonal noted as 119863Vlud (h) Fractional differential operator on

right downward diagonal noted as119863Vrdd

the fractional differential operator of two-dimensional dig-ital image before implementing FDM As for Grumwald-Letnikov definition of fractional calculus in (1) itmay removethe limit symbol when119873 is large enough we then introducethe signal value at nonnode to the definition improving theconvergence rate and accuracy that is (119889V119889119909V)119904(119909)|

119866-119871 cong

(119909minusV119873

VΓ(minusV)) sum119873minus1

119896=0(Γ(119896 minus V)Γ(119896 + 1))119904(119909 + (V1199092119873) minus

(119896119909119873)) Using Lagrange 3-point interpolation equationto perform fractional interpolation when V = 1 we canobtain the fractional differential operators of YiFeiPU-2respectively on the eight symmetric directions [35 38] whichis shown in Figure 3

In Figure 3 119862119904minus1

is the coefficient of operator 119904minus1= 119904(119909 +

119909119873) at noncausal pixel and 1198621199040

is the coefficient of operatorat pixel of interest 119904

0= 119904(119909) 119899 is often taken odd number

When 119896 rarr 119899 = 2119898 minus 1 we can implement fractionaldifferential operator of (2119898+1)times(2119898+1) whose coefficientsof the operator respectively are119862

119904minus1

= (V4)+(V28)1198621199040

=

1 minus (V22) minus (V38) 1198621199041

= minus(5V4) + (5V316) + (V416) 119862

119904119896

= (1Γ(minusV))[(Γ(119896minusV+1)(119896+1))sdot ((V4)+(V28))+(Γ(119896minusV)119896)sdot(1minus(V24))+(Γ(119896minusVminus1)(119896minus1))sdot(minus(V4)+(V28))] 119862

119904119899minus2

= (1Γ(minusV))[(Γ(119899 minus V minus 1)(119899 minus 1)) sdot ((V4) + (V28)) +(Γ(119899 minus V minus 2)(119899 minus 2)) sdot (1 minus (V24)) + (Γ(119899 minus V minus 3)(119899 minus 3)) sdot


(minus(V4) + (V28))] 119862119904119899minus1

= (Γ(119899 minus V minus 1)(119899 minus 1)Γ(minusV)) sdot

(1 minus (V24)) + (Γ(119899 minus V minus 2)(119899 minus 2)Γ(minusV)) sdot (minus(V4) + (V28))and 119862

119904119899

= (Γ(119899 minus V minus 1)(119899 minus 1)Γ(minusV)) sdot (minus(V4) + (V28))[35 38] Spatial filtering of convolution operator is adoptedas the numerical algorithm of fractional differential operatorWe take the maximum value of fractional partial differentialon the above 8 directions as the pixelrsquos fractional calculus Asfor fractional differential operator 119863V

= 1198941119863

V119909minus

+ 1198942119863

V119910minus

+

1198943119863

V119909+

+ 1198944119863

V119910+

+ 1198945119863

Vldd + 119894

6119863

Vrud + 119894

7119863

Vlud + 119894

8119863

Vrdd in (34)

and (35) when V = 1 119863V119909minus 119863V

119910minus 119863V

119909+ 119863V

119910+ 119863V

ldd 119863Vrud 119863

Vlud

119863Vrdd can perform numerical computing referring to Figure 3

and their concerning coefficients [35 38] and when V =

1 we take a special difference as the approximate of first-order differential to maintain calculation stability that is119863

1

119909119904(119909 119910) = (2[119904(119909 + 1 119910) minus 119904(119909 minus 1 119910)] + 119904(119909 + 1 119910 +

1) minus 119904(119909 minus 1 119910 + 1) + 119904(119909 + 1 119910 minus 1) minus 119904(119909 minus 1 119910 minus 1))4119863

1

119910119904(119909 119910) = (2[119904(119909 119910 + 1) minus 119904(119909 119910 minus 1)] + 119904(119909 + 1 119910 + 1) minus

119904(119909 + 1 119910 minus 1) + 119904(119909 minus 1 119910 + 1) minus 119904(119909 minus 1 119910 minus 1))4If time equal interval is Δ119905 time 119899 is noted as 119905

119899= 119899Δ119905

when 119899 = 0 1 1199050= 0 represents the initial time The

digital image at time 119899 is 119904119899119909119910

= 119904(119909 119910 119905119899) Thus we have

1199040

119909119910= 119904

0(119909 119910 119905

0) + 119899(119909 119910 119905

0) where 119904

0

119909119910is the original

image and 1199040is the desired clean image that is a constant

that is 1199040(119909 119910 119905

0) = 119904

0(119909 119910 119905

119899) We can derive the numerical

implementation of fractional calculus about time from (1) as

120597V119904

120597119905V= Δ119905

minusV[119904

119899+1

119909119910minus 119904

119899

119909119910+

2120583120578

Γ (3 minus V)(119904

119899

119909119910minus 119904

Vlowast119909119910)2

(119904119899

119909119910)minusV]

when V = 1

(36)

where 120583 and 120578 are the factors for ensuring iterative conver-gence Also the desired clean image 119904

0(119909 119910 119905

0) is unknown

but the intermediate denoising results is an approximate tothe desired clean image that is 119904119899

119909119910rarr 119904

0(119909 119910 119905

0) =

1199040(119909 119910 119905

119899) We consider that (119904 minus 119904

0)119909119910

cong 1199040

119909119910minus 119904

119899

119909119910for

making 119904 close to 1199040as much as possible We should compute

the fractional calculus on the 8 directions simultaneously inpractice in order to improve the calculation accuracy andantirotation capability

The best image 119904Vlowast119909119910

in (36) is unknown in numericaliteration but the intermediate result 119904119899

119909119910is an approximate

to 119904Vlowast119909119910

that is 119904119899119909119910

rarr 119904Vlowast119909119910

We consider that (119904119899119909119910

minus

119904Vlowast119909119910)2cong (119904

119899minus1

119909119910minus 119904

119899

119909119910)2 to make the iterative result approximate

(119904119899

119909119910minus 119904

Vlowast119909119910)2 as much as possible Take 120578 = 1 in (36) and

119896 = 0 1 in (34) and (35) then we can derive the numericalimplementation of (34) and (35) as

119904119899+1

119909119910= 119875 (119904

119899

119909119910) Δ119905

V3 minus

120582119899Δ119905

V3


3)119904119899

119909119910

+ 119904119899

119909119910minus

2120583

Γ (3 minus V3)(119904

119899minus1

119909119910minus 119904

119899

119909119910)2

(119904119899

119909119910)minusV3

when V3= 1 2 and 3

(37)

120582119899=Γ (1 minus V

3) Γ (2 minus V

3)

1205901198992

119904119899119909119910

sum

119909119910

119875 (119904119899

119909119910) (119904

0

119909119910minus 119904

119899

119909119910)2

(38)

where prod119899

120591=1(V

2minus 120591 + 1)

119899=0

= 1 1205901198992

= sum119909119910

(1199040

119909119910minus 119904

119899

119909119910)2

119875(119904119899

119909119910) = (minusΓ(1 minus V

1)Γ(minusV

1)Γ(minusV

3)) sum

1

119896=0(prod

2119896

120591=1(V

2minus

120591+1)(2119896))[1198631

119909(|119863

V1119904

119899

119909119910|V2minus2119896minus2

119863V1

119909119904119899

119909119910)+119863

1

119910(|119863

V1119904

119899

119909119910|V2minus2119896minus2

sdot119863V1

119910119904119899

119909119910)]

We should pay attention to the following when perform-ing numerical iterative implementation First 120583 is a smallnumber in (37) to ensure convergence and here we take120583 = 0005 Second we do not need to know or estimatethe variance of noise but we need to assume 12059012 to be asmall positive number in the first iteration We thereforeassume that 12059012

= 001 in the experiment below We take120590

12 to (38) and perform numerical iteration Each iterativeresult 120590119899

2

may be different but it is the approximate to thevariance of noise Third it is possible that |119863V

1119904119899

119909119910| = 0 in

numerical iterative calculation so we take |119863V1119904

119899

119909119910| = 00689

when |119863V1119904

119899

119909119910| le 00689 to ensure that (37) and (38) are

meaningful Fourth we take 119904119899119909119910

= 000001 when 119904119899

119909119910= 0

to make (119904119899119909119910)minusV3 have meaning Fifth to completely denoise

faint noise in very low-frequency and direct current FDMtakes the simple way by reducing the convex in the areawhere gradient is not changed obviouslyWe therefore need toperform low-pass filtering for very low-frequency and directcurrent in numerical iterating The practices of (37) and (38)are as follows For one-dimensional signal we consider that119904119899+1

119909= (119904

119899+1

119909minus1+2119904

119899+1

119909+119904

119899+1

119909+1)4when |119863V

1119904119899

119909| ≺ 120572

119860and120572

119873119860≺ 120572

119860

and if the noise is not severe in order to ensuring denoisingeffect and in the rest conditions we consider that 119904119899+1

119909=

119904119899+1

119909 where 120572

119860= (1119873

119909) sum

119873119909

119909|119863

V1119904

119899

119909| 120572

119873119860= (|119863

V1119904

119899

119909minus1| +

2|119863V1119904

119899

119909| + |119863

V1119904

119899

119909+1|)4 and if the noise is very strong we

consider that 119904119899+1

119909= (119904

119899+1

119909minus1+ 2119904

119899+1

119909+ 119904

119899+1

119909+1)4 to accelerate

the denoising speed For two-dimensional signal we consider119904119899+1

119909119910= (

119909119904119899+1

119909119910+

119910119904119899+1

119909119910+

119903119904119899+1

119909119910+

119897119904119899+1

119909119910)4 When 120572

119909

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119909

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910)4 and 119910

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119910

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119910

119904119899+1

119909119910= (119904

119899+1

119909119910minus1+ 2119904

119899+1

119909119910+

119904119899+1

119909119910+1)4 and 119909

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119903

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119903

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+1+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910minus1)4 and 119909

119904119899+1

119909119910=

119910119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 and in the

rest occasions it has 119909119904119899+1

119909119910=

119910119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910=

119904119899+1

119909119910 where 120572119909

119873119860= (|119863

V1119904

119899

119909minus1119910| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910|)4

120572119910

119873119860= (|119863

V1119904

119899

119909119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909119910+1|)4 and

120572119903

119873119860= (|119863

V1119904

119899

119909minus1119910+1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910minus1|)4 120572119897

119873119860=

(|119863V1119904

119899

119909minus1119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910+1|)4 Here 119909 denotes

119909-coordinate direction 119910 denotes 119910-coordinate direction 119903denotes right diagonal direction and 119897 denotes left diagonaldirection Sixth we expand the edge of the image for betterdenoising edge pixels Seventh since we perform low-pass


151

05

minus05minus1minus15

0

0 1000 2000 3000 4000 5000 6000

Original clean signalNoisy signal

(a)

151

05

minus05minus1

0

0 1000 2000 3000 4000 5000 6000

Original clean signalGaussian denoising

(b)

151

05


0

0 1000 2000 3000 4000 5000 6000

Original clean signalFourth-order TV denoising

(c)

151

05


0

0 1000 2000 3000 4000 5000 6000

Original clean signalBilateral filtering denoising

(d)

151

05


0

0 1000 2000 3000 4000 5000 6000

Original clean signalContourlet denoising

(e)

151

05


0

0 1000 2000 3000 4000 5000 6000

Original clean signalWavelet denoising

(f)

151

05


0

0 1000 2000 3000 4000 5000 6000

Original clean signalNLMF denoising

(g)

151

05


0

0 1000 2000 3000 4000 5000 6000

Original clean signalFractional-order anisotropicdiffusion denoising

(h)

151

05


0

0 1000 2000 3000 4000 5000 6000

Original clean signalFDM denoising

(i)

Figure 4 Denoising for integrated one-dimensional signal comprising rectangle wave sine wave and sawtooth wave (a) Original cleansignal and noisy signal (adds white Gaussian noise to the original clean signal peak signal-to-noise ratio (PSNR) = 252486) (b) Gaussiandenoising (c) fourth-order TV denoising [71ndash73] (d) bilateral filtering denoising [86ndash88] (e) contourlet denoising [89 90] (f) waveletdenoising [88 91] (g) nonlocal means noise filtering (NLMF) denoising [92 93] (h) fractional-order anisotropic diffusion denoising [82]and (i) FDM denoising (V

1= 1025 V

2= 225 V

3= 105 Δ119905 = 00296)

filtering in (37) and (38) in order to remove the possibledivergence point in numerical iteration we consider that119904119899+1

119909= 119904

119899

119909when |119904119899+1

119909| ≻ 6|119904

119899

119909| for one-dimensional signal and

we consider that 119904119899+1

119909119910= 119904

119899

119909119910when |119904

119899+1

119909119910| ≻ 6|119904

119899

119909119910| for two-

dimensional signal

42 Denoising Capabilities Analysis of the Fractional Par-tial Differential Equation-Based Denoising Model for TextureImage To analyze and explain the denoising capabilitiesof fractional partial differential equation-based denoisingmodel for texture image we perform the comparative experi-ments using the composite one-dimensional signal combinedby the rectangle wave the sine wave and the sawtooth waveThe numerical iteration will stop at the point where the

peak signal-to-noise ratio (PSNR) is the highest as shown inFigure 4

From subjective visual effect we know the following fromFigure 4 First the denoising effect of Gaussian denoisingand fourth-order TV denoising is comparatively worse thanothers and the high-frequency singularity component hasbeen greatly diffused and smoothedWe can see from Figures4(b) and 4(c) that the convexes of high-frequency edge ofrectangle wave and sawtooth wave are remarkably smoothedand their energy of high-frequency singularity is obviouslydiffused in neighboring Second the denoising capabilityof fractional-order anisotropic diffusion denoising is in themiddle that is the capability of maintaining high-frequencysingularity is better than that of Gaussian denoising fourth-order TV denoising and contourlet denoising but less than


bilateral filtering denoising wavelet and NLMF denoisingAlso the denoising is not completed We can see fromFigure 4(h) that the convexes of high-frequency edge of therectangle wave and the sawtooth wave are weakly smoothedand their energy is weakly diffused in neighboring Also thedenoised signal has tiny burrThird the denoising capabilitiesof bilateral filtering denoising contourlet denoising waveletdenoising and NLMF denoising are better which can wellmaintain the high-frequency singularity but the denoisingis still uncomplete We can see from Figures 4(d)ndash4(g) thatthe convexes of high-frequency edge of rectangle wave andsawtooth wave are well retained but the denoised signal stillhas many small burrs Finally the denoising capability ofFDM is the best which is not only well maintains the high-frequency edge of rectangle wave and sawtooth wave but alsodenoises completely We can see from Figure 4(i) that thehigh-frequency edge singularity of rectangle and sawtoothwave is well maintained and little burr is left

From the viewpoint of quantitative analysis we takePSNR and correlation coefficients between noisy signal ordenoised signal and original clean signal to measure thedenoising effect of the above algorithms as shown in Table 1

We know from Table 1 as follows First PSNR of Gaus-sian denoising fourth-order TV denoising and contourletdenoising are relatively small among the above approacheswith 269611 le PSNR le 282945 and their correlation coeffi-cients are also small with 09953 le correlation coefficients le09959 which indicates that their denoising capability isworse and the similarity between denoised signal and originalclean signal is also low Second PSNR of fractional-orderanisotropic diffusion denoising is in the middle with PSNR =

298692 and its correlation coefficients are in the middlewith correlation coefficients = 09975 which shows thatits denoising capability and the similarity between denoisedsignal and original clean signal is also in the middle ThirdPSNR of bilateral filtering denoising wavelet denoisingNLMF denoising and FDM denoising is relatively big with333088 le PSNR le 390434 and its correlation coefficientsare also big with 09975 le correlation coefficients le 09996which shows that their denoising capabilities are better andthe similarity between denoised signal and original cleansignal is also high PSNR and correlation coefficients of FDMdenoising is the highest which shows that its denoisingcapability is the best and the similarity between denoisedsignal and original clean signal is also the highest

From the subjective visual of Figure 4 and quanli-tative analysis of Table 1 we know the following FirstFDM has the best denoising capability no matter in high-frequency middle-frequency and low-frequency compo-nents the denoised signal fits for the edge of high-frequencyand the outline of middle-frequency component and alsothe noise in low-frequency component is removed clearlyand little blur is left Second the high-frequency edgesingularity of rectangle signal and sawtooth wave signalin Figure 4 has strong high-frequency component whichcorresponds to high-frequency edge and texture details oftwo-dimensional signal while the high-frequency singularityis small and the middle-frequency components are big in theslope of sine wave and sawtooth signal which corresponds to

Table 1 Denoising effect for composite one-dimensional signal thatcombined rectangle wave sine wave and sawtooth wave

Denoising algorithm Denoising effectPSNR correlation Coefficients

Noisy signal 252486 09951Gaussian denoising 269611 09953Fourth-order TV denoising 282589 09959Bilateral filtering denoising 350247 09990Contourlet denoising 282945 09959Wavelet denoising 321565 09975NLMF denoising 333088 09976Fractional-order anisotropicDiffusion denoising 298692 09975

FDM denoising 390434 09996

middle-frequency of two-dimensional signal The constantdirect current of regulation signal corresponds to low-frequency and direct current components and backgrounds

To analyze and explain the good denoising capabilityof FDM we choose the better models including bilateralfiltering denoising wavelet denoising NLMF denoisingand FDM denoising to perform the contrast experimentsfor texture-rich metallographic images of an iron ball Thenumerical iterative process will stop at the point where thePSNR is the highest as shown in Figure 5

From a subjective view of the visual effect we know thefollowing from Figure 5 First the denoising capabilities ofbilateral filtering and wavelet denoising are worse than theother methods because they obviously diffuse and smooththe high-frequency edge and texture details We can see thatthe edge and texture details are clear from Figures 5(e) and5(h) that is the denoised noise by bilateral filtering andwavelet denoising may not be the same as the added noiseAnd we can see from Figures 5(f) and 5(i) that the denoisedimage is blurryThe denoising is not completed in Figure 5(i)which shows that their capabilities for preserving edge andtexture details areworse Second the capability for preservingedge and texture details of NLMF denoising is better butits denoising capability for edge and texture neighboring isworse From Figure 5(k) we know that though the edge andtexture details of denoised image are weaker than those inFigures 5(e) and 5(h) but they still can be seen the denoisednoise by NLMF denoising is close to the added noise InFigure 5(l) the edge and texture details are small blurredthat is NLMF denoising can well preserve the edge andtexture details Also the neighboring of edge and texturedetails is smooth in Figure 5(k) while the residual noisein the edge and texture neighboring is stronger than otherparts in Figure 5(l) that is the denoising capability of NLMFdenoising is worse at the edge and texture neighboringThirdthe denoising capabilities of FDM denoising are the bestwhich preserve the high-frequency edge and texture detailswell and also denoise comparatively completed We can seeindistinctly the edge and texture detail from Figures 5(n) and5(q) which shows that the denoised noise by FDM denoising


(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)

Figure 5 Denoising effects of texture-rich metallographic image of iron ball (a) Original clean image (b) noisy image (adds white Gaussiannoise to the original clean image PSNR = 136872) (c) partial enlarged details of 14 party of (a) red box (d) denoised image of bilateralfiltering denoising [88 89 93] (e) residual plot of bilateral filtering denoising (f) partial enlarged details of 14 party of (d) red box (g)denoised image of wavelet denoising [92 93] (h) residual plot of wavelet denoising (i) partial enlarged details of 14 party of (g) red box(j) denoised image of NLMF denoising [94 95] (k) residual plot of NLMF denoising (l) partial enlarged details of 14 party of (j) red box(m) denoised image of FDM denoising (V

1= 175 V

2= 225 V

3= 105 Δ119905 = 10

minus10) (n) residual plot of FDM denoising (o) partial enlargeddetails of 14 party of (m) red box (p) denoised Image of FDM denoising (V

1= 225 V

2= 25 V

3= 125 Δ119905 = 10

minus10) (q) residual plot ofFDM denoising (r) partial enlarged details of 14 party of (p) red box

Table 2 Comprehensive denoising effects results for texture-rich metallographic images of an iron ball

Denoising algorithm Denoising effectPSNR Correlation coefficients Contrast Correlation Energy Homogeneity

Noisy image 136872 09991 59118 02491 00197 04580Bilateral filtering denoising 204845 09995 11200 08038 00787 07092Wavelet denoising 165708 09994 21387 03910 00473 05833NLMF denoising 210494 09996 12524 07150 00728 06822FDM denoising 214826 09996 14365 06876 00688 06646FDM denoising 215482 09997 14371 06646 00688 06608

is the most close to the added noise the edge and texturedetails are the smallest blurred and the noise is the clearestremoved in Figures 5(o) and 5(r) that is the capabilities forpreserving the edge and texture details of FDM are the best

From the viewpoint of quantitative analysis we take thePSNR the correlation coefficients between the noisy imageor the denoised image and the original clean image [94] andthe average gray level concurrencematrix to comprehensivelyestimate the denoised effect We calculate the gray levelconcurrence matrix coefficient in 5 pixel distance in Figure 5and export the typical coefficients contrast correlationenergy and homogeneity taking four directions of 0∘ 45∘ 90∘and 135

∘ Here 0∘ represents the projection in the positive

119910-coordinate direction and 90∘ represents the projection inthe 119909-coordinate directionWe then average the above valuesas seen in Table 2

We know the denoising capabilities of the above algo-rithms are as follows from Table 2 First the denoisingcapabilities of bilateral filtering and wavelet denoising areworse than the other methods and their PSNR values andcorrelation coefficients are relatively small This shows thatthe high-frequency edge and texture details are greatlydiffused and smoothed and the noise is not completelyremovedThe similarity between denoised image and originalclean image is small Also the contrast of average gray levelconcurrence matrix for bilateral filtering denoising is small


(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

Figure 6 Denoising effects of abdomen MRI of texture-rich internal organ when Gaussian noise is very strong (a) Original clean MRI(b) noisy image (adds white Gaussian noise to the original clean MRI PSNR = 54389) (c) denoised image of bilateral filtering denoising[88 89 93] (d) partial enlarged details of 14 party of (c) red box (e) denoised image of wavelet denoising [88 91] (f) partial enlarged detailsof 14 party of (e) red box (g) denoised image of NLMF denoising [94 95] (h) partial enlarged details 14 party of (g) red box (i) denoisedimage of FDM denoising (V

1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10

minus6) (j) partial enlarged details of 14 party of (i) red box (k) denoisedimage of FDM denoising (V

1= 225 V

2= 25 V

3= 125 Δ119905 = 5 times 10minus6) and (l) partial enlarged details of 14 party of (k) red box

which shows that fewer pixels have great contrast and thatthe texture depth is light and seems fuzzyThe contrast of theaverage gray level concurrence matrix of wavelet denoisingis the largest This shows that there are more pixels withgreat contrast but we cannot say that the texture depth isdeeper and the visual effects are clearer because the denoisingis incomplete Second the denoising capabilities of NLMFand FDM denoising are better Their PSNR and correlationcoefficients are comparatively higher which indicates thatthe high-frequency edge and texture details of the denoisedimage arewell preserved that the denoising is completed andthat the similarity between the denoised image and originalclean image is also great PSNR and correlation coefficientsof FDM denoising are the highest that is the denoisingis most completed and the similarity is also the greatestThe contrast of the average gray level concurrence matrixof FDM denoising is the highest This shows that there aremore pixels with great contrast that the texture depth is thedeepest and that the image looks clearer The correlationis small which shows that the partial gray correlation isweak and that the texture details are obvious The energy israther smaller which shows that the texture changing is not

uniform and regular and thus the texture details are obviousThe homogeneity is also small which indicates the regionalchanging is dramatic and that the texture details are obviousTherefore we can conclude that FDM denoising is the bestdenoising algorithm

To consider a scenario where the Gaussian noise is verystrong and especially when the original clean signal is com-pletely drowned in noises we perform comparison experi-ments using the well-performed algorithms discussed aboveincluding bilateral filtering denoising wavelet denoisingNLMF denoising and FDM denoising for further analysisof the denoising capability of FMD for robust noise Thenumerical iterative process will also stop at the point wherepeak signal-to-noise ratio is the highest as seen in Figure 6

From the viewpoint of visual effects we know the follow-ing from Figure 6 when noise is very strong especially whenMRI is drawn completely First the denoising capabilities ofbilateral filtering and wavelet denoising are worse than theother methods We can see indistinctly that the contour andthe texture details of inner organ can hardly be recognizedfrom Figures 6(c)ndash6(f) Second the denoising capability ofNLMF is better because we see that the contour is clearer but


Table 3 Denoising effects of abdomen MRI of texture-rich internal organ when Gaussian noise is very strong

Denoising algorithm Denoising EffectPSNR Correlation coefficients Contrast Correlation Energy Homogeneity

Noisy Image 54389 09857 212697 00013 00844 04811Bilateral filtering Denoising 114039 09894 11790 01796 01113 06562Wavelet denoising 99246 09892 18443 00175 00806 05941NLMF denoising 125855 09921 11795 01247 01083 06406FDM denoising 174317 09975 11847 00836 00913 05737FDM denoising 174572 09976 11853 00830 00893 05715

the edge and texture details are still blurry as in Figures 6(g)and 6(h) Finally the denoising capabilities of FDMdenoisingare the best because we can see from Figures 6(i)ndash6(l) thatthe contour is clearest and the edge and texture details can berecognized also

For quantitative analysis we measure the denoisingeffects in terms of the PSNR the correlation coefficientsbetween the noisy image or the denoised image and the orig-inal clean MRI [94] and the average gray level concurrencematrix as seen in Table 3

From Table 3 we know that the denoising capabilities ofthe above algorithms are as follows when noise is very strongespecially when MRI is completely drowned by the noiseFirst the denoising capabilities of bilateral filtering denoisingand wavelet denoising are rather poor and their PSNRand correlation coefficients are relatively small This showsthat the noise cannot be clearly denoised and the similaritybetween the denoised image and the original clean MRI issmall Also the contrast of average gray level concurrencematrix of wavelet algorithm is the greatest which indicatesgreater pixels with greater contrast but we cannot say thatthe texture depth is deeper and that the visual effects areclearer because the denoising is incomplete Second thedenoising capabilities of NLMF denoising and FDM denois-ing are better and their PSNR and correlation coefficients arecomparatively higher which shows that the noise is denoisedcompletely and the similarity between denoised image andoriginal clean MRI is great The PSNR and the correlationcoefficients of FDM denoising is the highest and its contrastof average gray level concurrence matrix is the greatest Alsoits correlation energy and homogeneity are smaller Wetherefore can conclude that FDM denoising is the best modelof the above models

When the noise is very strong and especially whenoriginal clean signal is completely drowned in noises we takea texture-richmeteorite crater remote sensing image ofmoonsatellite to perform further comparison experiments usingthe above well-performed algorithms including bilateralfiltering denoising wavelet denoising NLMF denoising andFDM denosing to test the denoising capability of FDMfor robust noise The added noise is the composite noisecombined bywhiteGaussian noise salt and pepper noise andspeckle noise Also the numerical iterative process will stopat the point where peak signal-to-noise ratio is the highest asshown in Figure 7

From view of visual effects we know from Figure 7the following when the composite noise is added by white

Gaussian noise salt and pepper noise and speckle togetherespecially when texture-rich meteorite crater remote sensingimage of moon satellite is completely drown in noises Firstthe denoising capabilities of bilateral filtering andwavelet andNLMF denoising are comparatively worse We can see fromFigures 7(c)ndash7(h) that the contour may be seen indistinctlyand the edge and texture details can hardly be recognizedSecond the denoising capability of FDMdenoising is the bestWe can see from Figures 7(i)ndash7(l) that the contour is not onlythe clearest and also the edge and texture details can be clearlyrecognized

For quantitative analysis we measure the denoisingeffects in terms of the PSNR correlation coefficients betweennoisy image or denoised image and original clean remotesensing image [94] and average gray level concurrencematrix as seen in Table 4

We know denoising capabilities of the above algorithmsfrom Table 4 are as follows when composite noise addedby white Gaussian noise salt and pepper noise and specklenoise especially when the original image is completelydrowned in noises First the denoising capabilities of bilateralfiltering wavelet and NLMF denoising are poor and theirPSNR and correlation coefficients are relatively small Thisshows that the added noise cannot be denoised completelyand the similarities between denoised image and originalclean MRI are small Also the contrasts of average gray levelconcurrence matrices of NLMF algorithm are the greatestwhich indicates that there are more pixels with great contrastbut we cannot say that the texture depth is deeper and thevisual effect is clearer because the denoising is uncompletedSecond the denoising capability of FDMdenoising is the bestIts PSNR and correlation coefficients are comparatively highwhich shows that the noise is completely denoised and thesimilarity between denoised image and original clean MRI isthe greatest Also the PSNR and the correlation coefficients ofFDM denoising are the highest The contrast of average graylevel concurrence matrix of FDM denoising is high while itscorrelation energy and homogeneity are small We thereforecan say that FDM denoising is the best denoising model ofthe above models

By comparing the visual effects of Figures 5 6 and 7and quantitative analysis in Figures 1 2 and 3 we findthe following First the denoising capability of FDM is thebest irrespective of the strength and type of added noiseIts PSNR and correlation coefficients are the highest and thedenoising is relatively completed The similarity between thedenoised image and the original image is the highest Second


(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

Figure 7 Denoising effects of meteorite crater texture-richmoon satellite remote sensing image when white Gaussian noise salt and peppernoise and speckle noise are added together (a)Original clean image (b) noisy image (addswhite Gaussian noise (its standard variance is 002)salt and pepper noise (its noise density is 02) and speckle noise (its standard variance is 01) to the original clean image PSNR = 88564)(c) denoised image of bilateral filtering denoising [88 89 93] (d) partial enlarged details of 14 party of (c) red box (e) denoised image ofwavelet denoising [92 93] (f) partial enlarged details of 14 party of (e) red box (g) denoised image of NLMF denoising [94 95] (h) partialenlarged details of 14 party of (g) red box (i) denoised image of FDM denoising (V

1= 175 V

2= 275 V

3= 105 and Δ119905 = 10minus10) (j) partial

enlarged details of 14 party of (i) red box (k) denoised image of FDM denoising (V1= 225 V

2= 25 V

3= 125 and Δ119905 = 10

minus10) and (l)Partial enlarged details of 14 party of (k) red box

Table 4 Denoising effects on texture-rich meteorite crater remote sensing image of moon satellite when white Gaussian noise salt andpepper noise and speckle noise are added together



FDM denoising can nonlinearly preserve the low-frequencycontour feature in the smooth area to the furthest degreenonlinearly enhance high-frequency edge information inthose areas where gray level changes obviously nonlinearlyenhance the high-frequency edge and texture details andalso nonlinearly maintain the fractional multiscale denoisingcapability in those areas where gray level does not changeobviously

5 Conclusions

We propose the introduction of a new mathematicalmethodmdashfractional calculus to the field of image processingand the implementation of fractional partial differentialequation First it presents three common-used definitions ofGrumwald-Letnikov Riemann-Liouville and Caputo whichis the premise of fractional partial differential equation-based


denoising model Second we derive fractional Greenrsquos for-mula for two-dimensional image processing by extendingclassical integer order to fractional order and then derivefractional Euler-Lagrange formula Based on the above frac-tional formulas a fractional partial differential equation isproposed Finally we show the denoising capability of theproposed model by comparing Gaussian denoising fourth-order TV denoising bilateral filtering denoising contourletdenoising wavelet Denoising nonlocal means noise filtering(NLMF) denoising and fractional-order anisotropic diffu-sion denoisingThe experimental results prove that FDM canpreserve the low-frequency contour feature in the smootharea nonlinearly maintain the high-frequency edge andtexture details in those areas where gray level change greatlyand also nonlinearly retain the texture details in those areaswhere gray level has little changed As for texture-rich imagesthe denoising capability of the proposed FDM denoisingmodel is obviously superior to traditional integral basedalgorithm when denoising

Acknowledgment

The work is supported by Foundation Franco-Chinoise PourLa Science Et Ses Applications (FFCSA) China NationalNature Science Foundation (Grants no 60972131 and no61201438) Returned Overseas Chinese Scholars State Edu-cation Ministry of China (Grant no 20111139) SichuanScience andTechnology Support Project of China (Grant nos2011GZ0201 and 2013SZ0071) Open fund of State key Lab-oratory of Networking and Switching Technology of BeijingUniversity of Posts and Telecommunications (SKLNST-2010-1-03) and Project of Sichuan Province of China under Grant2011GZ0201and Grant 2013SZ0071 the Soft Science Projectof Sichuan Province of China under Grant 2013ZR0010 andthe initial funding for young teachers of Sichuan Universityunder Grant 2011SCU11

References

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[2] K B Oldham and SpanierThe Fractional Calculus Integrationsand Differentiations of Arbitrary Order Academic Press NewYork NY USA 1974

[3] A C McBride Fractional Calculus Halsted Press New YorkNY USA 1986

[4] K Nishimoto Fractional Calculus University of New HavenPress New Haven Conn USA 1989

[5] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Publishers Yverdon Switzerland 1993

[6] K S Miller ldquoDerivatives of noninteger orderrdquo MathematicsMagazine vol 68 no 3 pp 183ndash192 1995

[7] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1987

[8] N Engheta ldquoOn fractional calculus and fractional multipolesin electromagnetismrdquo Institute of Electrical and Electronics

Engineers Transactions on Antennas and Propagation vol 44no 4 pp 554ndash566 1996

[9] N Engheta ldquoOn the role of fractional calculus in electromag-netic theoryrdquo IEEEAntennas and PropagationMagazine vol 39no 4 pp 35ndash46 1997

[10] M-P Chen andHM Srivastava ldquoFractional calculus operatorsand their applications involving power functions and summa-tion of seriesrdquo Applied Mathematics and Computation vol 81no 2-3 pp 287ndash304 1997

[11] P L Butzer and U Westphal ldquoAn introduction to fractionalcalculusrdquo in Applications of Fractional Calculus in Physicschapter 1 pp 1ndash85 World Scientific Singapore 2000

[12] S Kempfle I Schafer and H Beyer ldquoFractional calculusvia functional calculus theory and applicationsrdquo NonlinearDynamics vol 29 no 1ndash4 pp 99ndash127 2002

[13] R L Magin ldquoFractional calculus in bioengineeringrdquo CriticalReviews in Biomedical Engineering vol 32 no 3-4 pp 195ndash3772004

[14] A A Kilbas H M Srivastava and J J Trujiilo Theoryand Applications of Fractional Differential Equations ElsevierAmsterdam The Netherlands 2006

[15] J Sabatier O P Agrawal and J A Tenreiro Machado Advancesin Fractional Calculus Theoretical Developments and Appli-cations in Physics and Engineering Springer Dordrecht TheNetherlands 2007

[16] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[17] Y A Rossikhin and M V Shitikova ldquoApplications of fractionalcalculus to dynamic problems of linear and nonlinear heredi-tary mechanics of solidsrdquo Applied Mechanics Reviews vol 50no 1 pp 15ndash67 1997

[18] S Manabe ldquoA suggestion of fractional-order controller forflexible spacecraft attitude controlrdquoNonlinearDynamics vol 29no 1ndash4 pp 251ndash268 2002

[19] W Chen and S Holm ldquoFractional Laplacian time-spacemodelsfor linear and nonlinear lossy media exhibiting arbitrary fre-quency power-lawdependencyrdquo Journal of theAcoustical Societyof America vol 115 no 4 pp 1424ndash1430 2004

[20] E Perrin R Harba C Berzin-Joseph I Iribarren and ABonami ldquoNth-order fractional Brownianmotion and fractionalGaussian noisesrdquo IEEE Transactions on Signal Processing vol49 no 5 pp 1049ndash1059 2001

[21] C-C Tseng ldquoDesign of fractional order digital FIR differentia-torsrdquo IEEE Signal Processing Letters vol 8 no 3 pp 77ndash79 2001

[22] Y Q Chen and BM Vinagre ldquoA new IIR-type digital fractionalorder differentiatorrdquo Signal Processing vol 83 no 11 pp 2359ndash2365 2003

[23] Y-F Pu Research on application of fractional calculus to latestsignal analysis and processing [PhD thesis] Sichuan UniversityChengdu China 2006

[24] Y-F Pu X Yuan K Liao Z-L Chen and J-L Zhou ldquoFivenumerical algorithms of fractional calculus applied in modernsignal analyzing and processingrdquo Journal of Sichuan Universityvol 37 no 5 pp 118ndash124 2005

[25] Y-F Pu X Yuan K Liao et al ldquoStructuring analog fractancecircuit for 12 order fractional calculusrdquo in Proceedings of theIEEE 6th International Conference on ASIC (ASICON rsquo05) vol2 pp 1136ndash1139 Shanghai China October 2005

[26] Y-F Pu X Yuan K Liao and J-L Zhou ldquoImplement anyfractional order neural-type pulse oscillator with net-grid type


analog fractance circuitrdquo Journal of Sichuan University vol 38no 1 pp 128ndash132 2006

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[28] S Didas B Burgeth A Imiya and J Weickert ldquoRegularity andscale-space properties of fractional high order linear filteringrdquoin Proceedings of the 5th International Conference on Scale Spaceand PDE Methods in Computer Vision Scale-Space vol 3459pp 13ndash25 April 2005

[29] M Unser and T Blu ldquoFractional splines and waveletsrdquo SIAMReview vol 42 no 1 pp 43ndash67 2000

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[31] B Mathieu P Melchior A Oustaloup and C Ceyral ldquoFrac-tional differentiation for edge detectionrdquo Signal Processing vol83 no 11 pp 2421ndash2432 2003

[32] S-C Liu and S Chang ldquoDimension estimation of discrete-timefractional Brownian motion with applications to image textureclassificationrdquo IEEE Transactions on Image Processing vol 6 no8 pp 1176ndash1184 1997

[33] Y-F Pu ldquoFractional calculus approach to texture of digitalimagerdquo in Proceedings of the IEEE 8th International Conferenceon Signal Processing (ICSP rsquo06) pp 1002ndash1002 Beijing ChinaNovember 2006

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[35] Y-F Pu ldquoHigh precision fractional calculus filter of digitalimagerdquo Invention Patent of China NoZL2010101387422 2010

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[46] P Blomgren and T F Chan ldquoColor TV total variation methodsfor restoration of vector-valued imagesrdquo IEEE Transactions onImage Processing vol 7 no 3 pp 304ndash309 1998

[47] N P Galatsanos and A K Katsaggelos ldquoMethods for choosingthe regularization parameter and estimating the noise variancein image restoration and their relationrdquo IEEE Transactions onImage Processing vol 1 no 3 pp 322ndash336 1992

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[49] N Nguyen P Milanfar and G Golub ldquoEfficient generalizedcross-validation with applications to parametric image restora-tion and resolution enhancementrdquo IEEE Transactions on ImageProcessing vol 10 no 9 pp 1299ndash1308 2001

[50] D M Strong J F Aujol and T F Chan ldquoScale recognitionregularization parameter selection andMeyerrsquos G norm in totalvariation regularizationrdquoMultiscaleModelingamp Simulation vol5 no 1 pp 273ndash303 2006

[51] AMThompson J C Brown JW Kay andDM TitteringtonldquoA study of methods of choosing the smoothing parameterin image restoration by regularizationrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 13 no 4 pp 326ndash339 1991

[52] P Mrazek and M Navara ldquoSelection of optimal stoppingtime for nonlinear diffusion filteringrdquo International Journal ofComputer Vision vol 52 no 2-3 pp 189ndash203 2003

[53] G Gilboa N Sochen and Y Y Zeevi ldquoEstimation of optimalPDE-based denoising in the SNR senserdquo IEEE Transactions onImage Processing vol 15 no 8 pp 2269ndash2280 2006

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[55] J Darbon and M Sigelle ldquoExact optimization of discrete con-strained total variation minimization problemsrdquo in Proceedingsof the 10th nternational Workshop on Combinatorial ImageAnalysisI (WCIA rsquo04) pp 548ndash557 Auckland New Zealand2004

[56] J Darbon and M Sigelle ldquoImage restoration with discreteconstrained total variation I Fast and exact optimizationrdquoJournal of Mathematical Imaging and Vision vol 26 no 3 pp261ndash276 2006

[57] J Darbon and M Sigelle ldquoImage restoration with discreteconstrained total variation II Levelable functions convexpriors and non-convex casesrdquo Journal of Mathematical Imagingand Vision vol 26 no 3 pp 277ndash291 2006

[58] B Wohlberg and P Rodriguez ldquoAn iteratively reweighted normalgorithm forminimization of total variation functionalsrdquo IEEESignal Processing Letters vol 14 no 12 pp 948ndash951 2007

[59] F Catte P L Lions J M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992

[60] Y Meyer Oscillating Patterns in Image Processing and in SomeNonlinear Evolution Equations The American MathematicalSociety Providence RI USA 2001

[61] D Strong and T Chan ldquoEdge-preserving and scale-dependentproperties of total variation regularizationrdquo Inverse Problemsvol 19 no 6 pp S165ndashS187 2003


[62] S Alliney ldquoA property of theminimum vectors of a regularizingfunctional defined by means of the absolute normrdquo IEEETransactions on Signal Processing vol 45 no 4 pp 913ndash917 1997

[63] M Nikolova ldquoA variational approach to remove outliers andimpulse noiserdquo Journal ofMathematical Imaging andVision vol20 no 1-2 pp 99ndash120 2004

[64] T F Chan and S Esedoglu ldquoAspects of total variation regu-larized 119871

1 function approximationrdquo SIAM Journal on AppliedMathematics vol 65 no 5 pp 1817ndash1837 2005

[65] M Nikolova ldquoMinimizers of cost-functions involving nons-mooth data-fidelity termsrdquo SIAM Journal on Numerical Anal-ysis vol 40 no 3 pp 965ndash994 2002

[66] S Osher M Burger D Goldfarb J Xu and W Yin ldquoAniterative regularization method for total variation-based imagerestorationrdquoMultiscale Modeling amp Simulation vol 4 no 2 pp460ndash489 2005

[67] G Gilboa Y Y Zeevi and N Sochen ldquoTexture preserving vari-ational denoising using an adaptive fidelity termrdquo inProceedingsof the 2nd IEEE Workshop on Variational Geometric and LevelSet Methods in Computer Vision (VLSM rsquo03) pp 137ndash144 NiceFrance

[68] S Esedoglu and S J Osher ldquoDecomposition of images by theanisotropic Rudin-Osher-Fatemi modelrdquo Communications onPure and Applied Mathematics vol 57 no 12 pp 1609ndash16262004

[69] P Blomgren T F Chan and P Mulet ldquoExtensions to total vari-ation denoisingrdquo in Proceedings of the SPIE on Advanced SignalProcessing Algorithms Architectures and Implementations VIIvol 3162 pp 367ndash375 San Diego Calif USA July 1997

[70] P Blomgren P Mulet T F Chan and C K Wong ldquoTotal vari-ation image restoration numerical methods and extensionsrdquo inProceedings of the International Conference on Image Processing(ICIP rsquo97) pp 384ndash387 October 1997

[71] T Chan A Marquina and P Mulet ldquoHigh-order totalvariation-based image restorationrdquo SIAM Journal on ScientificComputing vol 22 no 2 pp 503ndash516 2000

[72] Y L You and M Kaveh ldquoFourth-order partial differentialequations for noise removalrdquo IEEE Transactions on ImageProcessing vol 9 no 10 pp 1723ndash1730 2000

[73] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15892003

[74] G Gilboa N Sochen and Y Y Zeevi ldquoImage enhancement anddenoising by complex diffusion processesrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 26 no 8 pp1020ndash1036 2004

[75] A Chambolle and P-L Lions ldquoImage recovery via total vari-ation minimization and related problemsrdquo Numerische Mathe-matik vol 76 no 2 pp 167ndash188 1997

[76] S Osher A Sole and L Vese ldquoImage decomposition andrestoration using total variation minimization and the 119867

minus1

normrdquoMultiscale Modeling amp Simulation vol 1 no 3 pp 349ndash370 2003

[77] M Lysaker andXC Tai ldquoIterative image restoration combiningtotal variation minimization and a second-order functionalrdquoInternational Journal of Computer Vision vol 66 no 1 pp 5ndash18 2006

[78] F Li C Shen J Fan andC Shen ldquoImage restoration combininga total variational filter and a fourth-order filterrdquo Journal of

Visual Communication and Image Representation vol 18 no 4pp 322ndash330 2007

[79] M Lysaker S Osher and X-C Tai ldquoNoise removal usingsmoothed normals and surface fittingrdquo IEEE Transactions onImage Processing vol 13 no 10 pp 1345ndash1357 2004

[80] F Dong Z Liu D Kong and K Liu ldquoAn improved LOTmodelfor image restorationrdquo Journal of Mathematical Imaging andVision vol 34 no 1 pp 89ndash97 2009

[81] P Guidotti and J V Lambers ldquoTwo new nonlinear nonlocaldiffusions for noise reductionrdquo Journal ofMathematical Imagingand Vision vol 33 no 1 pp 25ndash37 2009

[82] J Bai and X-C Feng ldquoFractional-order anisotropic diffusionfor image denoisingrdquo IEEE Transactions on Image Processingvol 16 no 10 pp 2492ndash2502 2007

[83] G Leitmann The Calculus of Variations and Optimal ControlAn Introduction Springer New York NY USA 1981

[84] V E Tarasov ldquoFractional vector calculus and fractionalMaxwellrsquos equationsrdquo Annals of Physics vol 323 no 11 pp2756ndash2778 2008

[85] W Rudin Functional Analysis 2nd Edition McGraw-Hill NewYork NY USA 1991

[86] C Tomasi and R Manduchi ldquoBilateral filtering for gray andcolor imagesrdquo in Proceedings of the 1998 IEEE 6th InternationalConference on Computer Vision pp 839ndash846 January 1998

[87] M Zhang and B K Gunturk ldquoMultiresolution bilateral filteringfor image denoisingrdquo IEEE Transactions on Image Processingvol 17 no 12 pp 2324ndash2333 2008

[88] H Yu L Zhao and HWang ldquoImage denoising using trivariateshrinkage filter in the wavelet domain and joint bilateral filterin the spatial domainrdquo IEEE Transactions on Image Processingvol 18 no 10 pp 2364ndash2369 2009

[89] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEETransactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[90] D D-Y Po and M N Do ldquoDirectional multiscale modeling ofimages using the contourlet transformrdquo IEEE Transactions onImage Processing vol 15 no 6 pp 1610ndash1620 2006

[91] G Y Chen and T D Bui ldquoMultiwavelets denoising usingneighboring coefficientsrdquo IEEE Signal Processing Letters vol 10no 7 pp 211ndash214 2003

[92] A Buades B Coll and J-M Morel ldquoA non-local algorithm forimage denoisingrdquo in Proceedings of the IEEE Computer SocietyConference on Computer Vision and Pattern Recognition (CVPRrsquo05) vol 2 pp 60ndash65 San Diego Calif USA June 2005

[93] A Buades B Coll and J-MMorel ldquoNonlocal image andmoviedenoisingrdquo International Journal of Computer Vision vol 76 no2 pp 123ndash139 2008

[94] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004

[95] M Abramowitz and I Stegun Handbook of MathematicalFunctionsWith Formulas Graphs andMathematical Tables USDepartment of Commerce Washington DC USA 1964

[96] P R HalmosMeasureTheory D van Nostrand Company NewYork NY USA 1950

[97] M E Munroe Introduction to Measure and Integration Addi-son Wesley London UK 1953

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Integer-order partial differential equation-based imageprocessing is an important branch in the field of image pro-cessing By exploring the essence of image and image process-ing people tend to reconstruct the traditional image process-ing approaches through strictly mathematical theories andit will be a great challenge to practical-oriented traditionalimage processing Image denoising is a significant researchbranch of integer-order partial differential equation-basedimage processing with two kinds of denoising approach thenonlinear diffusion-based method and the minimum energynorm-based variational method [39ndash42] They have twocorresponding models which are the anisotropic diffusionproposed by Perona and Malik [43] (Perona-Malik or PM)and the total variation model proposed by Rudin et al [44](Rudin-Osher-Fatemi or ROF) The PM model simulatesthe denoising process by a thermal energy diffusion processand the denoising result is the balanced state of thermaldiffusion while the ROF model describes the same thermalenergy by a total variation In further study some researchershave applied the PM model and the ROF model to colorimages [45 46] discussed the selection of the parameters forthe models [47ndash51] and found the optimal stopping pointin iteration process [52 53] Rudin and his team proposeda variable time step method to solve the Euler-Lagrangeequation [44] Vogel and Oman proposed improving thestability of ROF model by a fixed point iteration approach[54] Darbon and Sigelle decomposed the original problemsinto independent optimal Markov random fields by usinglevel set methods and obtained globally optimal solution byreconstruction [55ndash57] Wohlberg and Rodriguez proposedto solve the total variation by using an iterative weightednorm to improve the computing efficiency [58] MeanwhileCatte et al proposed to perform a Gaussian smoothingprocess in the initial stage to improve the suitability of thePM model [59] However PM model and ROF model havesome obvious defects in image denoising that is they caneasily lose the contrast and texture details and can producestaircase effects [39 60 61] Some improved models havebeen proposed to solve these problems To maintain thecontrast and texture details some scholars have proposedto replace the 119871

2 norm with the 1198711 norm [62ndash65] while

Osher et al proposed an iterative regularizationmethod [66]Gilboa et al proposed a denoising method using a numericaladaptive fidelity term that can change with the space [67]Esedoglu andOsher proposed to decompose images using theanisotropic ROFmodel and retaining certain edge directionalinformation [68] To remove the staircase effects Blomgren etal proposed to extend the total variation denoising model bychanging it with gradients [69 70] Some scholars introducedhigh-order derivative to energy norm [71ndash76] Lysaker etal integrated high-order deductive to original ROF model[77 78] while other scholars proposed a two-stage denoisingalgorithm which smoothes the corresponding vector fieldfirst and then fits it by using the curve surface [79 80]The above methods have provided some improvements inmaintaining contrast and texture details and removing thestaircase effect but they still have some drawbacks Firstthe improved algorithms have greatly increased calculationcomplexity for real-time processing and excessive storage and

computational requirements will lead them to be impracticalSecond the above algorithms are essentially integer-orderdifferential based algorithm and thus they may cause theedge field to be somewhat fuzzy and the texture-preservationeffect to be less effective than expected

We therefore propose to introduce a new mathematicalmethodmdashfractional calculus to the denoising field for textureimage and implementing a fractional partial differentialequation to solve the above problems by the integer-orderpartial differential equation-based denoising algorithms [2333ndash38] Guidotti and Lambers [81] and Bai and Feng [82]have pushed the classic anisotropic diffusion model to thefractional field extended gradient operator of the energynorm from first-order to fractional-order and numericallyimplemented the fractional developmental equation in thefrequency domain which has some effects on image denois-ing However the algorithm still has certain drawbacks Firstit simply took the gradient operator of the energy normfrom first order to fractional order and still cannot essentiallysolve the problem of how to nonlinearly maintain the texturedetails via the anisotropic diffusion Therefore the textureinformation is not retained well after denoising Second thealgorithm does not include the effects of the fractional powerof the energy norm and the fractional extreme value onnonlinearly maintaining texture details Third the methoddoes not deduce the corresponding fractional Euler-Lagrangeformula according to fractional calculus features and directlyreplace it according to the complex conjugate transposefeatures of the Hilbert adjoint operator It greatly increasedthe complex of the numerical implementation of the frac-tional developmental equation in frequency field Finally thetransition function of fractional calculus in Fourier transformdomain is (119894120596)V Its form looks simple but the Fourierrsquosinverse transform of (119894120596)V belongs to the first kind of Eulerintegral which is difficult to calculate theoretically Thealgorithm simply converted the first-order difference into thefractional-order difference in the frequency domain form andreplaced the fractional differential operator which has notsolved the computing problem of the Euler integral of the firstkind

The properties of fractional differential are as follows [2324 38] First the fractional differential of a constant is non-zero whereas it must be zero under integer-order differentialFractional calculus varies from a maximum at a singularleaping point to zero in the smooth areas where the signal isunchanged or not changed greatly note that by default anyinteger-order differential in a smooth area is approximatedto zero which is the remarkable difference between thefractional differential and integer-order differential Secondthe fractional differential at the starting point of a gradientof a signal phase or slope is nonzero which nonlinearlyenhances the singularity of high-frequency componentsWith the increasing fractional order the strengthening of thesingularity of high-frequency components is also greater Forexample when 0 lt V lt 1 the strengthening is less thanwhen V = 1 The integral differential is a special case of thefractional calculus Finally the fractional calculus along theslope is neither zero nor constant but is a nonlinear curvewhile integer-order differential along slope is the constant




2 Related Work



119863V119866-119871119904 (119909) =

119889V

[119889 (119909 minus 119886)]V 119904 (119909)

10038161003816100381610038161003816100381610038161003816119866-119871


((119909 minus 119886) 119873)

minusV

Γ (minusV)

times

119873minus1

sum

119896=0

Γ (119896 minus V)

Γ (119896 + 1)119904 (119909 minus 119896 (

119909 minus 119886

119873))

(1)





119863V119877-119871119904 (119909) =

119889V

[119889 (119909 minus 119886)]V 119904 (119909)

10038161003816100381610038161003816100381610038161003816119877-119871

=1

Γ (minusV)int

119909

119886


119904 (120578) 119889120578

=minus1

Γ (minusV)int

119909

119886

119904 (120578) 119889(119909 minus 120578)minusV V ≺ 0

(2)



119863V119877-119871119904 (119909) =

119889V

[119889 (119909 minus 119886)]V 119904 (119909)

10038161003816100381610038161003816100381610038161003816119877-119871

=119889

119899

119889119909119899

119889Vminus119899

[119889(119909 minus 119886]Vminus119899

119904 (119909)

10038161003816100381610038161003816100381610038161003816119877-119871

=

119899minus1

sum

119896=0

(119909 minus 119886)119896minusV119904(119896)(119886)

Γ (119896 minus V + 1)

+1


119909

119886

119904(119899)(120578)

(119909 minus 120578)Vminus119899+1

119889120578 0 le V ≺ 119899

(3)


FT [119863V119904 (119909)] = (119894120596)

VFT [119904 (119909)] minus119899minus1

sum

119896=0

(119894120596)119896 119889

Vminus1minus119896

119889119909Vminus1minus119896119904 (0)

(4)


FT [119863V119904 (119909)] = (119894120596)

VFT [119904 (119909)] (5)





d

c

a b

C1

C2

y = 1206011(x)

y = 1206012(x)

Ω

y

x

x = 1205951(y)

x = 1205952(y)A

B

0





V= 119863


119909119868V119910)Ω


119862(1198601198621119861)



997888997888997888997888rarr

1198601198621119861 119868V

119862minus


1(119909) 119910 = 120601

2(119909) 119886 le 119909 le 119887 or

119909 = 1205951(119910) 119909 = 120595



119863minusV1119863


V1119863

V2119875 = 119863

V1+V2119875 minus

119863V1+V2(119875 minus 119863

minusV1119863

V1119875) Thus when 119868

V2

119909119868V2

119910119863

V1

119910119875(119909 119910) =

119868V2

119909119863

V1minusV2

119910119875(119909 119910) minus 119863

V1minusV2

119910[119875(119909 119910)minus119863

minusV1

119910119863

V1

119910119875(119909 119910)] it has

(119868V2

119909119868V2

119910)Ω119863

V1

119910119875 (119909 119910)

=119887

119886119868V2

119909

1206012(119909)

1206011(119909)

119868V2

119910119863

V1

119910119875 (119909 119910)

=119887

119886119868V2

119909119863

V1minusV2

119910119875 (119909 119910)


119910[119875(119909 119910) minus 119863

minusV1

119910119863

V1

119910119875(119909 119910)]

10038161003816100381610038161003816

1206012(119909)

1206011(119909)

= minus 119868V2

119909

119862(1198611198622119860)

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

minus 119868V2

119909

119862(1198601198621119861)

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

= minus119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

(6)

Similarly it has

(119868V2

119909119868V2

119910)Ω119863

V1

119909119876 (119909 119910)

= 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

(7)


(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

+ 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

(8)



V1120593 = 0


V2120593 = 119863


to read

(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910) + 119868

V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)

(9)


2= V


V119909119876(119909 119910) minus 119863

V119910119875(119909 119910)) =

(119868V119909)119862minus119875(119909 119910) + (119868



= V2


1199091198681

119910)Ω(119863

1

119909119876(119909 119910) minus

1198631

119910119875(119909 119910)) = (119868

1

119909)119862minus119875(119909 119910) + (119868

1





119909+ 119895120593



V120597119909

V) + 119895(120597

V120597119910

V) = 119894119863

V119909+ 119895119863

V119910

= (119863V119909 119863




y

x

C3(x0 y1) C2(x1 y1)

C1(x1 y0)C0(x0 y0)

Ω

0




Γ(V)) int119909

119886119909

(119909 minus 120578)Vminus1119904(120578 119910)119889120578 119868V

119910119904(119909 119910) = (1Γ(V)) int

119910

119886119910

(119910 minus

120577)Vminus1119904(119909 120577)119889120577 and 119868

V119909119868V119910119904(119909 119910) = (1Γ

2(V)) int

119909

119886119909

int119910

119886119910

(119909 minus

120578)Vminus1(119910 minus 120577)



(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

+ 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

=1199091

1199090

119868V2

119909119863

V1minusV2

119910119875 minus 119863

V1minusV2

119910[119875 minus 119863

minusV1

119910119863

V1

119910119875]

+1199090

1199091

119868V2

119909119863

V1minusV2

119910119875 minus 119863

V1minusV2

119910[119875 minus 119863

minusV1

119910119863

V1

119910119875]

+1199101

1199100

119868V2

119910119863

V1minusV2

119909119876 minus 119863

V1minusV2

119909[119876 minus 119863

minusV1

119909119863

V1

119909119876]

+1199100

1199101

119868V2

119910119863

V1minusV2

119909119876 minus 119863

V1minusV2

119909[119876 minus 119863

minusV1

119909119863

V1

119909119876] equiv 0

(10)

Since suminfin

119898=0sum

119898

119899=0equiv sum

infin

119899=0sum

infin

119898=119899and ( V

119903+119899 ) (119903+119899

119899) equiv (

V119899 ) (

Vminus119899

119903)

we have

119863V119909minus119886

(119891119892) =

infin

sum

119899=0

[(V119899) (119863

Vminus119899

119909minus119886119891)119863

119899

119909minus119886119892] (11)


119909119868V2

119910)Ω[119863

V1

119909(119906120593

119909) +

119863V1

119910(119906120593

119910)] = (119868

V2

119909119868V2

119910)Ω[119863

V1

119909(119906120593

119909) minus 119863

V1

119910(minus119906120593

119910)] = 0 Then we

have

(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙

= (119868V2

119909119868V2

119910)Ω[(119863

V1

119909119906) 120593

119909+ (119863

V1

119910119906) 120593

119910]

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [(119863

V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910]

(12)


119863V∙ = 119863

V119909120593

119909+ 119863

V119910120593



infin

119899=1(V119899 ) [119894((119863

Vminus119899

119909119906)119906)119863

119899

119909+

119895((119863Vminus119899

119910119906)119906)119863

119899



V∙ = 119875

V119909120593

119909+ 119875

V119910120593




(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙ = ⟨119863

V1119906 ⟩

V2

= ⟨119906 (119863V1)

lowast

⟩V2

= (119868V2

119909119868V2

119910)Ω119906 ((119863

V1)

lowast

)

= minus(119868V2

119909119868V2

119910)Ω119906 (119875

V1 ∙ )

= minus(119868V1

119909119868V2

119910)Ω119906 (div119875V

1 )

(13)


2-order



(119863V)lowast



V1= V

2= 1 (12) becomes

(1198681

1199091198681

119910)Ω119863

1119906 ∙ = ⟨119863

1119906 ⟩

1

= ⟨119906 (1198631)lowast

⟩

1

= (1198681

1199091198681

119910)Ω119906 ((119863

1)lowast

) = minus1198681

1199091198681

119910

Ω

119906 (div1)

(15)



1∙ = 119863

1

119909120593

119909+ 119863

1

119910120593

119910represents



(1198631)lowast

= minusdiv1 (16)



119909119868V2

119910)Ω119863

V1119906 ∙ = 0 (14) becomes

(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [(119863

V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910]

= 0


V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910= (119863

V1minus119899

119909119906119863

V1minus119899

119910119906) ∙ (119863

119899

119909120593

119909

119863119899

119910120593



119909119906 and 119863

V1minus119899



119909120593

119909 119863

119899

119910120593

119910) = (0 0) is required


1

119909120593

119909

1198631

119910120593

119910) = (0 0) to make (119863

119899

119909120593

119909 119863

119899

119910120593

119910) = (0 0) estab-


V2

119909119868V2

119910)Ω(V1

1) [(119863

V1minus1

119909119906)119863

1

119909120593

119909+ (119863

V1minus1

119910119906)119863

1

119910120593

119910] =

minus(119868V2

119909119868V2

119910)Ω(V1

1) (119863

V1minus1

119909119906119863

V1minus1

119910119906) ∙ (119863

1

119909120593

119909 119863

1

119910120593

119910) = 0


minus(V1

1) [119863

1

119909120593

119909+ 119863

1

119910120593

119910] =

Γ (1 minus V1)

Γ (minusV1)[119863

1

119909120593

119909+ 119863

1

119910120593

119910] = 0


119909119868V2

119910)Ω119863

V1119906 ∙ = 0







119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙ =


(119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙

= (119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)119863

V1119906 ∙ (Φ

2() )

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [ (Φ

1(119863

V1119906)119863

V1minus119899

119909119906)

times 119863119899

119909(Φ

2() 120593

119909)

+ (Φ1(119863

V1119906)119863

V1minus119899

119910119906)

times119863119899

119910(Φ

2() 120593

119910)] = 0

(19)


minus (V1

1) [119863

1

119909(Φ

2() 120593

119909) + 119863

1

119910(Φ

2() 120593

119910)]

=Γ (1 minus V

1)

Γ (minusV1)[119863

1

119909(Φ

2() 120593

119909)

+1198631

119910(Φ

2() 120593

119910)] = 0

(20)


119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙ = 0




(119868V2

119909119868V2


integral (119868V119909119868V119910)Ω







0(119909 119910) + 119899(119909 119910)


0(119909 119910) + 119899(119909 119910) we


119904 to be |119863V1119904|

V2= (radic(119863

V1

119909 119904)2

+ (119863V1

119910 119904)2

)

V2

where V2is any


V2 is


119864FTV (119904) = (1198681

1199091198681

119910)Ω[119891 (

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

)] = ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

) 119889119909 119889119910

(21)






and Ψ2(120573) = ∬

Ω120582[119904 + (120573 minus 1)120585 minus 119904

0]119904

0119889119909 119889119910 where Ψ

2(120573)











Ψ (120573) = Ψ1(120573) + Ψ

2(120573)

= (1198681

1199091198681

119910)Ω[119891 (

1003816100381610038161003816119863V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+ 120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0]


= ∬Ω

[119891 (1003816100381610038161003816119863

V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0] 119889119909 119889119910

(22)



119891(|119863V1119904 + (120573 minus 1)119863

V1120585|

V2) exists Ψ

1(120573) has the V

3-order


119863V3

120573Ψ

1(120573)

100381610038161003816100381610038161003816120573=1

=120597V3

120597120573V3∬

Ω

119891 (10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= 0

(23)

where it has 2= ||

2= [radic()

2]

2


V1119904+(120573minus1)119863








119863V120573minus119886

119891 [119892 (120573)] =(120573 minus 119886)

minusV

Γ (1 minus V)119891

+

infin

sum

119899=1

(V119899)

(120573 minus 119886)119899minusV

Γ (119899 minus V + 1)119899

times

119899

sum

119898=1

119863119898

119892119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573minus119886119892

119896

]

]

119875119896

(24)



119896satisfies

119899

sum

119896=1

119896119875119896= 119899

119899

sum

119896=1

119875119896= 119898 (25)


119899

119896=1(1119875

119896)[119863

119896

120573minus119886119892119896]

119875119896

|119898=1rarr119899


119875119896|119898=1rarr119899


∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120573minusV3119891 (

10038161003816100381610038161003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(V3

119899)

120573119899minusV3

Γ (119899 minus V3+ 1)

119899

times

119899

sum

119898=1

119863119898

| |V2119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)

119896

]

]

119875119896

119889119909 119889119910

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

sum

119898=1

119863119898

| |V2119891

100381610038161003816100381610038161003816120573=1

sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

times 119889119909119889119910 = 0

(26)


120578119891(120578) = 1 and119863119898

120578119891(120578)

119898ge2


∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

119898=1

119889119909 119889119910 = 0

(27)


119899= 1 and

1198751= 119875



∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119899119889119909 119889119910 = 0

(28)


120573(||

V2)|


119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896+1

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus119899minus1

times1003816100381610038161003816119863

V11205851003816100381610038161003816

119899minus1

(119863V1120585) ∙ (119863

V1119904)

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus1198991003816100381610038161003816119863

V11205851003816100381610038161003816

119899

(29)



∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus2119896minus21003816100381610038161003816119863

V11205851003816100381610038161003816

2119896

Γ (minusV3) (2119896)

(119863V1119904)

∙ 119863V1 [

Γ (2119896 minus V3)

Γ (2119896 minus V3+ 1)

119904 minus(V

2minus 2119896) Γ (2119896 minus V

3+ 1)

(2119896 + 1) Γ (2119896 minus V3+ 2)

120585]

times 119889119909 119889119910 = 0

(30)


120591=1(V

2minus 120591 + 1)

119899=0




3+1))119904 minus ((V

2minus

2119896)Γ(2119896minusV3+1)(2119896+1)Γ(2119896minusV



Γ (1 minus V1)


3)

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904) + 119863

1

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)] = 0

(31)


120591=1(V

2minus 120591 + 1)

119899=0



119863V3

120573Ψ

2(120573)

100381610038161003816100381610038161003816120573=1

= 119863V3

120573(119868

1

1199091198681

119910)Ω120582 [119904 + (120573 minus 1) 120585 minus 119904

0] 119904

0

100381610038161003816100381610038161003816120573=1

= ∬Ω

1205821199040


3)



3) 120585] 119889119909 119889119910 = 0

(32)


0minus



1205821199040


3)= 0 (33)




3= 1


by

120597V3119904

120597119905V3=



3)

times

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

minus120582119904

0


3)

(34)


120591=1(V

2minus 120591 + 1)

119899=0


Ω119899(119909 119910)119889119909 119889119910 = ∬

Ω(119904minus119904

0)119889119909 119889119910 = 0When 120597V3119904120597119905V3 = 0



120582 (119905) =minusΓ (1 minus V

1) Γ (1 minus V

3) Γ (2 minus V

3)


3) 119904

0

times∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

times (119904 minus 1199040)2

119889119909 119889119910

(35)



120591=1(V

2minus120591+1)




3) in the


3= 0 the












Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(a)


Cs119896Cs1

Cs0Csminus1

Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(b)

Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1

Cs119899




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(c)


Cs119896Cs1Cs0

Csminus1Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(d)







0

0

0

0 0

0

00

0 0

0

0

0

0Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

(e)







0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(f)



middotmiddotmiddot





0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middotmiddotmiddot



middotmiddotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(g)


middotmiddotmiddot

middotmiddotmiddot






0

0

0

00

0

0 0

0 0

0

0

0

0 Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Cs119899minus2

Cs119899minus1

(h)










119866-119871 cong

(119909minusV119873


119896=0(Γ(119896 minus V)Γ(119896 + 1))119904(119909 + (V1199092119873) minus








119904minus1

= (V4)+(V28)1198621199040

=

1 minus (V22) minus (V38) 1198621199041

= minus(5V4) + (5V316) + (V416) 119862

119904119896


119904119899minus2



(minus(V4) + (V28))] 119862119904119899minus1



119904119899


= 1198941119863

V119909minus

+ 1198942119863

V119910minus

+

1198943119863

V119909+

+ 1198944119863

V119910+

+ 1198945119863

Vldd + 119894

6119863

Vrud + 119894

7119863

Vlud + 119894

8119863

Vrdd in (34)


119910minus 119863V

119909+ 119863V

119910+ 119863V

ldd 119863Vrud 119863

Vlud




1

119909119904(119909 119910) = (2[119904(119909 + 1 119910) minus 119904(119909 minus 1 119910)] + 119904(119909 + 1 119910 +


1



119899= 119899Δ119905



= 119904(119909 119910 119905119899) Thus we have

1199040

119909119910= 119904

0(119909 119910 119905

0) + 119899(119909 119910 119905

0) where 119904

0



that is 1199040(119909 119910 119905

0) = 119904

0(119909 119910 119905



120597V119904

120597119905V= Δ119905

minusV[119904

119899+1

119909119910minus 119904

119899

119909119910+

2120583120578

Γ (3 minus V)(119904

119899

119909119910minus 119904

Vlowast119909119910)2

(119904119899

119909119910)minusV]

when V = 1

(36)


0(119909 119910 119905

0) is unknown


119909119910rarr 119904

0(119909 119910 119905

0) =

1199040(119909 119910 119905


0)119909119910

cong 1199040

119909119910minus 119904

119899

119909119910for






to 119904Vlowast119909119910

that is 119904119899119909119910

rarr 119904Vlowast119909119910

We consider that (119904119899119909119910

minus

119904Vlowast119909119910)2cong (119904

119899minus1

119909119910minus 119904

119899


(119904119899

119909119910minus 119904



119904119899+1

119909119910= 119875 (119904

119899

119909119910) Δ119905

V3 minus

120582119899Δ119905

V3


3)119904119899

119909119910

+ 119904119899

119909119910minus

2120583

Γ (3 minus V3)(119904

119899minus1

119909119910minus 119904

119899

119909119910)2

(119904119899

119909119910)minusV3

when V3= 1 2 and 3

(37)

120582119899=Γ (1 minus V

3) Γ (2 minus V

3)

1205901198992

119904119899119909119910

sum

119909119910

119875 (119904119899

119909119910) (119904

0

119909119910minus 119904

119899

119909119910)2

(38)

where prod119899

120591=1(V

2minus 120591 + 1)

119899=0

= 1 1205901198992

= sum119909119910

(1199040

119909119910minus 119904

119899

119909119910)2

119875(119904119899

119909119910) = (minusΓ(1 minus V

1)Γ(minusV

1)Γ(minusV

3)) sum

1

119896=0(prod

2119896

120591=1(V

2minus

120591+1)(2119896))[1198631

119909(|119863

V1119904

119899

119909119910|V2minus2119896minus2

119863V1

119909119904119899

119909119910)+119863

1

119910(|119863

V1119904

119899

119909119910|V2minus2119896minus2

sdot119863V1

119910119904119899

119909119910)]




2


1119904119899

119909119910| = 0 in


119899

119909119910| = 00689

when |119863V1119904

119899



= 000001 when 119904119899

119909119910= 0



119909= (119904

119899+1

119909minus1+2119904

119899+1

119909+119904

119899+1

119909+1)4when |119863V

1119904119899

119909| ≺ 120572

119860and120572

119873119860≺ 120572

119860


119909=

119904119899+1

119909 where 120572

119860= (1119873

119909) sum

119873119909

119909|119863

V1119904

119899

119909| 120572

119873119860= (|119863

V1119904

119899

119909minus1| +

2|119863V1119904

119899

119909| + |119863

V1119904

119899



119909= (119904

119899+1

119909minus1+ 2119904

119899+1

119909+ 119904

119899+1



119909119910= (

119909119904119899+1

119909119910+

119910119904119899+1

119909119910+

119903119904119899+1

119909119910+

119897119904119899+1

119909119910)4 When 120572

119909

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119909

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910)4 and 119910

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119910

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119910

119904119899+1

119909119910= (119904

119899+1

119909119910minus1+ 2119904

119899+1

119909119910+

119904119899+1

119909119910+1)4 and 119909

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119903

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119903

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+1+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910minus1)4 and 119909

119904119899+1

119909119910=

119910119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 and in the


119909119910=

119910119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910=

119904119899+1

119909119910 where 120572119909

119873119860= (|119863

V1119904

119899

119909minus1119910| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910|)4

120572119910

119873119860= (|119863

V1119904

119899

119909119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909119910+1|)4 and

120572119903

119873119860= (|119863

V1119904

119899

119909minus1119910+1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910minus1|)4 120572119897

119873119860=

(|119863V1119904

119899

119909minus1119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910+1|)4 Here 119909 denotes



151

05


0

0 1000 2000 3000 4000 5000 6000


(a)

151

05

minus05minus1

0

0 1000 2000 3000 4000 5000 6000


(b)

151

05


0

0 1000 2000 3000 4000 5000 6000


(c)

151

05


0

0 1000 2000 3000 4000 5000 6000


(d)

151

05


0

0 1000 2000 3000 4000 5000 6000


(e)

151

05


0

0 1000 2000 3000 4000 5000 6000


(f)

151

05


0

0 1000 2000 3000 4000 5000 6000


(g)

151

05


0

0 1000 2000 3000 4000 5000 6000


(h)

151

05


0

0 1000 2000 3000 4000 5000 6000


(i)


1= 1025 V

2= 225 V

3= 105 Δ119905 = 00296)


119909= 119904

119899

119909when |119904119899+1

119909| ≻ 6|119904

119899



119909119910= 119904

119899

119909119910when |119904

119899+1

119909119910| ≻ 6|119904

119899

119909119910| for two-

dimensional signal


















(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)


1= 175 V

2= 225 V

3= 105 Δ119905 = 10


1= 225 V

2= 25 V

3= 125 Δ119905 = 10











(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2 Related Work



119863V119866-119871119904 (119909) =

119889V

[119889 (119909 minus 119886)]V 119904 (119909)

10038161003816100381610038161003816100381610038161003816119866-119871


((119909 minus 119886) 119873)

minusV

Γ (minusV)

times

119873minus1

sum

119896=0

Γ (119896 minus V)

Γ (119896 + 1)119904 (119909 minus 119896 (

119909 minus 119886

119873))

(1)





119863V119877-119871119904 (119909) =

119889V

[119889 (119909 minus 119886)]V 119904 (119909)

10038161003816100381610038161003816100381610038161003816119877-119871

=1

Γ (minusV)int

119909

119886


119904 (120578) 119889120578

=minus1

Γ (minusV)int

119909

119886

119904 (120578) 119889(119909 minus 120578)minusV V ≺ 0

(2)



119863V119877-119871119904 (119909) =

119889V

[119889 (119909 minus 119886)]V 119904 (119909)

10038161003816100381610038161003816100381610038161003816119877-119871

=119889

119899

119889119909119899

119889Vminus119899

[119889(119909 minus 119886]Vminus119899

119904 (119909)

10038161003816100381610038161003816100381610038161003816119877-119871

=

119899minus1

sum

119896=0

(119909 minus 119886)119896minusV119904(119896)(119886)

Γ (119896 minus V + 1)

+1


119909

119886

119904(119899)(120578)

(119909 minus 120578)Vminus119899+1

119889120578 0 le V ≺ 119899

(3)


FT [119863V119904 (119909)] = (119894120596)

VFT [119904 (119909)] minus119899minus1

sum

119896=0

(119894120596)119896 119889

Vminus1minus119896

119889119909Vminus1minus119896119904 (0)

(4)


FT [119863V119904 (119909)] = (119894120596)

VFT [119904 (119909)] (5)





d

c

a b

C1

C2

y = 1206011(x)

y = 1206012(x)

Ω

y

x

x = 1205951(y)

x = 1205952(y)A

B

0





V= 119863


119909119868V119910)Ω


119862(1198601198621119861)



997888997888997888997888rarr

1198601198621119861 119868V

119862minus


1(119909) 119910 = 120601

2(119909) 119886 le 119909 le 119887 or

119909 = 1205951(119910) 119909 = 120595



119863minusV1119863


V1119863

V2119875 = 119863

V1+V2119875 minus

119863V1+V2(119875 minus 119863

minusV1119863

V1119875) Thus when 119868

V2

119909119868V2

119910119863

V1

119910119875(119909 119910) =

119868V2

119909119863

V1minusV2

119910119875(119909 119910) minus 119863

V1minusV2

119910[119875(119909 119910)minus119863

minusV1

119910119863

V1

119910119875(119909 119910)] it has

(119868V2

119909119868V2

119910)Ω119863

V1

119910119875 (119909 119910)

=119887

119886119868V2

119909

1206012(119909)

1206011(119909)

119868V2

119910119863

V1

119910119875 (119909 119910)

=119887

119886119868V2

119909119863

V1minusV2

119910119875 (119909 119910)


119910[119875(119909 119910) minus 119863

minusV1

119910119863

V1

119910119875(119909 119910)]

10038161003816100381610038161003816

1206012(119909)

1206011(119909)

= minus 119868V2

119909

119862(1198611198622119860)

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

minus 119868V2

119909

119862(1198601198621119861)

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

= minus119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

(6)

Similarly it has

(119868V2

119909119868V2

119910)Ω119863

V1

119909119876 (119909 119910)

= 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

(7)


(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

+ 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

(8)



V1120593 = 0


V2120593 = 119863


to read

(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910) + 119868

V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)

(9)


2= V


V119909119876(119909 119910) minus 119863

V119910119875(119909 119910)) =

(119868V119909)119862minus119875(119909 119910) + (119868



= V2


1199091198681

119910)Ω(119863

1

119909119876(119909 119910) minus

1198631

119910119875(119909 119910)) = (119868

1

119909)119862minus119875(119909 119910) + (119868

1





119909+ 119895120593



V120597119909

V) + 119895(120597

V120597119910

V) = 119894119863

V119909+ 119895119863

V119910

= (119863V119909 119863




y

x

C3(x0 y1) C2(x1 y1)

C1(x1 y0)C0(x0 y0)

Ω

0




Γ(V)) int119909

119886119909

(119909 minus 120578)Vminus1119904(120578 119910)119889120578 119868V

119910119904(119909 119910) = (1Γ(V)) int

119910

119886119910

(119910 minus

120577)Vminus1119904(119909 120577)119889120577 and 119868

V119909119868V119910119904(119909 119910) = (1Γ

2(V)) int

119909

119886119909

int119910

119886119910

(119909 minus

120578)Vminus1(119910 minus 120577)



(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

+ 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

=1199091

1199090

119868V2

119909119863

V1minusV2

119910119875 minus 119863

V1minusV2

119910[119875 minus 119863

minusV1

119910119863

V1

119910119875]

+1199090

1199091

119868V2

119909119863

V1minusV2

119910119875 minus 119863

V1minusV2

119910[119875 minus 119863

minusV1

119910119863

V1

119910119875]

+1199101

1199100

119868V2

119910119863

V1minusV2

119909119876 minus 119863

V1minusV2

119909[119876 minus 119863

minusV1

119909119863

V1

119909119876]

+1199100

1199101

119868V2

119910119863

V1minusV2

119909119876 minus 119863

V1minusV2

119909[119876 minus 119863

minusV1

119909119863

V1

119909119876] equiv 0

(10)

Since suminfin

119898=0sum

119898

119899=0equiv sum

infin

119899=0sum

infin

119898=119899and ( V

119903+119899 ) (119903+119899

119899) equiv (

V119899 ) (

Vminus119899

119903)

we have

119863V119909minus119886

(119891119892) =

infin

sum

119899=0

[(V119899) (119863

Vminus119899

119909minus119886119891)119863

119899

119909minus119886119892] (11)


119909119868V2

119910)Ω[119863

V1

119909(119906120593

119909) +

119863V1

119910(119906120593

119910)] = (119868

V2

119909119868V2

119910)Ω[119863

V1

119909(119906120593

119909) minus 119863

V1

119910(minus119906120593

119910)] = 0 Then we

have

(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙

= (119868V2

119909119868V2

119910)Ω[(119863

V1

119909119906) 120593

119909+ (119863

V1

119910119906) 120593

119910]

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [(119863

V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910]

(12)


119863V∙ = 119863

V119909120593

119909+ 119863

V119910120593



infin

119899=1(V119899 ) [119894((119863

Vminus119899

119909119906)119906)119863

119899

119909+

119895((119863Vminus119899

119910119906)119906)119863

119899



V∙ = 119875

V119909120593

119909+ 119875

V119910120593




(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙ = ⟨119863

V1119906 ⟩

V2

= ⟨119906 (119863V1)

lowast

⟩V2

= (119868V2

119909119868V2

119910)Ω119906 ((119863

V1)

lowast

)

= minus(119868V2

119909119868V2

119910)Ω119906 (119875

V1 ∙ )

= minus(119868V1

119909119868V2

119910)Ω119906 (div119875V

1 )

(13)


2-order



(119863V)lowast



V1= V

2= 1 (12) becomes

(1198681

1199091198681

119910)Ω119863

1119906 ∙ = ⟨119863

1119906 ⟩

1

= ⟨119906 (1198631)lowast

⟩

1

= (1198681

1199091198681

119910)Ω119906 ((119863

1)lowast

) = minus1198681

1199091198681

119910

Ω

119906 (div1)

(15)



1∙ = 119863

1

119909120593

119909+ 119863

1

119910120593

119910represents



(1198631)lowast

= minusdiv1 (16)



119909119868V2

119910)Ω119863

V1119906 ∙ = 0 (14) becomes

(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [(119863

V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910]

= 0


V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910= (119863

V1minus119899

119909119906119863

V1minus119899

119910119906) ∙ (119863

119899

119909120593

119909

119863119899

119910120593



119909119906 and 119863

V1minus119899



119909120593

119909 119863

119899

119910120593

119910) = (0 0) is required


1

119909120593

119909

1198631

119910120593

119910) = (0 0) to make (119863

119899

119909120593

119909 119863

119899

119910120593

119910) = (0 0) estab-


V2

119909119868V2

119910)Ω(V1

1) [(119863

V1minus1

119909119906)119863

1

119909120593

119909+ (119863

V1minus1

119910119906)119863

1

119910120593

119910] =

minus(119868V2

119909119868V2

119910)Ω(V1

1) (119863

V1minus1

119909119906119863

V1minus1

119910119906) ∙ (119863

1

119909120593

119909 119863

1

119910120593

119910) = 0


minus(V1

1) [119863

1

119909120593

119909+ 119863

1

119910120593

119910] =

Γ (1 minus V1)

Γ (minusV1)[119863

1

119909120593

119909+ 119863

1

119910120593

119910] = 0


119909119868V2

119910)Ω119863

V1119906 ∙ = 0







119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙ =


(119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙

= (119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)119863

V1119906 ∙ (Φ

2() )

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [ (Φ

1(119863

V1119906)119863

V1minus119899

119909119906)

times 119863119899

119909(Φ

2() 120593

119909)

+ (Φ1(119863

V1119906)119863

V1minus119899

119910119906)

times119863119899

119910(Φ

2() 120593

119910)] = 0

(19)


minus (V1

1) [119863

1

119909(Φ

2() 120593

119909) + 119863

1

119910(Φ

2() 120593

119910)]

=Γ (1 minus V

1)

Γ (minusV1)[119863

1

119909(Φ

2() 120593

119909)

+1198631

119910(Φ

2() 120593

119910)] = 0

(20)


119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙ = 0




(119868V2

119909119868V2


integral (119868V119909119868V119910)Ω







0(119909 119910) + 119899(119909 119910)


0(119909 119910) + 119899(119909 119910) we


119904 to be |119863V1119904|

V2= (radic(119863

V1

119909 119904)2

+ (119863V1

119910 119904)2

)

V2

where V2is any


V2 is


119864FTV (119904) = (1198681

1199091198681

119910)Ω[119891 (

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

)] = ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

) 119889119909 119889119910

(21)






and Ψ2(120573) = ∬

Ω120582[119904 + (120573 minus 1)120585 minus 119904

0]119904

0119889119909 119889119910 where Ψ

2(120573)











Ψ (120573) = Ψ1(120573) + Ψ

2(120573)

= (1198681

1199091198681

119910)Ω[119891 (

1003816100381610038161003816119863V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+ 120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0]


= ∬Ω

[119891 (1003816100381610038161003816119863

V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0] 119889119909 119889119910

(22)



119891(|119863V1119904 + (120573 minus 1)119863

V1120585|

V2) exists Ψ

1(120573) has the V

3-order


119863V3

120573Ψ

1(120573)

100381610038161003816100381610038161003816120573=1

=120597V3

120597120573V3∬

Ω

119891 (10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= 0

(23)

where it has 2= ||

2= [radic()

2]

2


V1119904+(120573minus1)119863








119863V120573minus119886

119891 [119892 (120573)] =(120573 minus 119886)

minusV

Γ (1 minus V)119891

+

infin

sum

119899=1

(V119899)

(120573 minus 119886)119899minusV

Γ (119899 minus V + 1)119899

times

119899

sum

119898=1

119863119898

119892119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573minus119886119892

119896

]

]

119875119896

(24)



119896satisfies

119899

sum

119896=1

119896119875119896= 119899

119899

sum

119896=1

119875119896= 119898 (25)


119899

119896=1(1119875

119896)[119863

119896

120573minus119886119892119896]

119875119896

|119898=1rarr119899


119875119896|119898=1rarr119899


∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120573minusV3119891 (

10038161003816100381610038161003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(V3

119899)

120573119899minusV3

Γ (119899 minus V3+ 1)

119899

times

119899

sum

119898=1

119863119898

| |V2119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)

119896

]

]

119875119896

119889119909 119889119910

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

sum

119898=1

119863119898

| |V2119891

100381610038161003816100381610038161003816120573=1

sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

times 119889119909119889119910 = 0

(26)


120578119891(120578) = 1 and119863119898

120578119891(120578)

119898ge2


∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

119898=1

119889119909 119889119910 = 0

(27)


119899= 1 and

1198751= 119875



∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119899119889119909 119889119910 = 0

(28)


120573(||

V2)|


119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896+1

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus119899minus1

times1003816100381610038161003816119863

V11205851003816100381610038161003816

119899minus1

(119863V1120585) ∙ (119863

V1119904)

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus1198991003816100381610038161003816119863

V11205851003816100381610038161003816

119899

(29)



∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus2119896minus21003816100381610038161003816119863

V11205851003816100381610038161003816

2119896

Γ (minusV3) (2119896)

(119863V1119904)

∙ 119863V1 [

Γ (2119896 minus V3)

Γ (2119896 minus V3+ 1)

119904 minus(V

2minus 2119896) Γ (2119896 minus V

3+ 1)

(2119896 + 1) Γ (2119896 minus V3+ 2)

120585]

times 119889119909 119889119910 = 0

(30)


120591=1(V

2minus 120591 + 1)

119899=0




3+1))119904 minus ((V

2minus

2119896)Γ(2119896minusV3+1)(2119896+1)Γ(2119896minusV



Γ (1 minus V1)


3)

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904) + 119863

1

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)] = 0

(31)


120591=1(V

2minus 120591 + 1)

119899=0



119863V3

120573Ψ

2(120573)

100381610038161003816100381610038161003816120573=1

= 119863V3

120573(119868

1

1199091198681

119910)Ω120582 [119904 + (120573 minus 1) 120585 minus 119904

0] 119904

0

100381610038161003816100381610038161003816120573=1

= ∬Ω

1205821199040


3)



3) 120585] 119889119909 119889119910 = 0

(32)


0minus



1205821199040


3)= 0 (33)




3= 1


by

120597V3119904

120597119905V3=



3)

times

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

minus120582119904

0


3)

(34)


120591=1(V

2minus 120591 + 1)

119899=0


Ω119899(119909 119910)119889119909 119889119910 = ∬

Ω(119904minus119904

0)119889119909 119889119910 = 0When 120597V3119904120597119905V3 = 0



120582 (119905) =minusΓ (1 minus V

1) Γ (1 minus V

3) Γ (2 minus V

3)


3) 119904

0

times∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

times (119904 minus 1199040)2

119889119909 119889119910

(35)



120591=1(V

2minus120591+1)




3) in the


3= 0 the












Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(a)


Cs119896Cs1

Cs0Csminus1

Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(b)

Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1

Cs119899




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(c)


Cs119896Cs1Cs0

Csminus1Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(d)







0

0

0

0 0

0

00

0 0

0

0

0

0Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

(e)







0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(f)



middotmiddotmiddot





0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middotmiddotmiddot



middotmiddotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(g)


middotmiddotmiddot

middotmiddotmiddot






0

0

0

00

0

0 0

0 0

0

0

0

0 Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Cs119899minus2

Cs119899minus1

(h)










119866-119871 cong

(119909minusV119873


119896=0(Γ(119896 minus V)Γ(119896 + 1))119904(119909 + (V1199092119873) minus








119904minus1

= (V4)+(V28)1198621199040

=

1 minus (V22) minus (V38) 1198621199041

= minus(5V4) + (5V316) + (V416) 119862

119904119896


119904119899minus2



(minus(V4) + (V28))] 119862119904119899minus1



119904119899


= 1198941119863

V119909minus

+ 1198942119863

V119910minus

+

1198943119863

V119909+

+ 1198944119863

V119910+

+ 1198945119863

Vldd + 119894

6119863

Vrud + 119894

7119863

Vlud + 119894

8119863

Vrdd in (34)


119910minus 119863V

119909+ 119863V

119910+ 119863V

ldd 119863Vrud 119863

Vlud




1

119909119904(119909 119910) = (2[119904(119909 + 1 119910) minus 119904(119909 minus 1 119910)] + 119904(119909 + 1 119910 +


1



119899= 119899Δ119905



= 119904(119909 119910 119905119899) Thus we have

1199040

119909119910= 119904

0(119909 119910 119905

0) + 119899(119909 119910 119905

0) where 119904

0



that is 1199040(119909 119910 119905

0) = 119904

0(119909 119910 119905



120597V119904

120597119905V= Δ119905

minusV[119904

119899+1

119909119910minus 119904

119899

119909119910+

2120583120578

Γ (3 minus V)(119904

119899

119909119910minus 119904

Vlowast119909119910)2

(119904119899

119909119910)minusV]

when V = 1

(36)


0(119909 119910 119905

0) is unknown


119909119910rarr 119904

0(119909 119910 119905

0) =

1199040(119909 119910 119905


0)119909119910

cong 1199040

119909119910minus 119904

119899

119909119910for






to 119904Vlowast119909119910

that is 119904119899119909119910

rarr 119904Vlowast119909119910

We consider that (119904119899119909119910

minus

119904Vlowast119909119910)2cong (119904

119899minus1

119909119910minus 119904

119899


(119904119899

119909119910minus 119904



119904119899+1

119909119910= 119875 (119904

119899

119909119910) Δ119905

V3 minus

120582119899Δ119905

V3


3)119904119899

119909119910

+ 119904119899

119909119910minus

2120583

Γ (3 minus V3)(119904

119899minus1

119909119910minus 119904

119899

119909119910)2

(119904119899

119909119910)minusV3

when V3= 1 2 and 3

(37)

120582119899=Γ (1 minus V

3) Γ (2 minus V

3)

1205901198992

119904119899119909119910

sum

119909119910

119875 (119904119899

119909119910) (119904

0

119909119910minus 119904

119899

119909119910)2

(38)

where prod119899

120591=1(V

2minus 120591 + 1)

119899=0

= 1 1205901198992

= sum119909119910

(1199040

119909119910minus 119904

119899

119909119910)2

119875(119904119899

119909119910) = (minusΓ(1 minus V

1)Γ(minusV

1)Γ(minusV

3)) sum

1

119896=0(prod

2119896

120591=1(V

2minus

120591+1)(2119896))[1198631

119909(|119863

V1119904

119899

119909119910|V2minus2119896minus2

119863V1

119909119904119899

119909119910)+119863

1

119910(|119863

V1119904

119899

119909119910|V2minus2119896minus2

sdot119863V1

119910119904119899

119909119910)]




2


1119904119899

119909119910| = 0 in


119899

119909119910| = 00689

when |119863V1119904

119899



= 000001 when 119904119899

119909119910= 0



119909= (119904

119899+1

119909minus1+2119904

119899+1

119909+119904

119899+1

119909+1)4when |119863V

1119904119899

119909| ≺ 120572

119860and120572

119873119860≺ 120572

119860


119909=

119904119899+1

119909 where 120572

119860= (1119873

119909) sum

119873119909

119909|119863

V1119904

119899

119909| 120572

119873119860= (|119863

V1119904

119899

119909minus1| +

2|119863V1119904

119899

119909| + |119863

V1119904

119899



119909= (119904

119899+1

119909minus1+ 2119904

119899+1

119909+ 119904

119899+1



119909119910= (

119909119904119899+1

119909119910+

119910119904119899+1

119909119910+

119903119904119899+1

119909119910+

119897119904119899+1

119909119910)4 When 120572

119909

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119909

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910)4 and 119910

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119910

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119910

119904119899+1

119909119910= (119904

119899+1

119909119910minus1+ 2119904

119899+1

119909119910+

119904119899+1

119909119910+1)4 and 119909

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119903

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119903

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+1+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910minus1)4 and 119909

119904119899+1

119909119910=

119910119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 and in the


119909119910=

119910119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910=

119904119899+1

119909119910 where 120572119909

119873119860= (|119863

V1119904

119899

119909minus1119910| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910|)4

120572119910

119873119860= (|119863

V1119904

119899

119909119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909119910+1|)4 and

120572119903

119873119860= (|119863

V1119904

119899

119909minus1119910+1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910minus1|)4 120572119897

119873119860=

(|119863V1119904

119899

119909minus1119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910+1|)4 Here 119909 denotes



151

05


0

0 1000 2000 3000 4000 5000 6000


(a)

151

05

minus05minus1

0

0 1000 2000 3000 4000 5000 6000


(b)

151

05


0

0 1000 2000 3000 4000 5000 6000


(c)

151

05


0

0 1000 2000 3000 4000 5000 6000


(d)

151

05


0

0 1000 2000 3000 4000 5000 6000


(e)

151

05


0

0 1000 2000 3000 4000 5000 6000


(f)

151

05


0

0 1000 2000 3000 4000 5000 6000


(g)

151

05


0

0 1000 2000 3000 4000 5000 6000


(h)

151

05


0

0 1000 2000 3000 4000 5000 6000


(i)


1= 1025 V

2= 225 V

3= 105 Δ119905 = 00296)


119909= 119904

119899

119909when |119904119899+1

119909| ≻ 6|119904

119899



119909119910= 119904

119899

119909119910when |119904

119899+1

119909119910| ≻ 6|119904

119899

119909119910| for two-

dimensional signal


















(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)


1= 175 V

2= 225 V

3= 105 Δ119905 = 10


1= 225 V

2= 25 V

3= 125 Δ119905 = 10











(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










d

c

a b

C1

C2

y = 1206011(x)

y = 1206012(x)

Ω

y

x

x = 1205951(y)

x = 1205952(y)A

B

0





V= 119863


119909119868V119910)Ω


119862(1198601198621119861)



997888997888997888997888rarr

1198601198621119861 119868V

119862minus


1(119909) 119910 = 120601

2(119909) 119886 le 119909 le 119887 or

119909 = 1205951(119910) 119909 = 120595



119863minusV1119863


V1119863

V2119875 = 119863

V1+V2119875 minus

119863V1+V2(119875 minus 119863

minusV1119863

V1119875) Thus when 119868

V2

119909119868V2

119910119863

V1

119910119875(119909 119910) =

119868V2

119909119863

V1minusV2

119910119875(119909 119910) minus 119863

V1minusV2

119910[119875(119909 119910)minus119863

minusV1

119910119863

V1

119910119875(119909 119910)] it has

(119868V2

119909119868V2

119910)Ω119863

V1

119910119875 (119909 119910)

=119887

119886119868V2

119909

1206012(119909)

1206011(119909)

119868V2

119910119863

V1

119910119875 (119909 119910)

=119887

119886119868V2

119909119863

V1minusV2

119910119875 (119909 119910)


119910[119875(119909 119910) minus 119863

minusV1

119910119863

V1

119910119875(119909 119910)]

10038161003816100381610038161003816

1206012(119909)

1206011(119909)

= minus 119868V2

119909

119862(1198611198622119860)

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

minus 119868V2

119909

119862(1198601198621119861)

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

= minus119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

(6)

Similarly it has

(119868V2

119909119868V2

119910)Ω119863

V1

119909119876 (119909 119910)

= 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

(7)


(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

+ 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

(8)



V1120593 = 0


V2120593 = 119863


to read

(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910) + 119868

V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)

(9)


2= V


V119909119876(119909 119910) minus 119863

V119910119875(119909 119910)) =

(119868V119909)119862minus119875(119909 119910) + (119868



= V2


1199091198681

119910)Ω(119863

1

119909119876(119909 119910) minus

1198631

119910119875(119909 119910)) = (119868

1

119909)119862minus119875(119909 119910) + (119868

1





119909+ 119895120593



V120597119909

V) + 119895(120597

V120597119910

V) = 119894119863

V119909+ 119895119863

V119910

= (119863V119909 119863




y

x

C3(x0 y1) C2(x1 y1)

C1(x1 y0)C0(x0 y0)

Ω

0




Γ(V)) int119909

119886119909

(119909 minus 120578)Vminus1119904(120578 119910)119889120578 119868V

119910119904(119909 119910) = (1Γ(V)) int

119910

119886119910

(119910 minus

120577)Vminus1119904(119909 120577)119889120577 and 119868

V119909119868V119910119904(119909 119910) = (1Γ

2(V)) int

119909

119886119909

int119910

119886119910

(119909 minus

120578)Vminus1(119910 minus 120577)



(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

+ 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

=1199091

1199090

119868V2

119909119863

V1minusV2

119910119875 minus 119863

V1minusV2

119910[119875 minus 119863

minusV1

119910119863

V1

119910119875]

+1199090

1199091

119868V2

119909119863

V1minusV2

119910119875 minus 119863

V1minusV2

119910[119875 minus 119863

minusV1

119910119863

V1

119910119875]

+1199101

1199100

119868V2

119910119863

V1minusV2

119909119876 minus 119863

V1minusV2

119909[119876 minus 119863

minusV1

119909119863

V1

119909119876]

+1199100

1199101

119868V2

119910119863

V1minusV2

119909119876 minus 119863

V1minusV2

119909[119876 minus 119863

minusV1

119909119863

V1

119909119876] equiv 0

(10)

Since suminfin

119898=0sum

119898

119899=0equiv sum

infin

119899=0sum

infin

119898=119899and ( V

119903+119899 ) (119903+119899

119899) equiv (

V119899 ) (

Vminus119899

119903)

we have

119863V119909minus119886

(119891119892) =

infin

sum

119899=0

[(V119899) (119863

Vminus119899

119909minus119886119891)119863

119899

119909minus119886119892] (11)


119909119868V2

119910)Ω[119863

V1

119909(119906120593

119909) +

119863V1

119910(119906120593

119910)] = (119868

V2

119909119868V2

119910)Ω[119863

V1

119909(119906120593

119909) minus 119863

V1

119910(minus119906120593

119910)] = 0 Then we

have

(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙

= (119868V2

119909119868V2

119910)Ω[(119863

V1

119909119906) 120593

119909+ (119863

V1

119910119906) 120593

119910]

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [(119863

V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910]

(12)


119863V∙ = 119863

V119909120593

119909+ 119863

V119910120593



infin

119899=1(V119899 ) [119894((119863

Vminus119899

119909119906)119906)119863

119899

119909+

119895((119863Vminus119899

119910119906)119906)119863

119899



V∙ = 119875

V119909120593

119909+ 119875

V119910120593




(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙ = ⟨119863

V1119906 ⟩

V2

= ⟨119906 (119863V1)

lowast

⟩V2

= (119868V2

119909119868V2

119910)Ω119906 ((119863

V1)

lowast

)

= minus(119868V2

119909119868V2

119910)Ω119906 (119875

V1 ∙ )

= minus(119868V1

119909119868V2

119910)Ω119906 (div119875V

1 )

(13)


2-order



(119863V)lowast



V1= V

2= 1 (12) becomes

(1198681

1199091198681

119910)Ω119863

1119906 ∙ = ⟨119863

1119906 ⟩

1

= ⟨119906 (1198631)lowast

⟩

1

= (1198681

1199091198681

119910)Ω119906 ((119863

1)lowast

) = minus1198681

1199091198681

119910

Ω

119906 (div1)

(15)



1∙ = 119863

1

119909120593

119909+ 119863

1

119910120593

119910represents



(1198631)lowast

= minusdiv1 (16)



119909119868V2

119910)Ω119863

V1119906 ∙ = 0 (14) becomes

(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [(119863

V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910]

= 0


V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910= (119863

V1minus119899

119909119906119863

V1minus119899

119910119906) ∙ (119863

119899

119909120593

119909

119863119899

119910120593



119909119906 and 119863

V1minus119899



119909120593

119909 119863

119899

119910120593

119910) = (0 0) is required


1

119909120593

119909

1198631

119910120593

119910) = (0 0) to make (119863

119899

119909120593

119909 119863

119899

119910120593

119910) = (0 0) estab-


V2

119909119868V2

119910)Ω(V1

1) [(119863

V1minus1

119909119906)119863

1

119909120593

119909+ (119863

V1minus1

119910119906)119863

1

119910120593

119910] =

minus(119868V2

119909119868V2

119910)Ω(V1

1) (119863

V1minus1

119909119906119863

V1minus1

119910119906) ∙ (119863

1

119909120593

119909 119863

1

119910120593

119910) = 0


minus(V1

1) [119863

1

119909120593

119909+ 119863

1

119910120593

119910] =

Γ (1 minus V1)

Γ (minusV1)[119863

1

119909120593

119909+ 119863

1

119910120593

119910] = 0


119909119868V2

119910)Ω119863

V1119906 ∙ = 0







119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙ =


(119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙

= (119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)119863

V1119906 ∙ (Φ

2() )

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [ (Φ

1(119863

V1119906)119863

V1minus119899

119909119906)

times 119863119899

119909(Φ

2() 120593

119909)

+ (Φ1(119863

V1119906)119863

V1minus119899

119910119906)

times119863119899

119910(Φ

2() 120593

119910)] = 0

(19)


minus (V1

1) [119863

1

119909(Φ

2() 120593

119909) + 119863

1

119910(Φ

2() 120593

119910)]

=Γ (1 minus V

1)

Γ (minusV1)[119863

1

119909(Φ

2() 120593

119909)

+1198631

119910(Φ

2() 120593

119910)] = 0

(20)


119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙ = 0




(119868V2

119909119868V2


integral (119868V119909119868V119910)Ω







0(119909 119910) + 119899(119909 119910)


0(119909 119910) + 119899(119909 119910) we


119904 to be |119863V1119904|

V2= (radic(119863

V1

119909 119904)2

+ (119863V1

119910 119904)2

)

V2

where V2is any


V2 is


119864FTV (119904) = (1198681

1199091198681

119910)Ω[119891 (

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

)] = ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

) 119889119909 119889119910

(21)






and Ψ2(120573) = ∬

Ω120582[119904 + (120573 minus 1)120585 minus 119904

0]119904

0119889119909 119889119910 where Ψ

2(120573)











Ψ (120573) = Ψ1(120573) + Ψ

2(120573)

= (1198681

1199091198681

119910)Ω[119891 (

1003816100381610038161003816119863V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+ 120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0]


= ∬Ω

[119891 (1003816100381610038161003816119863

V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0] 119889119909 119889119910

(22)



119891(|119863V1119904 + (120573 minus 1)119863

V1120585|

V2) exists Ψ

1(120573) has the V

3-order


119863V3

120573Ψ

1(120573)

100381610038161003816100381610038161003816120573=1

=120597V3

120597120573V3∬

Ω

119891 (10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= 0

(23)

where it has 2= ||

2= [radic()

2]

2


V1119904+(120573minus1)119863








119863V120573minus119886

119891 [119892 (120573)] =(120573 minus 119886)

minusV

Γ (1 minus V)119891

+

infin

sum

119899=1

(V119899)

(120573 minus 119886)119899minusV

Γ (119899 minus V + 1)119899

times

119899

sum

119898=1

119863119898

119892119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573minus119886119892

119896

]

]

119875119896

(24)



119896satisfies

119899

sum

119896=1

119896119875119896= 119899

119899

sum

119896=1

119875119896= 119898 (25)


119899

119896=1(1119875

119896)[119863

119896

120573minus119886119892119896]

119875119896

|119898=1rarr119899


119875119896|119898=1rarr119899


∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120573minusV3119891 (

10038161003816100381610038161003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(V3

119899)

120573119899minusV3

Γ (119899 minus V3+ 1)

119899

times

119899

sum

119898=1

119863119898

| |V2119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)

119896

]

]

119875119896

119889119909 119889119910

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

sum

119898=1

119863119898

| |V2119891

100381610038161003816100381610038161003816120573=1

sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

times 119889119909119889119910 = 0

(26)


120578119891(120578) = 1 and119863119898

120578119891(120578)

119898ge2


∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

119898=1

119889119909 119889119910 = 0

(27)


119899= 1 and

1198751= 119875



∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119899119889119909 119889119910 = 0

(28)


120573(||

V2)|


119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896+1

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus119899minus1

times1003816100381610038161003816119863

V11205851003816100381610038161003816

119899minus1

(119863V1120585) ∙ (119863

V1119904)

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus1198991003816100381610038161003816119863

V11205851003816100381610038161003816

119899

(29)



∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus2119896minus21003816100381610038161003816119863

V11205851003816100381610038161003816

2119896

Γ (minusV3) (2119896)

(119863V1119904)

∙ 119863V1 [

Γ (2119896 minus V3)

Γ (2119896 minus V3+ 1)

119904 minus(V

2minus 2119896) Γ (2119896 minus V

3+ 1)

(2119896 + 1) Γ (2119896 minus V3+ 2)

120585]

times 119889119909 119889119910 = 0

(30)


120591=1(V

2minus 120591 + 1)

119899=0




3+1))119904 minus ((V

2minus

2119896)Γ(2119896minusV3+1)(2119896+1)Γ(2119896minusV



Γ (1 minus V1)


3)

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904) + 119863

1

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)] = 0

(31)


120591=1(V

2minus 120591 + 1)

119899=0



119863V3

120573Ψ

2(120573)

100381610038161003816100381610038161003816120573=1

= 119863V3

120573(119868

1

1199091198681

119910)Ω120582 [119904 + (120573 minus 1) 120585 minus 119904

0] 119904

0

100381610038161003816100381610038161003816120573=1

= ∬Ω

1205821199040


3)



3) 120585] 119889119909 119889119910 = 0

(32)


0minus



1205821199040


3)= 0 (33)




3= 1


by

120597V3119904

120597119905V3=



3)

times

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

minus120582119904

0


3)

(34)


120591=1(V

2minus 120591 + 1)

119899=0


Ω119899(119909 119910)119889119909 119889119910 = ∬

Ω(119904minus119904

0)119889119909 119889119910 = 0When 120597V3119904120597119905V3 = 0



120582 (119905) =minusΓ (1 minus V

1) Γ (1 minus V

3) Γ (2 minus V

3)


3) 119904

0

times∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

times (119904 minus 1199040)2

119889119909 119889119910

(35)



120591=1(V

2minus120591+1)




3) in the


3= 0 the












Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(a)


Cs119896Cs1

Cs0Csminus1

Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(b)

Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1

Cs119899




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(c)


Cs119896Cs1Cs0

Csminus1Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(d)







0

0

0

0 0

0

00

0 0

0

0

0

0Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

(e)







0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(f)



middotmiddotmiddot





0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middotmiddotmiddot



middotmiddotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(g)


middotmiddotmiddot

middotmiddotmiddot






0

0

0

00

0

0 0

0 0

0

0

0

0 Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Cs119899minus2

Cs119899minus1

(h)










119866-119871 cong

(119909minusV119873


119896=0(Γ(119896 minus V)Γ(119896 + 1))119904(119909 + (V1199092119873) minus








119904minus1

= (V4)+(V28)1198621199040

=

1 minus (V22) minus (V38) 1198621199041

= minus(5V4) + (5V316) + (V416) 119862

119904119896


119904119899minus2



(minus(V4) + (V28))] 119862119904119899minus1



119904119899


= 1198941119863

V119909minus

+ 1198942119863

V119910minus

+

1198943119863

V119909+

+ 1198944119863

V119910+

+ 1198945119863

Vldd + 119894

6119863

Vrud + 119894

7119863

Vlud + 119894

8119863

Vrdd in (34)


119910minus 119863V

119909+ 119863V

119910+ 119863V

ldd 119863Vrud 119863

Vlud




1

119909119904(119909 119910) = (2[119904(119909 + 1 119910) minus 119904(119909 minus 1 119910)] + 119904(119909 + 1 119910 +


1



119899= 119899Δ119905



= 119904(119909 119910 119905119899) Thus we have

1199040

119909119910= 119904

0(119909 119910 119905

0) + 119899(119909 119910 119905

0) where 119904

0



that is 1199040(119909 119910 119905

0) = 119904

0(119909 119910 119905



120597V119904

120597119905V= Δ119905

minusV[119904

119899+1

119909119910minus 119904

119899

119909119910+

2120583120578

Γ (3 minus V)(119904

119899

119909119910minus 119904

Vlowast119909119910)2

(119904119899

119909119910)minusV]

when V = 1

(36)


0(119909 119910 119905

0) is unknown


119909119910rarr 119904

0(119909 119910 119905

0) =

1199040(119909 119910 119905


0)119909119910

cong 1199040

119909119910minus 119904

119899

119909119910for






to 119904Vlowast119909119910

that is 119904119899119909119910

rarr 119904Vlowast119909119910

We consider that (119904119899119909119910

minus

119904Vlowast119909119910)2cong (119904

119899minus1

119909119910minus 119904

119899


(119904119899

119909119910minus 119904



119904119899+1

119909119910= 119875 (119904

119899

119909119910) Δ119905

V3 minus

120582119899Δ119905

V3


3)119904119899

119909119910

+ 119904119899

119909119910minus

2120583

Γ (3 minus V3)(119904

119899minus1

119909119910minus 119904

119899

119909119910)2

(119904119899

119909119910)minusV3

when V3= 1 2 and 3

(37)

120582119899=Γ (1 minus V

3) Γ (2 minus V

3)

1205901198992

119904119899119909119910

sum

119909119910

119875 (119904119899

119909119910) (119904

0

119909119910minus 119904

119899

119909119910)2

(38)

where prod119899

120591=1(V

2minus 120591 + 1)

119899=0

= 1 1205901198992

= sum119909119910

(1199040

119909119910minus 119904

119899

119909119910)2

119875(119904119899

119909119910) = (minusΓ(1 minus V

1)Γ(minusV

1)Γ(minusV

3)) sum

1

119896=0(prod

2119896

120591=1(V

2minus

120591+1)(2119896))[1198631

119909(|119863

V1119904

119899

119909119910|V2minus2119896minus2

119863V1

119909119904119899

119909119910)+119863

1

119910(|119863

V1119904

119899

119909119910|V2minus2119896minus2

sdot119863V1

119910119904119899

119909119910)]




2


1119904119899

119909119910| = 0 in


119899

119909119910| = 00689

when |119863V1119904

119899



= 000001 when 119904119899

119909119910= 0



119909= (119904

119899+1

119909minus1+2119904

119899+1

119909+119904

119899+1

119909+1)4when |119863V

1119904119899

119909| ≺ 120572

119860and120572

119873119860≺ 120572

119860


119909=

119904119899+1

119909 where 120572

119860= (1119873

119909) sum

119873119909

119909|119863

V1119904

119899

119909| 120572

119873119860= (|119863

V1119904

119899

119909minus1| +

2|119863V1119904

119899

119909| + |119863

V1119904

119899



119909= (119904

119899+1

119909minus1+ 2119904

119899+1

119909+ 119904

119899+1



119909119910= (

119909119904119899+1

119909119910+

119910119904119899+1

119909119910+

119903119904119899+1

119909119910+

119897119904119899+1

119909119910)4 When 120572

119909

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119909

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910)4 and 119910

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119910

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119910

119904119899+1

119909119910= (119904

119899+1

119909119910minus1+ 2119904

119899+1

119909119910+

119904119899+1

119909119910+1)4 and 119909

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119903

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119903

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+1+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910minus1)4 and 119909

119904119899+1

119909119910=

119910119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 and in the


119909119910=

119910119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910=

119904119899+1

119909119910 where 120572119909

119873119860= (|119863

V1119904

119899

119909minus1119910| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910|)4

120572119910

119873119860= (|119863

V1119904

119899

119909119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909119910+1|)4 and

120572119903

119873119860= (|119863

V1119904

119899

119909minus1119910+1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910minus1|)4 120572119897

119873119860=

(|119863V1119904

119899

119909minus1119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910+1|)4 Here 119909 denotes



151

05


0

0 1000 2000 3000 4000 5000 6000


(a)

151

05

minus05minus1

0

0 1000 2000 3000 4000 5000 6000


(b)

151

05


0

0 1000 2000 3000 4000 5000 6000


(c)

151

05


0

0 1000 2000 3000 4000 5000 6000


(d)

151

05


0

0 1000 2000 3000 4000 5000 6000


(e)

151

05


0

0 1000 2000 3000 4000 5000 6000


(f)

151

05


0

0 1000 2000 3000 4000 5000 6000


(g)

151

05


0

0 1000 2000 3000 4000 5000 6000


(h)

151

05


0

0 1000 2000 3000 4000 5000 6000


(i)


1= 1025 V

2= 225 V

3= 105 Δ119905 = 00296)


119909= 119904

119899

119909when |119904119899+1

119909| ≻ 6|119904

119899



119909119910= 119904

119899

119909119910when |119904

119899+1

119909119910| ≻ 6|119904

119899

119909119910| for two-

dimensional signal


















(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)


1= 175 V

2= 225 V

3= 105 Δ119905 = 10


1= 225 V

2= 25 V

3= 125 Δ119905 = 10











(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










y

x

C3(x0 y1) C2(x1 y1)

C1(x1 y0)C0(x0 y0)

Ω

0




Γ(V)) int119909

119886119909

(119909 minus 120578)Vminus1119904(120578 119910)119889120578 119868V

119910119904(119909 119910) = (1Γ(V)) int

119910

119886119910

(119910 minus

120577)Vminus1119904(119909 120577)119889120577 and 119868

V119909119868V119910119904(119909 119910) = (1Γ

2(V)) int

119909

119886119909

int119910

119886119910

(119909 minus

120578)Vminus1(119910 minus 120577)



(119868V2

119909119868V2

119910)Ω(119863

V1

119909119876 (119909 119910) minus 119863

V1

119910119875 (119909 119910))

= 119868V2

119909

119862minus

119863V1minusV2

119910119875 (119909 119910)


119910[119875 (119909 119910) minus 119863

minusV1

119910119863

V1

119910119875 (119909 119910)]

+ 119868V2

119910

119862minus

119863V1minusV2

119909119876 (119909 119910)


119909[119876 (119909 119910) minus 119863

minusV1

119909119863

V1

119909119876 (119909 119910)]

=1199091

1199090

119868V2

119909119863

V1minusV2

119910119875 minus 119863

V1minusV2

119910[119875 minus 119863

minusV1

119910119863

V1

119910119875]

+1199090

1199091

119868V2

119909119863

V1minusV2

119910119875 minus 119863

V1minusV2

119910[119875 minus 119863

minusV1

119910119863

V1

119910119875]

+1199101

1199100

119868V2

119910119863

V1minusV2

119909119876 minus 119863

V1minusV2

119909[119876 minus 119863

minusV1

119909119863

V1

119909119876]

+1199100

1199101

119868V2

119910119863

V1minusV2

119909119876 minus 119863

V1minusV2

119909[119876 minus 119863

minusV1

119909119863

V1

119909119876] equiv 0

(10)

Since suminfin

119898=0sum

119898

119899=0equiv sum

infin

119899=0sum

infin

119898=119899and ( V

119903+119899 ) (119903+119899

119899) equiv (

V119899 ) (

Vminus119899

119903)

we have

119863V119909minus119886

(119891119892) =

infin

sum

119899=0

[(V119899) (119863

Vminus119899

119909minus119886119891)119863

119899

119909minus119886119892] (11)


119909119868V2

119910)Ω[119863

V1

119909(119906120593

119909) +

119863V1

119910(119906120593

119910)] = (119868

V2

119909119868V2

119910)Ω[119863

V1

119909(119906120593

119909) minus 119863

V1

119910(minus119906120593

119910)] = 0 Then we

have

(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙

= (119868V2

119909119868V2

119910)Ω[(119863

V1

119909119906) 120593

119909+ (119863

V1

119910119906) 120593

119910]

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [(119863

V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910]

(12)


119863V∙ = 119863

V119909120593

119909+ 119863

V119910120593



infin

119899=1(V119899 ) [119894((119863

Vminus119899

119909119906)119906)119863

119899

119909+

119895((119863Vminus119899

119910119906)119906)119863

119899



V∙ = 119875

V119909120593

119909+ 119875

V119910120593




(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙ = ⟨119863

V1119906 ⟩

V2

= ⟨119906 (119863V1)

lowast

⟩V2

= (119868V2

119909119868V2

119910)Ω119906 ((119863

V1)

lowast

)

= minus(119868V2

119909119868V2

119910)Ω119906 (119875

V1 ∙ )

= minus(119868V1

119909119868V2

119910)Ω119906 (div119875V

1 )

(13)


2-order



(119863V)lowast



V1= V

2= 1 (12) becomes

(1198681

1199091198681

119910)Ω119863

1119906 ∙ = ⟨119863

1119906 ⟩

1

= ⟨119906 (1198631)lowast

⟩

1

= (1198681

1199091198681

119910)Ω119906 ((119863

1)lowast

) = minus1198681

1199091198681

119910

Ω

119906 (div1)

(15)



1∙ = 119863

1

119909120593

119909+ 119863

1

119910120593

119910represents



(1198631)lowast

= minusdiv1 (16)



119909119868V2

119910)Ω119863

V1119906 ∙ = 0 (14) becomes

(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [(119863

V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910]

= 0


V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910= (119863

V1minus119899

119909119906119863

V1minus119899

119910119906) ∙ (119863

119899

119909120593

119909

119863119899

119910120593



119909119906 and 119863

V1minus119899



119909120593

119909 119863

119899

119910120593

119910) = (0 0) is required


1

119909120593

119909

1198631

119910120593

119910) = (0 0) to make (119863

119899

119909120593

119909 119863

119899

119910120593

119910) = (0 0) estab-


V2

119909119868V2

119910)Ω(V1

1) [(119863

V1minus1

119909119906)119863

1

119909120593

119909+ (119863

V1minus1

119910119906)119863

1

119910120593

119910] =

minus(119868V2

119909119868V2

119910)Ω(V1

1) (119863

V1minus1

119909119906119863

V1minus1

119910119906) ∙ (119863

1

119909120593

119909 119863

1

119910120593

119910) = 0


minus(V1

1) [119863

1

119909120593

119909+ 119863

1

119910120593

119910] =

Γ (1 minus V1)

Γ (minusV1)[119863

1

119909120593

119909+ 119863

1

119910120593

119910] = 0


119909119868V2

119910)Ω119863

V1119906 ∙ = 0







119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙ =


(119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙

= (119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)119863

V1119906 ∙ (Φ

2() )

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [ (Φ

1(119863

V1119906)119863

V1minus119899

119909119906)

times 119863119899

119909(Φ

2() 120593

119909)

+ (Φ1(119863

V1119906)119863

V1minus119899

119910119906)

times119863119899

119910(Φ

2() 120593

119910)] = 0

(19)


minus (V1

1) [119863

1

119909(Φ

2() 120593

119909) + 119863

1

119910(Φ

2() 120593

119910)]

=Γ (1 minus V

1)

Γ (minusV1)[119863

1

119909(Φ

2() 120593

119909)

+1198631

119910(Φ

2() 120593

119910)] = 0

(20)


119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙ = 0




(119868V2

119909119868V2


integral (119868V119909119868V119910)Ω







0(119909 119910) + 119899(119909 119910)


0(119909 119910) + 119899(119909 119910) we


119904 to be |119863V1119904|

V2= (radic(119863

V1

119909 119904)2

+ (119863V1

119910 119904)2

)

V2

where V2is any


V2 is


119864FTV (119904) = (1198681

1199091198681

119910)Ω[119891 (

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

)] = ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

) 119889119909 119889119910

(21)






and Ψ2(120573) = ∬

Ω120582[119904 + (120573 minus 1)120585 minus 119904

0]119904

0119889119909 119889119910 where Ψ

2(120573)











Ψ (120573) = Ψ1(120573) + Ψ

2(120573)

= (1198681

1199091198681

119910)Ω[119891 (

1003816100381610038161003816119863V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+ 120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0]


= ∬Ω

[119891 (1003816100381610038161003816119863

V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0] 119889119909 119889119910

(22)



119891(|119863V1119904 + (120573 minus 1)119863

V1120585|

V2) exists Ψ

1(120573) has the V

3-order


119863V3

120573Ψ

1(120573)

100381610038161003816100381610038161003816120573=1

=120597V3

120597120573V3∬

Ω

119891 (10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= 0

(23)

where it has 2= ||

2= [radic()

2]

2


V1119904+(120573minus1)119863








119863V120573minus119886

119891 [119892 (120573)] =(120573 minus 119886)

minusV

Γ (1 minus V)119891

+

infin

sum

119899=1

(V119899)

(120573 minus 119886)119899minusV

Γ (119899 minus V + 1)119899

times

119899

sum

119898=1

119863119898

119892119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573minus119886119892

119896

]

]

119875119896

(24)



119896satisfies

119899

sum

119896=1

119896119875119896= 119899

119899

sum

119896=1

119875119896= 119898 (25)


119899

119896=1(1119875

119896)[119863

119896

120573minus119886119892119896]

119875119896

|119898=1rarr119899


119875119896|119898=1rarr119899


∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120573minusV3119891 (

10038161003816100381610038161003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(V3

119899)

120573119899minusV3

Γ (119899 minus V3+ 1)

119899

times

119899

sum

119898=1

119863119898

| |V2119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)

119896

]

]

119875119896

119889119909 119889119910

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

sum

119898=1

119863119898

| |V2119891

100381610038161003816100381610038161003816120573=1

sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

times 119889119909119889119910 = 0

(26)


120578119891(120578) = 1 and119863119898

120578119891(120578)

119898ge2


∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

119898=1

119889119909 119889119910 = 0

(27)


119899= 1 and

1198751= 119875



∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119899119889119909 119889119910 = 0

(28)


120573(||

V2)|


119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896+1

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus119899minus1

times1003816100381610038161003816119863

V11205851003816100381610038161003816

119899minus1

(119863V1120585) ∙ (119863

V1119904)

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus1198991003816100381610038161003816119863

V11205851003816100381610038161003816

119899

(29)



∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus2119896minus21003816100381610038161003816119863

V11205851003816100381610038161003816

2119896

Γ (minusV3) (2119896)

(119863V1119904)

∙ 119863V1 [

Γ (2119896 minus V3)

Γ (2119896 minus V3+ 1)

119904 minus(V

2minus 2119896) Γ (2119896 minus V

3+ 1)

(2119896 + 1) Γ (2119896 minus V3+ 2)

120585]

times 119889119909 119889119910 = 0

(30)


120591=1(V

2minus 120591 + 1)

119899=0




3+1))119904 minus ((V

2minus

2119896)Γ(2119896minusV3+1)(2119896+1)Γ(2119896minusV



Γ (1 minus V1)


3)

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904) + 119863

1

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)] = 0

(31)


120591=1(V

2minus 120591 + 1)

119899=0



119863V3

120573Ψ

2(120573)

100381610038161003816100381610038161003816120573=1

= 119863V3

120573(119868

1

1199091198681

119910)Ω120582 [119904 + (120573 minus 1) 120585 minus 119904

0] 119904

0

100381610038161003816100381610038161003816120573=1

= ∬Ω

1205821199040


3)



3) 120585] 119889119909 119889119910 = 0

(32)


0minus



1205821199040


3)= 0 (33)




3= 1


by

120597V3119904

120597119905V3=



3)

times

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

minus120582119904

0


3)

(34)


120591=1(V

2minus 120591 + 1)

119899=0


Ω119899(119909 119910)119889119909 119889119910 = ∬

Ω(119904minus119904

0)119889119909 119889119910 = 0When 120597V3119904120597119905V3 = 0



120582 (119905) =minusΓ (1 minus V

1) Γ (1 minus V

3) Γ (2 minus V

3)


3) 119904

0

times∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

times (119904 minus 1199040)2

119889119909 119889119910

(35)



120591=1(V

2minus120591+1)




3) in the


3= 0 the












Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(a)


Cs119896Cs1

Cs0Csminus1

Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(b)

Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1

Cs119899




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(c)


Cs119896Cs1Cs0

Csminus1Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(d)







0

0

0

0 0

0

00

0 0

0

0

0

0Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

(e)







0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(f)



middotmiddotmiddot





0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middotmiddotmiddot



middotmiddotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(g)


middotmiddotmiddot

middotmiddotmiddot






0

0

0

00

0

0 0

0 0

0

0

0

0 Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Cs119899minus2

Cs119899minus1

(h)










119866-119871 cong

(119909minusV119873


119896=0(Γ(119896 minus V)Γ(119896 + 1))119904(119909 + (V1199092119873) minus








119904minus1

= (V4)+(V28)1198621199040

=

1 minus (V22) minus (V38) 1198621199041

= minus(5V4) + (5V316) + (V416) 119862

119904119896


119904119899minus2



(minus(V4) + (V28))] 119862119904119899minus1



119904119899


= 1198941119863

V119909minus

+ 1198942119863

V119910minus

+

1198943119863

V119909+

+ 1198944119863

V119910+

+ 1198945119863

Vldd + 119894

6119863

Vrud + 119894

7119863

Vlud + 119894

8119863

Vrdd in (34)


119910minus 119863V

119909+ 119863V

119910+ 119863V

ldd 119863Vrud 119863

Vlud




1

119909119904(119909 119910) = (2[119904(119909 + 1 119910) minus 119904(119909 minus 1 119910)] + 119904(119909 + 1 119910 +


1



119899= 119899Δ119905



= 119904(119909 119910 119905119899) Thus we have

1199040

119909119910= 119904

0(119909 119910 119905

0) + 119899(119909 119910 119905

0) where 119904

0



that is 1199040(119909 119910 119905

0) = 119904

0(119909 119910 119905



120597V119904

120597119905V= Δ119905

minusV[119904

119899+1

119909119910minus 119904

119899

119909119910+

2120583120578

Γ (3 minus V)(119904

119899

119909119910minus 119904

Vlowast119909119910)2

(119904119899

119909119910)minusV]

when V = 1

(36)


0(119909 119910 119905

0) is unknown


119909119910rarr 119904

0(119909 119910 119905

0) =

1199040(119909 119910 119905


0)119909119910

cong 1199040

119909119910minus 119904

119899

119909119910for






to 119904Vlowast119909119910

that is 119904119899119909119910

rarr 119904Vlowast119909119910

We consider that (119904119899119909119910

minus

119904Vlowast119909119910)2cong (119904

119899minus1

119909119910minus 119904

119899


(119904119899

119909119910minus 119904



119904119899+1

119909119910= 119875 (119904

119899

119909119910) Δ119905

V3 minus

120582119899Δ119905

V3


3)119904119899

119909119910

+ 119904119899

119909119910minus

2120583

Γ (3 minus V3)(119904

119899minus1

119909119910minus 119904

119899

119909119910)2

(119904119899

119909119910)minusV3

when V3= 1 2 and 3

(37)

120582119899=Γ (1 minus V

3) Γ (2 minus V

3)

1205901198992

119904119899119909119910

sum

119909119910

119875 (119904119899

119909119910) (119904

0

119909119910minus 119904

119899

119909119910)2

(38)

where prod119899

120591=1(V

2minus 120591 + 1)

119899=0

= 1 1205901198992

= sum119909119910

(1199040

119909119910minus 119904

119899

119909119910)2

119875(119904119899

119909119910) = (minusΓ(1 minus V

1)Γ(minusV

1)Γ(minusV

3)) sum

1

119896=0(prod

2119896

120591=1(V

2minus

120591+1)(2119896))[1198631

119909(|119863

V1119904

119899

119909119910|V2minus2119896minus2

119863V1

119909119904119899

119909119910)+119863

1

119910(|119863

V1119904

119899

119909119910|V2minus2119896minus2

sdot119863V1

119910119904119899

119909119910)]




2


1119904119899

119909119910| = 0 in


119899

119909119910| = 00689

when |119863V1119904

119899



= 000001 when 119904119899

119909119910= 0



119909= (119904

119899+1

119909minus1+2119904

119899+1

119909+119904

119899+1

119909+1)4when |119863V

1119904119899

119909| ≺ 120572

119860and120572

119873119860≺ 120572

119860


119909=

119904119899+1

119909 where 120572

119860= (1119873

119909) sum

119873119909

119909|119863

V1119904

119899

119909| 120572

119873119860= (|119863

V1119904

119899

119909minus1| +

2|119863V1119904

119899

119909| + |119863

V1119904

119899



119909= (119904

119899+1

119909minus1+ 2119904

119899+1

119909+ 119904

119899+1



119909119910= (

119909119904119899+1

119909119910+

119910119904119899+1

119909119910+

119903119904119899+1

119909119910+

119897119904119899+1

119909119910)4 When 120572

119909

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119909

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910)4 and 119910

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119910

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119910

119904119899+1

119909119910= (119904

119899+1

119909119910minus1+ 2119904

119899+1

119909119910+

119904119899+1

119909119910+1)4 and 119909

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119903

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119903

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+1+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910minus1)4 and 119909

119904119899+1

119909119910=

119910119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 and in the


119909119910=

119910119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910=

119904119899+1

119909119910 where 120572119909

119873119860= (|119863

V1119904

119899

119909minus1119910| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910|)4

120572119910

119873119860= (|119863

V1119904

119899

119909119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909119910+1|)4 and

120572119903

119873119860= (|119863

V1119904

119899

119909minus1119910+1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910minus1|)4 120572119897

119873119860=

(|119863V1119904

119899

119909minus1119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910+1|)4 Here 119909 denotes



151

05


0

0 1000 2000 3000 4000 5000 6000


(a)

151

05

minus05minus1

0

0 1000 2000 3000 4000 5000 6000


(b)

151

05


0

0 1000 2000 3000 4000 5000 6000


(c)

151

05


0

0 1000 2000 3000 4000 5000 6000


(d)

151

05


0

0 1000 2000 3000 4000 5000 6000


(e)

151

05


0

0 1000 2000 3000 4000 5000 6000


(f)

151

05


0

0 1000 2000 3000 4000 5000 6000


(g)

151

05


0

0 1000 2000 3000 4000 5000 6000


(h)

151

05


0

0 1000 2000 3000 4000 5000 6000


(i)


1= 1025 V

2= 225 V

3= 105 Δ119905 = 00296)


119909= 119904

119899

119909when |119904119899+1

119909| ≻ 6|119904

119899



119909119910= 119904

119899

119909119910when |119904

119899+1

119909119910| ≻ 6|119904

119899

119909119910| for two-

dimensional signal


















(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)


1= 175 V

2= 225 V

3= 105 Δ119905 = 10


1= 225 V

2= 25 V

3= 125 Δ119905 = 10











(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909119868V2

119910)Ω119863

V1119906 ∙ = 0 (14) becomes

(119868V2

119909119868V2

119910)Ω119863

V1119906 ∙

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [(119863

V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910]

= 0


V1minus119899

119909119906)119863

119899

119909120593

119909+ (119863

V1minus119899

119910119906)119863

119899

119910120593

119910= (119863

V1minus119899

119909119906119863

V1minus119899

119910119906) ∙ (119863

119899

119909120593

119909

119863119899

119910120593



119909119906 and 119863

V1minus119899



119909120593

119909 119863

119899

119910120593

119910) = (0 0) is required


1

119909120593

119909

1198631

119910120593

119910) = (0 0) to make (119863

119899

119909120593

119909 119863

119899

119910120593

119910) = (0 0) estab-


V2

119909119868V2

119910)Ω(V1

1) [(119863

V1minus1

119909119906)119863

1

119909120593

119909+ (119863

V1minus1

119910119906)119863

1

119910120593

119910] =

minus(119868V2

119909119868V2

119910)Ω(V1

1) (119863

V1minus1

119909119906119863

V1minus1

119910119906) ∙ (119863

1

119909120593

119909 119863

1

119910120593

119910) = 0


minus(V1

1) [119863

1

119909120593

119909+ 119863

1

119910120593

119910] =

Γ (1 minus V1)

Γ (minusV1)[119863

1

119909120593

119909+ 119863

1

119910120593

119910] = 0


119909119868V2

119910)Ω119863

V1119906 ∙ = 0







119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙ =


(119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙

= (119868V2

119909119868V2

119910)ΩΦ

1(119863

V1119906)119863

V1119906 ∙ (Φ

2() )

= minus(119868V2

119909119868V2

119910)Ω

infin

sum

119899=1

(V1

119899) [ (Φ

1(119863

V1119906)119863

V1minus119899

119909119906)

times 119863119899

119909(Φ

2() 120593

119909)

+ (Φ1(119863

V1119906)119863

V1minus119899

119910119906)

times119863119899

119910(Φ

2() 120593

119910)] = 0

(19)


minus (V1

1) [119863

1

119909(Φ

2() 120593

119909) + 119863

1

119910(Φ

2() 120593

119910)]

=Γ (1 minus V

1)

Γ (minusV1)[119863

1

119909(Φ

2() 120593

119909)

+1198631

119910(Φ

2() 120593

119910)] = 0

(20)


119909119868V2

119910)ΩΦ

1(119863

V1119906)Φ

2()119863

V1119906 ∙ = 0




(119868V2

119909119868V2


integral (119868V119909119868V119910)Ω







0(119909 119910) + 119899(119909 119910)


0(119909 119910) + 119899(119909 119910) we


119904 to be |119863V1119904|

V2= (radic(119863

V1

119909 119904)2

+ (119863V1

119910 119904)2

)

V2

where V2is any


V2 is


119864FTV (119904) = (1198681

1199091198681

119910)Ω[119891 (

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

)] = ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

) 119889119909 119889119910

(21)






and Ψ2(120573) = ∬

Ω120582[119904 + (120573 minus 1)120585 minus 119904

0]119904

0119889119909 119889119910 where Ψ

2(120573)











Ψ (120573) = Ψ1(120573) + Ψ

2(120573)

= (1198681

1199091198681

119910)Ω[119891 (

1003816100381610038161003816119863V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+ 120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0]


= ∬Ω

[119891 (1003816100381610038161003816119863

V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0] 119889119909 119889119910

(22)



119891(|119863V1119904 + (120573 minus 1)119863

V1120585|

V2) exists Ψ

1(120573) has the V

3-order


119863V3

120573Ψ

1(120573)

100381610038161003816100381610038161003816120573=1

=120597V3

120597120573V3∬

Ω

119891 (10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= 0

(23)

where it has 2= ||

2= [radic()

2]

2


V1119904+(120573minus1)119863








119863V120573minus119886

119891 [119892 (120573)] =(120573 minus 119886)

minusV

Γ (1 minus V)119891

+

infin

sum

119899=1

(V119899)

(120573 minus 119886)119899minusV

Γ (119899 minus V + 1)119899

times

119899

sum

119898=1

119863119898

119892119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573minus119886119892

119896

]

]

119875119896

(24)



119896satisfies

119899

sum

119896=1

119896119875119896= 119899

119899

sum

119896=1

119875119896= 119898 (25)


119899

119896=1(1119875

119896)[119863

119896

120573minus119886119892119896]

119875119896

|119898=1rarr119899


119875119896|119898=1rarr119899


∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120573minusV3119891 (

10038161003816100381610038161003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(V3

119899)

120573119899minusV3

Γ (119899 minus V3+ 1)

119899

times

119899

sum

119898=1

119863119898

| |V2119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)

119896

]

]

119875119896

119889119909 119889119910

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

sum

119898=1

119863119898

| |V2119891

100381610038161003816100381610038161003816120573=1

sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

times 119889119909119889119910 = 0

(26)


120578119891(120578) = 1 and119863119898

120578119891(120578)

119898ge2


∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

119898=1

119889119909 119889119910 = 0

(27)


119899= 1 and

1198751= 119875



∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119899119889119909 119889119910 = 0

(28)


120573(||

V2)|


119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896+1

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus119899minus1

times1003816100381610038161003816119863

V11205851003816100381610038161003816

119899minus1

(119863V1120585) ∙ (119863

V1119904)

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus1198991003816100381610038161003816119863

V11205851003816100381610038161003816

119899

(29)



∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus2119896minus21003816100381610038161003816119863

V11205851003816100381610038161003816

2119896

Γ (minusV3) (2119896)

(119863V1119904)

∙ 119863V1 [

Γ (2119896 minus V3)

Γ (2119896 minus V3+ 1)

119904 minus(V

2minus 2119896) Γ (2119896 minus V

3+ 1)

(2119896 + 1) Γ (2119896 minus V3+ 2)

120585]

times 119889119909 119889119910 = 0

(30)


120591=1(V

2minus 120591 + 1)

119899=0




3+1))119904 minus ((V

2minus

2119896)Γ(2119896minusV3+1)(2119896+1)Γ(2119896minusV



Γ (1 minus V1)


3)

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904) + 119863

1

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)] = 0

(31)


120591=1(V

2minus 120591 + 1)

119899=0



119863V3

120573Ψ

2(120573)

100381610038161003816100381610038161003816120573=1

= 119863V3

120573(119868

1

1199091198681

119910)Ω120582 [119904 + (120573 minus 1) 120585 minus 119904

0] 119904

0

100381610038161003816100381610038161003816120573=1

= ∬Ω

1205821199040


3)



3) 120585] 119889119909 119889119910 = 0

(32)


0minus



1205821199040


3)= 0 (33)




3= 1


by

120597V3119904

120597119905V3=



3)

times

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

minus120582119904

0


3)

(34)


120591=1(V

2minus 120591 + 1)

119899=0


Ω119899(119909 119910)119889119909 119889119910 = ∬

Ω(119904minus119904

0)119889119909 119889119910 = 0When 120597V3119904120597119905V3 = 0



120582 (119905) =minusΓ (1 minus V

1) Γ (1 minus V

3) Γ (2 minus V

3)


3) 119904

0

times∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

times (119904 minus 1199040)2

119889119909 119889119910

(35)



120591=1(V

2minus120591+1)




3) in the


3= 0 the












Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(a)


Cs119896Cs1

Cs0Csminus1

Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(b)

Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1

Cs119899




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(c)


Cs119896Cs1Cs0

Csminus1Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(d)







0

0

0

0 0

0

00

0 0

0

0

0

0Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

(e)







0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(f)



middotmiddotmiddot





0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middotmiddotmiddot



middotmiddotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(g)


middotmiddotmiddot

middotmiddotmiddot






0

0

0

00

0

0 0

0 0

0

0

0

0 Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Cs119899minus2

Cs119899minus1

(h)










119866-119871 cong

(119909minusV119873


119896=0(Γ(119896 minus V)Γ(119896 + 1))119904(119909 + (V1199092119873) minus








119904minus1

= (V4)+(V28)1198621199040

=

1 minus (V22) minus (V38) 1198621199041

= minus(5V4) + (5V316) + (V416) 119862

119904119896


119904119899minus2



(minus(V4) + (V28))] 119862119904119899minus1



119904119899


= 1198941119863

V119909minus

+ 1198942119863

V119910minus

+

1198943119863

V119909+

+ 1198944119863

V119910+

+ 1198945119863

Vldd + 119894

6119863

Vrud + 119894

7119863

Vlud + 119894

8119863

Vrdd in (34)


119910minus 119863V

119909+ 119863V

119910+ 119863V

ldd 119863Vrud 119863

Vlud




1

119909119904(119909 119910) = (2[119904(119909 + 1 119910) minus 119904(119909 minus 1 119910)] + 119904(119909 + 1 119910 +


1



119899= 119899Δ119905



= 119904(119909 119910 119905119899) Thus we have

1199040

119909119910= 119904

0(119909 119910 119905

0) + 119899(119909 119910 119905

0) where 119904

0



that is 1199040(119909 119910 119905

0) = 119904

0(119909 119910 119905



120597V119904

120597119905V= Δ119905

minusV[119904

119899+1

119909119910minus 119904

119899

119909119910+

2120583120578

Γ (3 minus V)(119904

119899

119909119910minus 119904

Vlowast119909119910)2

(119904119899

119909119910)minusV]

when V = 1

(36)


0(119909 119910 119905

0) is unknown


119909119910rarr 119904

0(119909 119910 119905

0) =

1199040(119909 119910 119905


0)119909119910

cong 1199040

119909119910minus 119904

119899

119909119910for






to 119904Vlowast119909119910

that is 119904119899119909119910

rarr 119904Vlowast119909119910

We consider that (119904119899119909119910

minus

119904Vlowast119909119910)2cong (119904

119899minus1

119909119910minus 119904

119899


(119904119899

119909119910minus 119904



119904119899+1

119909119910= 119875 (119904

119899

119909119910) Δ119905

V3 minus

120582119899Δ119905

V3


3)119904119899

119909119910

+ 119904119899

119909119910minus

2120583

Γ (3 minus V3)(119904

119899minus1

119909119910minus 119904

119899

119909119910)2

(119904119899

119909119910)minusV3

when V3= 1 2 and 3

(37)

120582119899=Γ (1 minus V

3) Γ (2 minus V

3)

1205901198992

119904119899119909119910

sum

119909119910

119875 (119904119899

119909119910) (119904

0

119909119910minus 119904

119899

119909119910)2

(38)

where prod119899

120591=1(V

2minus 120591 + 1)

119899=0

= 1 1205901198992

= sum119909119910

(1199040

119909119910minus 119904

119899

119909119910)2

119875(119904119899

119909119910) = (minusΓ(1 minus V

1)Γ(minusV

1)Γ(minusV

3)) sum

1

119896=0(prod

2119896

120591=1(V

2minus

120591+1)(2119896))[1198631

119909(|119863

V1119904

119899

119909119910|V2minus2119896minus2

119863V1

119909119904119899

119909119910)+119863

1

119910(|119863

V1119904

119899

119909119910|V2minus2119896minus2

sdot119863V1

119910119904119899

119909119910)]




2


1119904119899

119909119910| = 0 in


119899

119909119910| = 00689

when |119863V1119904

119899



= 000001 when 119904119899

119909119910= 0



119909= (119904

119899+1

119909minus1+2119904

119899+1

119909+119904

119899+1

119909+1)4when |119863V

1119904119899

119909| ≺ 120572

119860and120572

119873119860≺ 120572

119860


119909=

119904119899+1

119909 where 120572

119860= (1119873

119909) sum

119873119909

119909|119863

V1119904

119899

119909| 120572

119873119860= (|119863

V1119904

119899

119909minus1| +

2|119863V1119904

119899

119909| + |119863

V1119904

119899



119909= (119904

119899+1

119909minus1+ 2119904

119899+1

119909+ 119904

119899+1



119909119910= (

119909119904119899+1

119909119910+

119910119904119899+1

119909119910+

119903119904119899+1

119909119910+

119897119904119899+1

119909119910)4 When 120572

119909

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119909

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910)4 and 119910

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119910

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119910

119904119899+1

119909119910= (119904

119899+1

119909119910minus1+ 2119904

119899+1

119909119910+

119904119899+1

119909119910+1)4 and 119909

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119903

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119903

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+1+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910minus1)4 and 119909

119904119899+1

119909119910=

119910119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 and in the


119909119910=

119910119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910=

119904119899+1

119909119910 where 120572119909

119873119860= (|119863

V1119904

119899

119909minus1119910| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910|)4

120572119910

119873119860= (|119863

V1119904

119899

119909119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909119910+1|)4 and

120572119903

119873119860= (|119863

V1119904

119899

119909minus1119910+1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910minus1|)4 120572119897

119873119860=

(|119863V1119904

119899

119909minus1119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910+1|)4 Here 119909 denotes



151

05


0

0 1000 2000 3000 4000 5000 6000


(a)

151

05

minus05minus1

0

0 1000 2000 3000 4000 5000 6000


(b)

151

05


0

0 1000 2000 3000 4000 5000 6000


(c)

151

05


0

0 1000 2000 3000 4000 5000 6000


(d)

151

05


0

0 1000 2000 3000 4000 5000 6000


(e)

151

05


0

0 1000 2000 3000 4000 5000 6000


(f)

151

05


0

0 1000 2000 3000 4000 5000 6000


(g)

151

05


0

0 1000 2000 3000 4000 5000 6000


(h)

151

05


0

0 1000 2000 3000 4000 5000 6000


(i)


1= 1025 V

2= 225 V

3= 105 Δ119905 = 00296)


119909= 119904

119899

119909when |119904119899+1

119909| ≻ 6|119904

119899



119909119910= 119904

119899

119909119910when |119904

119899+1

119909119910| ≻ 6|119904

119899

119909119910| for two-

dimensional signal


















(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)


1= 175 V

2= 225 V

3= 105 Δ119905 = 10


1= 225 V

2= 25 V

3= 125 Δ119905 = 10











(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= ∬Ω

[119891 (1003816100381610038161003816119863

V1119904 + (120573 minus 1)119863

V11205851003816100381610038161003816

V2

)

+120582 [119904 + (120573 minus 1) 120585 minus 1199040] 119904

0] 119889119909 119889119910

(22)



119891(|119863V1119904 + (120573 minus 1)119863

V1120585|

V2) exists Ψ

1(120573) has the V

3-order


119863V3

120573Ψ

1(120573)

100381610038161003816100381610038161003816120573=1

=120597V3

120597120573V3∬

Ω

119891 (10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= 0

(23)

where it has 2= ||

2= [radic()

2]

2


V1119904+(120573minus1)119863








119863V120573minus119886

119891 [119892 (120573)] =(120573 minus 119886)

minusV

Γ (1 minus V)119891

+

infin

sum

119899=1

(V119899)

(120573 minus 119886)119899minusV

Γ (119899 minus V + 1)119899

times

119899

sum

119898=1

119863119898

119892119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573minus119886119892

119896

]

]

119875119896

(24)



119896satisfies

119899

sum

119896=1

119896119875119896= 119899

119899

sum

119896=1

119875119896= 119898 (25)


119899

119896=1(1119875

119896)[119863

119896

120573minus119886119892119896]

119875119896

|119898=1rarr119899


119875119896|119898=1rarr119899


∬Ω

120597V3

120597120573V3119891 (

10038161003816100381610038161003816100381610038161003816

V2

) 119889119909 119889119910

10038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

120573minusV3119891 (

10038161003816100381610038161003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(V3

119899)

120573119899minusV3

Γ (119899 minus V3+ 1)

119899

times

119899

sum

119898=1

119863119898

| |V2119891sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)

119896

]

]

119875119896

119889119909 119889119910

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816120573=1

= ∬Ω

119891 (1003816100381610038161003816119863

V11199041003816100381610038161003816

V2

)

Γ (1 minus V3)

+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

sum

119898=1

119863119898

| |V2119891

100381610038161003816100381610038161003816120573=1

sum

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

times 119889119909119889119910 = 0

(26)


120578119891(120578) = 1 and119863119898

120578119891(120578)

119898ge2


∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119899

prod

119896=1

1

119875119896

[

[

119863119896

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119896

]

]

119875119896

119898=1

119889119909 119889119910 = 0

(27)


119899= 1 and

1198751= 119875



∬Ω

1003816100381610038161003816119863V11199041003816100381610038161003816

V2

Γ (1 minus V3)+

infin

sum

119899=1

(minus1)119899Γ (119899 minus V

3)


3+ 1)

times

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816120573=1

119899119889119909 119889119910 = 0

(28)


120573(||

V2)|


119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896+1

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus119899minus1

times1003816100381610038161003816119863

V11205851003816100381610038161003816

119899minus1

(119863V1120585) ∙ (119863

V1119904)

119863119899

120573(10038161003816100381610038161003816100381610038161003816

V2

)10038161003816100381610038161003816

119899=2119896

120573=1=

119899

prod

120591=1

(V2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus1198991003816100381610038161003816119863

V11205851003816100381610038161003816

119899

(29)



∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus2119896minus21003816100381610038161003816119863

V11205851003816100381610038161003816

2119896

Γ (minusV3) (2119896)

(119863V1119904)

∙ 119863V1 [

Γ (2119896 minus V3)

Γ (2119896 minus V3+ 1)

119904 minus(V

2minus 2119896) Γ (2119896 minus V

3+ 1)

(2119896 + 1) Γ (2119896 minus V3+ 2)

120585]

times 119889119909 119889119910 = 0

(30)


120591=1(V

2minus 120591 + 1)

119899=0




3+1))119904 minus ((V

2minus

2119896)Γ(2119896minusV3+1)(2119896+1)Γ(2119896minusV



Γ (1 minus V1)


3)

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904) + 119863

1

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)] = 0

(31)


120591=1(V

2minus 120591 + 1)

119899=0



119863V3

120573Ψ

2(120573)

100381610038161003816100381610038161003816120573=1

= 119863V3

120573(119868

1

1199091198681

119910)Ω120582 [119904 + (120573 minus 1) 120585 minus 119904

0] 119904

0

100381610038161003816100381610038161003816120573=1

= ∬Ω

1205821199040


3)



3) 120585] 119889119909 119889119910 = 0

(32)


0minus



1205821199040


3)= 0 (33)




3= 1


by

120597V3119904

120597119905V3=



3)

times

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

minus120582119904

0


3)

(34)


120591=1(V

2minus 120591 + 1)

119899=0


Ω119899(119909 119910)119889119909 119889119910 = ∬

Ω(119904minus119904

0)119889119909 119889119910 = 0When 120597V3119904120597119905V3 = 0



120582 (119905) =minusΓ (1 minus V

1) Γ (1 minus V

3) Γ (2 minus V

3)


3) 119904

0

times∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

times (119904 minus 1199040)2

119889119909 119889119910

(35)



120591=1(V

2minus120591+1)




3) in the


3= 0 the












Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(a)


Cs119896Cs1

Cs0Csminus1

Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(b)

Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1

Cs119899




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(c)


Cs119896Cs1Cs0

Csminus1Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(d)







0

0

0

0 0

0

00

0 0

0

0

0

0Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

(e)







0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(f)



middotmiddotmiddot





0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middotmiddotmiddot



middotmiddotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(g)


middotmiddotmiddot

middotmiddotmiddot






0

0

0

00

0

0 0

0 0

0

0

0

0 Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Cs119899minus2

Cs119899minus1

(h)










119866-119871 cong

(119909minusV119873


119896=0(Γ(119896 minus V)Γ(119896 + 1))119904(119909 + (V1199092119873) minus








119904minus1

= (V4)+(V28)1198621199040

=

1 minus (V22) minus (V38) 1198621199041

= minus(5V4) + (5V316) + (V416) 119862

119904119896


119904119899minus2



(minus(V4) + (V28))] 119862119904119899minus1



119904119899


= 1198941119863

V119909minus

+ 1198942119863

V119910minus

+

1198943119863

V119909+

+ 1198944119863

V119910+

+ 1198945119863

Vldd + 119894

6119863

Vrud + 119894

7119863

Vlud + 119894

8119863

Vrdd in (34)


119910minus 119863V

119909+ 119863V

119910+ 119863V

ldd 119863Vrud 119863

Vlud




1

119909119904(119909 119910) = (2[119904(119909 + 1 119910) minus 119904(119909 minus 1 119910)] + 119904(119909 + 1 119910 +


1



119899= 119899Δ119905



= 119904(119909 119910 119905119899) Thus we have

1199040

119909119910= 119904

0(119909 119910 119905

0) + 119899(119909 119910 119905

0) where 119904

0



that is 1199040(119909 119910 119905

0) = 119904

0(119909 119910 119905



120597V119904

120597119905V= Δ119905

minusV[119904

119899+1

119909119910minus 119904

119899

119909119910+

2120583120578

Γ (3 minus V)(119904

119899

119909119910minus 119904

Vlowast119909119910)2

(119904119899

119909119910)minusV]

when V = 1

(36)


0(119909 119910 119905

0) is unknown


119909119910rarr 119904

0(119909 119910 119905

0) =

1199040(119909 119910 119905


0)119909119910

cong 1199040

119909119910minus 119904

119899

119909119910for






to 119904Vlowast119909119910

that is 119904119899119909119910

rarr 119904Vlowast119909119910

We consider that (119904119899119909119910

minus

119904Vlowast119909119910)2cong (119904

119899minus1

119909119910minus 119904

119899


(119904119899

119909119910minus 119904



119904119899+1

119909119910= 119875 (119904

119899

119909119910) Δ119905

V3 minus

120582119899Δ119905

V3


3)119904119899

119909119910

+ 119904119899

119909119910minus

2120583

Γ (3 minus V3)(119904

119899minus1

119909119910minus 119904

119899

119909119910)2

(119904119899

119909119910)minusV3

when V3= 1 2 and 3

(37)

120582119899=Γ (1 minus V

3) Γ (2 minus V

3)

1205901198992

119904119899119909119910

sum

119909119910

119875 (119904119899

119909119910) (119904

0

119909119910minus 119904

119899

119909119910)2

(38)

where prod119899

120591=1(V

2minus 120591 + 1)

119899=0

= 1 1205901198992

= sum119909119910

(1199040

119909119910minus 119904

119899

119909119910)2

119875(119904119899

119909119910) = (minusΓ(1 minus V

1)Γ(minusV

1)Γ(minusV

3)) sum

1

119896=0(prod

2119896

120591=1(V

2minus

120591+1)(2119896))[1198631

119909(|119863

V1119904

119899

119909119910|V2minus2119896minus2

119863V1

119909119904119899

119909119910)+119863

1

119910(|119863

V1119904

119899

119909119910|V2minus2119896minus2

sdot119863V1

119910119904119899

119909119910)]




2


1119904119899

119909119910| = 0 in


119899

119909119910| = 00689

when |119863V1119904

119899



= 000001 when 119904119899

119909119910= 0



119909= (119904

119899+1

119909minus1+2119904

119899+1

119909+119904

119899+1

119909+1)4when |119863V

1119904119899

119909| ≺ 120572

119860and120572

119873119860≺ 120572

119860


119909=

119904119899+1

119909 where 120572

119860= (1119873

119909) sum

119873119909

119909|119863

V1119904

119899

119909| 120572

119873119860= (|119863

V1119904

119899

119909minus1| +

2|119863V1119904

119899

119909| + |119863

V1119904

119899



119909= (119904

119899+1

119909minus1+ 2119904

119899+1

119909+ 119904

119899+1



119909119910= (

119909119904119899+1

119909119910+

119910119904119899+1

119909119910+

119903119904119899+1

119909119910+

119897119904119899+1

119909119910)4 When 120572

119909

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119909

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910)4 and 119910

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119910

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119910

119904119899+1

119909119910= (119904

119899+1

119909119910minus1+ 2119904

119899+1

119909119910+

119904119899+1

119909119910+1)4 and 119909

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119903

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119903

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+1+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910minus1)4 and 119909

119904119899+1

119909119910=

119910119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 and in the


119909119910=

119910119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910=

119904119899+1

119909119910 where 120572119909

119873119860= (|119863

V1119904

119899

119909minus1119910| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910|)4

120572119910

119873119860= (|119863

V1119904

119899

119909119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909119910+1|)4 and

120572119903

119873119860= (|119863

V1119904

119899

119909minus1119910+1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910minus1|)4 120572119897

119873119860=

(|119863V1119904

119899

119909minus1119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910+1|)4 Here 119909 denotes



151

05


0

0 1000 2000 3000 4000 5000 6000


(a)

151

05

minus05minus1

0

0 1000 2000 3000 4000 5000 6000


(b)

151

05


0

0 1000 2000 3000 4000 5000 6000


(c)

151

05


0

0 1000 2000 3000 4000 5000 6000


(d)

151

05


0

0 1000 2000 3000 4000 5000 6000


(e)

151

05


0

0 1000 2000 3000 4000 5000 6000


(f)

151

05


0

0 1000 2000 3000 4000 5000 6000


(g)

151

05


0

0 1000 2000 3000 4000 5000 6000


(h)

151

05


0

0 1000 2000 3000 4000 5000 6000


(i)


1= 1025 V

2= 225 V

3= 105 Δ119905 = 00296)


119909= 119904

119899

119909when |119904119899+1

119909| ≻ 6|119904

119899



119909119910= 119904

119899

119909119910when |119904

119899+1

119909119910| ≻ 6|119904

119899

119909119910| for two-

dimensional signal


















(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)


1= 175 V

2= 225 V

3= 105 Δ119905 = 10


1= 225 V

2= 25 V

3= 125 Δ119905 = 10











(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

1003816100381610038161003816119863V11199041003816100381610038161003816

V2minus2119896minus21003816100381610038161003816119863

V11205851003816100381610038161003816

2119896

Γ (minusV3) (2119896)

(119863V1119904)

∙ 119863V1 [

Γ (2119896 minus V3)

Γ (2119896 minus V3+ 1)

119904 minus(V

2minus 2119896) Γ (2119896 minus V

3+ 1)

(2119896 + 1) Γ (2119896 minus V3+ 2)

120585]

times 119889119909 119889119910 = 0

(30)


120591=1(V

2minus 120591 + 1)

119899=0




3+1))119904 minus ((V

2minus

2119896)Γ(2119896minusV3+1)(2119896+1)Γ(2119896minusV



Γ (1 minus V1)


3)

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904) + 119863

1

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)] = 0

(31)


120591=1(V

2minus 120591 + 1)

119899=0



119863V3

120573Ψ

2(120573)

100381610038161003816100381610038161003816120573=1

= 119863V3

120573(119868

1

1199091198681

119910)Ω120582 [119904 + (120573 minus 1) 120585 minus 119904

0] 119904

0

100381610038161003816100381610038161003816120573=1

= ∬Ω

1205821199040


3)



3) 120585] 119889119909 119889119910 = 0

(32)


0minus



1205821199040


3)= 0 (33)




3= 1


by

120597V3119904

120597119905V3=



3)

times

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

minus120582119904

0


3)

(34)


120591=1(V

2minus 120591 + 1)

119899=0


Ω119899(119909 119910)119889119909 119889119910 = ∬

Ω(119904minus119904

0)119889119909 119889119910 = 0When 120597V3119904120597119905V3 = 0



120582 (119905) =minusΓ (1 minus V

1) Γ (1 minus V

3) Γ (2 minus V

3)


3) 119904

0

times∬Ω

infin

sum

119896=0

prod2119896

120591=1(V

2minus 120591 + 1)

(2119896)

times [1198631

119909(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119909119904)

+1198631

119910(1003816100381610038161003816119863

V11199041003816100381610038161003816


119863V1

119910119904)]

times (119904 minus 1199040)2

119889119909 119889119910

(35)



120591=1(V

2minus120591+1)




3) in the


3= 0 the












Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(a)


Cs119896Cs1

Cs0Csminus1

Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(b)

Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1

Cs119899




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(c)


Cs119896Cs1Cs0

Csminus1Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(d)







0

0

0

0 0

0

00

0 0

0

0

0

0Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

(e)







0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(f)



middotmiddotmiddot





0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middotmiddotmiddot



middotmiddotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(g)


middotmiddotmiddot

middotmiddotmiddot






0

0

0

00

0

0 0

0 0

0

0

0

0 Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Cs119899minus2

Cs119899minus1

(h)










119866-119871 cong

(119909minusV119873


119896=0(Γ(119896 minus V)Γ(119896 + 1))119904(119909 + (V1199092119873) minus








119904minus1

= (V4)+(V28)1198621199040

=

1 minus (V22) minus (V38) 1198621199041

= minus(5V4) + (5V316) + (V416) 119862

119904119896


119904119899minus2



(minus(V4) + (V28))] 119862119904119899minus1



119904119899


= 1198941119863

V119909minus

+ 1198942119863

V119910minus

+

1198943119863

V119909+

+ 1198944119863

V119910+

+ 1198945119863

Vldd + 119894

6119863

Vrud + 119894

7119863

Vlud + 119894

8119863

Vrdd in (34)


119910minus 119863V

119909+ 119863V

119910+ 119863V

ldd 119863Vrud 119863

Vlud




1

119909119904(119909 119910) = (2[119904(119909 + 1 119910) minus 119904(119909 minus 1 119910)] + 119904(119909 + 1 119910 +


1



119899= 119899Δ119905



= 119904(119909 119910 119905119899) Thus we have

1199040

119909119910= 119904

0(119909 119910 119905

0) + 119899(119909 119910 119905

0) where 119904

0



that is 1199040(119909 119910 119905

0) = 119904

0(119909 119910 119905



120597V119904

120597119905V= Δ119905

minusV[119904

119899+1

119909119910minus 119904

119899

119909119910+

2120583120578

Γ (3 minus V)(119904

119899

119909119910minus 119904

Vlowast119909119910)2

(119904119899

119909119910)minusV]

when V = 1

(36)


0(119909 119910 119905

0) is unknown


119909119910rarr 119904

0(119909 119910 119905

0) =

1199040(119909 119910 119905


0)119909119910

cong 1199040

119909119910minus 119904

119899

119909119910for






to 119904Vlowast119909119910

that is 119904119899119909119910

rarr 119904Vlowast119909119910

We consider that (119904119899119909119910

minus

119904Vlowast119909119910)2cong (119904

119899minus1

119909119910minus 119904

119899


(119904119899

119909119910minus 119904



119904119899+1

119909119910= 119875 (119904

119899

119909119910) Δ119905

V3 minus

120582119899Δ119905

V3


3)119904119899

119909119910

+ 119904119899

119909119910minus

2120583

Γ (3 minus V3)(119904

119899minus1

119909119910minus 119904

119899

119909119910)2

(119904119899

119909119910)minusV3

when V3= 1 2 and 3

(37)

120582119899=Γ (1 minus V

3) Γ (2 minus V

3)

1205901198992

119904119899119909119910

sum

119909119910

119875 (119904119899

119909119910) (119904

0

119909119910minus 119904

119899

119909119910)2

(38)

where prod119899

120591=1(V

2minus 120591 + 1)

119899=0

= 1 1205901198992

= sum119909119910

(1199040

119909119910minus 119904

119899

119909119910)2

119875(119904119899

119909119910) = (minusΓ(1 minus V

1)Γ(minusV

1)Γ(minusV

3)) sum

1

119896=0(prod

2119896

120591=1(V

2minus

120591+1)(2119896))[1198631

119909(|119863

V1119904

119899

119909119910|V2minus2119896minus2

119863V1

119909119904119899

119909119910)+119863

1

119910(|119863

V1119904

119899

119909119910|V2minus2119896minus2

sdot119863V1

119910119904119899

119909119910)]




2


1119904119899

119909119910| = 0 in


119899

119909119910| = 00689

when |119863V1119904

119899



= 000001 when 119904119899

119909119910= 0



119909= (119904

119899+1

119909minus1+2119904

119899+1

119909+119904

119899+1

119909+1)4when |119863V

1119904119899

119909| ≺ 120572

119860and120572

119873119860≺ 120572

119860


119909=

119904119899+1

119909 where 120572

119860= (1119873

119909) sum

119873119909

119909|119863

V1119904

119899

119909| 120572

119873119860= (|119863

V1119904

119899

119909minus1| +

2|119863V1119904

119899

119909| + |119863

V1119904

119899



119909= (119904

119899+1

119909minus1+ 2119904

119899+1

119909+ 119904

119899+1



119909119910= (

119909119904119899+1

119909119910+

119910119904119899+1

119909119910+

119903119904119899+1

119909119910+

119897119904119899+1

119909119910)4 When 120572

119909

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119909

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910)4 and 119910

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119910

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119910

119904119899+1

119909119910= (119904

119899+1

119909119910minus1+ 2119904

119899+1

119909119910+

119904119899+1

119909119910+1)4 and 119909

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119903

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119903

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+1+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910minus1)4 and 119909

119904119899+1

119909119910=

119910119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 and in the


119909119910=

119910119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910=

119904119899+1

119909119910 where 120572119909

119873119860= (|119863

V1119904

119899

119909minus1119910| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910|)4

120572119910

119873119860= (|119863

V1119904

119899

119909119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909119910+1|)4 and

120572119903

119873119860= (|119863

V1119904

119899

119909minus1119910+1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910minus1|)4 120572119897

119873119860=

(|119863V1119904

119899

119909minus1119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910+1|)4 Here 119909 denotes



151

05


0

0 1000 2000 3000 4000 5000 6000


(a)

151

05

minus05minus1

0

0 1000 2000 3000 4000 5000 6000


(b)

151

05


0

0 1000 2000 3000 4000 5000 6000


(c)

151

05


0

0 1000 2000 3000 4000 5000 6000


(d)

151

05


0

0 1000 2000 3000 4000 5000 6000


(e)

151

05


0

0 1000 2000 3000 4000 5000 6000


(f)

151

05


0

0 1000 2000 3000 4000 5000 6000


(g)

151

05


0

0 1000 2000 3000 4000 5000 6000


(h)

151

05


0

0 1000 2000 3000 4000 5000 6000


(i)


1= 1025 V

2= 225 V

3= 105 Δ119905 = 00296)


119909= 119904

119899

119909when |119904119899+1

119909| ≻ 6|119904

119899



119909119910= 119904

119899

119909119910when |119904

119899+1

119909119910| ≻ 6|119904

119899

119909119910| for two-

dimensional signal


















(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)


1= 175 V

2= 225 V

3= 105 Δ119905 = 10


1= 225 V

2= 25 V

3= 125 Δ119905 = 10











(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(a)


Cs119896Cs1

Cs0Csminus1

Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(b)

Cs119899minus2

Cs119899minus1

Cs119896

Cs1

Cs0

Csminus1

Cs119899




middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot












0

0

0

0

0

0

0

middotmiddotmiddot

middotmiddotmiddot

0

0

0

0

0

0

0

(c)


Cs119896Cs1Cs0

Csminus1Cs119899

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


0 0 0 0 0 0 0

(d)







0

0

0

0 0

0

00

0 0

0

0

0

0Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

(e)







0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middot middotmiddot

middot middotmiddot

middot middotmiddot

middot middotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(f)



middotmiddotmiddot





0

0

0

0

0

00

0

0

0

0

0

0

Csminus1

Cs0

Csk

Cs1

middotmiddotmiddot



middotmiddotmiddot

Cs119899minus2

Cs119899minus1

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot


(g)


middotmiddotmiddot

middotmiddotmiddot






0

0

0

00

0

0 0

0 0

0

0

0

0 Cs0

Csk

Csn


middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Cs119899minus2

Cs119899minus1

(h)










119866-119871 cong

(119909minusV119873


119896=0(Γ(119896 minus V)Γ(119896 + 1))119904(119909 + (V1199092119873) minus








119904minus1

= (V4)+(V28)1198621199040

=

1 minus (V22) minus (V38) 1198621199041

= minus(5V4) + (5V316) + (V416) 119862

119904119896


119904119899minus2



(minus(V4) + (V28))] 119862119904119899minus1



119904119899


= 1198941119863

V119909minus

+ 1198942119863

V119910minus

+

1198943119863

V119909+

+ 1198944119863

V119910+

+ 1198945119863

Vldd + 119894

6119863

Vrud + 119894

7119863

Vlud + 119894

8119863

Vrdd in (34)


119910minus 119863V

119909+ 119863V

119910+ 119863V

ldd 119863Vrud 119863

Vlud




1

119909119904(119909 119910) = (2[119904(119909 + 1 119910) minus 119904(119909 minus 1 119910)] + 119904(119909 + 1 119910 +


1



119899= 119899Δ119905



= 119904(119909 119910 119905119899) Thus we have

1199040

119909119910= 119904

0(119909 119910 119905

0) + 119899(119909 119910 119905

0) where 119904

0



that is 1199040(119909 119910 119905

0) = 119904

0(119909 119910 119905



120597V119904

120597119905V= Δ119905

minusV[119904

119899+1

119909119910minus 119904

119899

119909119910+

2120583120578

Γ (3 minus V)(119904

119899

119909119910minus 119904

Vlowast119909119910)2

(119904119899

119909119910)minusV]

when V = 1

(36)


0(119909 119910 119905

0) is unknown


119909119910rarr 119904

0(119909 119910 119905

0) =

1199040(119909 119910 119905


0)119909119910

cong 1199040

119909119910minus 119904

119899

119909119910for






to 119904Vlowast119909119910

that is 119904119899119909119910

rarr 119904Vlowast119909119910

We consider that (119904119899119909119910

minus

119904Vlowast119909119910)2cong (119904

119899minus1

119909119910minus 119904

119899


(119904119899

119909119910minus 119904



119904119899+1

119909119910= 119875 (119904

119899

119909119910) Δ119905

V3 minus

120582119899Δ119905

V3


3)119904119899

119909119910

+ 119904119899

119909119910minus

2120583

Γ (3 minus V3)(119904

119899minus1

119909119910minus 119904

119899

119909119910)2

(119904119899

119909119910)minusV3

when V3= 1 2 and 3

(37)

120582119899=Γ (1 minus V

3) Γ (2 minus V

3)

1205901198992

119904119899119909119910

sum

119909119910

119875 (119904119899

119909119910) (119904

0

119909119910minus 119904

119899

119909119910)2

(38)

where prod119899

120591=1(V

2minus 120591 + 1)

119899=0

= 1 1205901198992

= sum119909119910

(1199040

119909119910minus 119904

119899

119909119910)2

119875(119904119899

119909119910) = (minusΓ(1 minus V

1)Γ(minusV

1)Γ(minusV

3)) sum

1

119896=0(prod

2119896

120591=1(V

2minus

120591+1)(2119896))[1198631

119909(|119863

V1119904

119899

119909119910|V2minus2119896minus2

119863V1

119909119904119899

119909119910)+119863

1

119910(|119863

V1119904

119899

119909119910|V2minus2119896minus2

sdot119863V1

119910119904119899

119909119910)]




2


1119904119899

119909119910| = 0 in


119899

119909119910| = 00689

when |119863V1119904

119899



= 000001 when 119904119899

119909119910= 0



119909= (119904

119899+1

119909minus1+2119904

119899+1

119909+119904

119899+1

119909+1)4when |119863V

1119904119899

119909| ≺ 120572

119860and120572

119873119860≺ 120572

119860


119909=

119904119899+1

119909 where 120572

119860= (1119873

119909) sum

119873119909

119909|119863

V1119904

119899

119909| 120572

119873119860= (|119863

V1119904

119899

119909minus1| +

2|119863V1119904

119899

119909| + |119863

V1119904

119899



119909= (119904

119899+1

119909minus1+ 2119904

119899+1

119909+ 119904

119899+1



119909119910= (

119909119904119899+1

119909119910+

119910119904119899+1

119909119910+

119903119904119899+1

119909119910+

119897119904119899+1

119909119910)4 When 120572

119909

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119909

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910)4 and 119910

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119910

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119910

119904119899+1

119909119910= (119904

119899+1

119909119910minus1+ 2119904

119899+1

119909119910+

119904119899+1

119909119910+1)4 and 119909

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119903

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119903

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+1+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910minus1)4 and 119909

119904119899+1

119909119910=

119910119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 and in the


119909119910=

119910119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910=

119904119899+1

119909119910 where 120572119909

119873119860= (|119863

V1119904

119899

119909minus1119910| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910|)4

120572119910

119873119860= (|119863

V1119904

119899

119909119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909119910+1|)4 and

120572119903

119873119860= (|119863

V1119904

119899

119909minus1119910+1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910minus1|)4 120572119897

119873119860=

(|119863V1119904

119899

119909minus1119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910+1|)4 Here 119909 denotes



151

05


0

0 1000 2000 3000 4000 5000 6000


(a)

151

05

minus05minus1

0

0 1000 2000 3000 4000 5000 6000


(b)

151

05


0

0 1000 2000 3000 4000 5000 6000


(c)

151

05


0

0 1000 2000 3000 4000 5000 6000


(d)

151

05


0

0 1000 2000 3000 4000 5000 6000


(e)

151

05


0

0 1000 2000 3000 4000 5000 6000


(f)

151

05


0

0 1000 2000 3000 4000 5000 6000


(g)

151

05


0

0 1000 2000 3000 4000 5000 6000


(h)

151

05


0

0 1000 2000 3000 4000 5000 6000


(i)


1= 1025 V

2= 225 V

3= 105 Δ119905 = 00296)


119909= 119904

119899

119909when |119904119899+1

119909| ≻ 6|119904

119899



119909119910= 119904

119899

119909119910when |119904

119899+1

119909119910| ≻ 6|119904

119899

119909119910| for two-

dimensional signal


















(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)


1= 175 V

2= 225 V

3= 105 Δ119905 = 10


1= 225 V

2= 25 V

3= 125 Δ119905 = 10











(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(minus(V4) + (V28))] 119862119904119899minus1



119904119899


= 1198941119863

V119909minus

+ 1198942119863

V119910minus

+

1198943119863

V119909+

+ 1198944119863

V119910+

+ 1198945119863

Vldd + 119894

6119863

Vrud + 119894

7119863

Vlud + 119894

8119863

Vrdd in (34)


119910minus 119863V

119909+ 119863V

119910+ 119863V

ldd 119863Vrud 119863

Vlud




1

119909119904(119909 119910) = (2[119904(119909 + 1 119910) minus 119904(119909 minus 1 119910)] + 119904(119909 + 1 119910 +


1



119899= 119899Δ119905



= 119904(119909 119910 119905119899) Thus we have

1199040

119909119910= 119904

0(119909 119910 119905

0) + 119899(119909 119910 119905

0) where 119904

0



that is 1199040(119909 119910 119905

0) = 119904

0(119909 119910 119905



120597V119904

120597119905V= Δ119905

minusV[119904

119899+1

119909119910minus 119904

119899

119909119910+

2120583120578

Γ (3 minus V)(119904

119899

119909119910minus 119904

Vlowast119909119910)2

(119904119899

119909119910)minusV]

when V = 1

(36)


0(119909 119910 119905

0) is unknown


119909119910rarr 119904

0(119909 119910 119905

0) =

1199040(119909 119910 119905


0)119909119910

cong 1199040

119909119910minus 119904

119899

119909119910for






to 119904Vlowast119909119910

that is 119904119899119909119910

rarr 119904Vlowast119909119910

We consider that (119904119899119909119910

minus

119904Vlowast119909119910)2cong (119904

119899minus1

119909119910minus 119904

119899


(119904119899

119909119910minus 119904



119904119899+1

119909119910= 119875 (119904

119899

119909119910) Δ119905

V3 minus

120582119899Δ119905

V3


3)119904119899

119909119910

+ 119904119899

119909119910minus

2120583

Γ (3 minus V3)(119904

119899minus1

119909119910minus 119904

119899

119909119910)2

(119904119899

119909119910)minusV3

when V3= 1 2 and 3

(37)

120582119899=Γ (1 minus V

3) Γ (2 minus V

3)

1205901198992

119904119899119909119910

sum

119909119910

119875 (119904119899

119909119910) (119904

0

119909119910minus 119904

119899

119909119910)2

(38)

where prod119899

120591=1(V

2minus 120591 + 1)

119899=0

= 1 1205901198992

= sum119909119910

(1199040

119909119910minus 119904

119899

119909119910)2

119875(119904119899

119909119910) = (minusΓ(1 minus V

1)Γ(minusV

1)Γ(minusV

3)) sum

1

119896=0(prod

2119896

120591=1(V

2minus

120591+1)(2119896))[1198631

119909(|119863

V1119904

119899

119909119910|V2minus2119896minus2

119863V1

119909119904119899

119909119910)+119863

1

119910(|119863

V1119904

119899

119909119910|V2minus2119896minus2

sdot119863V1

119910119904119899

119909119910)]




2


1119904119899

119909119910| = 0 in


119899

119909119910| = 00689

when |119863V1119904

119899



= 000001 when 119904119899

119909119910= 0



119909= (119904

119899+1

119909minus1+2119904

119899+1

119909+119904

119899+1

119909+1)4when |119863V

1119904119899

119909| ≺ 120572

119860and120572

119873119860≺ 120572

119860


119909=

119904119899+1

119909 where 120572

119860= (1119873

119909) sum

119873119909

119909|119863

V1119904

119899

119909| 120572

119873119860= (|119863

V1119904

119899

119909minus1| +

2|119863V1119904

119899

119909| + |119863

V1119904

119899



119909= (119904

119899+1

119909minus1+ 2119904

119899+1

119909+ 119904

119899+1



119909119910= (

119909119904119899+1

119909119910+

119910119904119899+1

119909119910+

119903119904119899+1

119909119910+

119897119904119899+1

119909119910)4 When 120572

119909

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119909

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910)4 and 119910

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119910

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119910

119904119899+1

119909119910= (119904

119899+1

119909119910minus1+ 2119904

119899+1

119909119910+

119904119899+1

119909119910+1)4 and 119909

119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 when 120572119903

119873119860=

min(120572119909

119873119860 120572

119910

119873119860 120572

119903

119873119860 120572

119897

119873119860) they are 119903

119904119899+1

119909119910= (119904

119899+1

119909minus1119910+1+ 2119904

119899+1

119909119910+

119904119899+1

119909+1119910minus1)4 and 119909

119904119899+1

119909119910=

119910119904119899+1

119909119910=

119897119904119899+1

119909119910= 119904

119899+1

119909119910 and in the


119909119910=

119910119904119899+1

119909119910=

119903119904119899+1

119909119910=

119897119904119899+1

119909119910=

119904119899+1

119909119910 where 120572119909

119873119860= (|119863

V1119904

119899

119909minus1119910| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910|)4

120572119910

119873119860= (|119863

V1119904

119899

119909119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909119910+1|)4 and

120572119903

119873119860= (|119863

V1119904

119899

119909minus1119910+1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910minus1|)4 120572119897

119873119860=

(|119863V1119904

119899

119909minus1119910minus1| + 2|119863

V1119904

119899

119909119910| + |119863

V1119904

119899

119909+1119910+1|)4 Here 119909 denotes



151

05


0

0 1000 2000 3000 4000 5000 6000


(a)

151

05

minus05minus1

0

0 1000 2000 3000 4000 5000 6000


(b)

151

05


0

0 1000 2000 3000 4000 5000 6000


(c)

151

05


0

0 1000 2000 3000 4000 5000 6000


(d)

151

05


0

0 1000 2000 3000 4000 5000 6000


(e)

151

05


0

0 1000 2000 3000 4000 5000 6000


(f)

151

05


0

0 1000 2000 3000 4000 5000 6000


(g)

151

05


0

0 1000 2000 3000 4000 5000 6000


(h)

151

05


0

0 1000 2000 3000 4000 5000 6000


(i)


1= 1025 V

2= 225 V

3= 105 Δ119905 = 00296)


119909= 119904

119899

119909when |119904119899+1

119909| ≻ 6|119904

119899



119909119910= 119904

119899

119909119910when |119904

119899+1

119909119910| ≻ 6|119904

119899

119909119910| for two-

dimensional signal


















(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)


1= 175 V

2= 225 V

3= 105 Δ119905 = 10


1= 225 V

2= 25 V

3= 125 Δ119905 = 10











(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










151

05


0

0 1000 2000 3000 4000 5000 6000


(a)

151

05

minus05minus1

0

0 1000 2000 3000 4000 5000 6000


(b)

151

05


0

0 1000 2000 3000 4000 5000 6000


(c)

151

05


0

0 1000 2000 3000 4000 5000 6000


(d)

151

05


0

0 1000 2000 3000 4000 5000 6000


(e)

151

05


0

0 1000 2000 3000 4000 5000 6000


(f)

151

05


0

0 1000 2000 3000 4000 5000 6000


(g)

151

05


0

0 1000 2000 3000 4000 5000 6000


(h)

151

05


0

0 1000 2000 3000 4000 5000 6000


(i)


1= 1025 V

2= 225 V

3= 105 Δ119905 = 00296)


119909= 119904

119899

119909when |119904119899+1

119909| ≻ 6|119904

119899



119909119910= 119904

119899

119909119910when |119904

119899+1

119909119910| ≻ 6|119904

119899

119909119910| for two-

dimensional signal


















(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)


1= 175 V

2= 225 V

3= 105 Δ119905 = 10


1= 225 V

2= 25 V

3= 125 Δ119905 = 10











(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)


1= 175 V

2= 225 V

3= 105 Δ119905 = 10


1= 225 V

2= 25 V

3= 125 Δ119905 = 10











(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

(m) (n) (o) (p) (q) (r)


1= 175 V

2= 225 V

3= 105 Δ119905 = 10


1= 225 V

2= 25 V

3= 125 Δ119905 = 10











(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V

3= 105 Δ119905 = 5 times 10


1= 225 V

2= 25 V




















(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)


1= 175 V

2= 275 V



2= 25 V

3= 125 and Δ119905 = 10






5 Conclusions




Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Acknowledgment


References


















































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































































minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























minus1































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Fractional Partial Differential Equation