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Research ArticleA Total Variation Model Based on the Strictly ConvexModification for Image Denoising
Boying Wu,1 Elisha Achieng Ogada,1,2 Jiebao Sun,1 and Zhichang Guo1
1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China2Department of Mathematics, Egerton University, P.O. Box 536-20115, Egerton, Kenya
Correspondence should be addressed to Jiebao Sun; sunjiebao@126.com
Received 18 April 2014; Revised 16 May 2014; Accepted 26 May 2014; Published 18 June 2014
Academic Editor: Adrian Petrusel
Copyright Β© 2014 Boying Wu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We propose a strictly convex functional in which the regular term consists of the total variation term and an adaptive logarithmbased convex modification term. We prove the existence and uniqueness of the minimizer for the proposed variational problem.The existence, uniqueness, and long-time behavior of the solution of the associated evolution system is also established. Finally, wepresent experimental results to illustrate the effectiveness of the model in noise reduction, and a comparison is made in relation tothe more classical methods of the traditional total variation (TV), the Perona-Malik (PM), and the more recent D-πΌ-PM method.Additional distinction from the other methods is that the parameters, for manual manipulation, in the proposed algorithm arereduced to basically only one.
1. Introduction
Noise removal, edge detection, contrast enhancement, in-painting, and segmentation have been the subject of intensemathematical image analysis and processing research fornearly three decades. Several methods have been pursuedover the passage of time. These include wavelet transform[1, 2], curvelet shrinkage methods [3β6], and variationalpartial differential equation (PDE) based methods [7, 8].These methods generate processes that can easily be dividedinto either linear and nonlinear processes or isotropic andnonisotropic processes [9].
Due its ability to preserve crucial image features, suchas edges, nonlinear anisotropic diffusion is favored overisotropic diffusion [10]. Much interest, therefore, has focusedon understanding operations andmathematical properties ofthe nonlinear anisotropic diffusion and associated variationalformulations [11, 12], formulation of well-posed and stableequations [8, 11], extending and modifying anisotropic dif-fusion [11, 13, 14], and studying the relationships that existbetween the various image processing techniques [11, 15, 16].
The objective of any image denoising process should notfocus only on the removal of noise, but it should also ensurethat no spurious details are created on the restored image and
that the edges are preserved or sharpened [7, 17, 18]. It is,therefore, necessary to develop formulations which are sen-sitive to the local image structure, especially edges/contours[19]. Consequently, a number of edge indicators have beenproposed and logically grafted into the partial differentialequation (PDE) based evolution equations [7, 11, 20].
Some of these PDEs originate from variational problems.For instance, Rudin et al. [8] proposed a minimizationfunctional, widely referred to as the total variation (TV)functional, of the form
minπ’
{πΉ (π’) = β«
Ξ©
|βπ’| ππ₯ + πβ«
Ξ©
(π’ β π)2
ππ₯} , (1)
where π is the fidelity parameter, π = π(π₯) denotes thenoise image, and Ξ© is an open bounded subset of R2. TVfunctionals are defined in the space of functions of boundedvariation (BV) and, therefore, do not necessarily requireimage functions to be continuous and smooth. This factmakes them allow jumps or discontinuities, and hence theyare able to preserve edges.
The original TV formulation has certain weaknesses.Firstly, the formulation is susceptible to backward diffusionsince it is not strictly convex. Secondly, the numerical imple-mentation cannot be accomplished without additional small
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 948392, 16 pageshttp://dx.doi.org/10.1155/2014/948392
2 Abstract and Applied Analysis
perturbation quantity, say π, at the denominator [21β23].Otherwise a spike/singularity is suddenly generated when|βπ’| = 0 in the homogeneous regions.This perturbation phe-nomenon is believed to contribute to some loss of accuracy inthe results of the restoration process.Moreover, the additionalparameter unnecessarily increases the number of parameters,therebymaking it difficult to determinewhich permutation ofparameter values will give optimal result.
Additionally, given that the method is more efficient inpreserving edges of uniform and small curvature, it mayexcessively smoothen and possibly destroy small scale fea-tures having more pronounced curvature edges [24]. TVregularization approach may also result in a loss of contrastand geometry of the final output images, even in noise freeobserved images [9, 25]. Furthermore, TV regularization hasdifficulties recovering texture, and there is also evidence ofenhanced noise when the fidelity parameter is chosen so thattexture is not removed [26]. Lastly, the formulation favorspiecewise constant solutions. This has the material effectof causing staircases (false edges) on the resultant image,especially from images severely degraded by noise [25].
However, given the strength of TV based techniques,especially in edge preservation, various modifications havebeen proposed. For instance, Vogel in [22, 27] proposed atotal variation penalty method of the form
minπ’
β«
Ξ©
(π΄π’ β π§)2
ππ₯ + πΌβ«
Ξ©
(β|βπ’|2
+ π½)ππ₯, (2)
where π½ β₯ 0, π΄ is a linear operator, and πΌ > 0 is thepenalty parameter. The formulation becomes total variationformulation of theRudin et al. form [8]whenπ½ = 0 and there-fore largely suffers the same shortcoming of the ordinary TVformulation. Hence, it has no significant practical advantageover TV.
Strong and Chan [28] proposed an adaptive total varia-tion based regularization model of the form
minπ’βBV(Ξ©)
β«
Ξ©
πΌ (π₯) |βπ’| , (3)
where 0 β€ πΌ(π₯) β€ 1 is a control factor which controlsthe speed of diffusion depending on whether the region ishomogeneous or an edge. This model demonstrated fairlygood results. However, since it is also not strictly convex,it is still susceptible to backward diffusion, which has thepotential of introducing blurs in the restored image.
Chambolle and Lions [29] proposed to minimize acombination of total variation and the integral of the squarednorm of the gradient and thus have
minπ’βBV(Ξ©)
1
2π
β«
|βπ’|β€π
|βπ’|2
+ β«
|βπ’|β₯π
|βπ’| β
π
2
, (4)
where π is a parameter. In this formulation |βπ’| β₯ π signalsedges, while |βπ’| β€ π signals homogeneous regions. Thismodel is successful in restoring images where homogeneousregions are separated by distinct edges but may becomesensitive to the thresholding parameter π in the event ofnonuniform image intensities or heavy degradation [24].
A variable exponent adaptive model was proposed byChen et al. in [24]. It has the form
minπ’βBV(Ξ©)
β«
Ξ©
|π·π’|π(π₯)
+
π
2
(π’ β πΌ)2
, π > 0, (5)
where 1 < πΌ β€ π(π₯) β€ 2, with π(π₯) proposed as π(π₯) =1 + 1/(1 + π|βπΊ
πβ πΌ(π₯)|
2
), πΊπis the Gaussian filter, and
π > 0, π > 0 are fixed parameters. It is observed that themethod exploits the benefits of Gaussian smoothing whenπ(π₯) = 2 and the strength of TV regularization when π(π₯) =1. This method also demonstrates good results and is indeeda great improvement of the earlier models. However, theconvolution of the image with a Gaussian kernel before theevolution process diminishes the accuracy of these models.This is because of the introduction of the scale variance of theGaussian; the scale variance is itself an additional parameterthat is subject to manual manipulation [12]. Furthermore, thediffusion process becomes ill-posed if scale variance is toosmall, while image features become smeared if the Gaussianvariance is too large [30]. Therefore, optimal selection of thescale variance remains a challenge. Parameter permutationgiving optimal result is a challenge due to the number of suchparameters involved.
Further literature surveys attest to the fact that research ineffective regularization functionals which have the ability togenerate diffusion processes that restore images, while simul-taneously preserving critical images features, the analysis, andpractical implementation of suchmodels, is still an extremelyactive concern.
Consequently, in this paper, we propose a new adaptivetotal variation (TV) formulation for image denoising, whichis strictly convex. The only parameter πΎ, which is a thresh-olding parameter to be tweaked, is such that it depends on theevolution parameter π‘ and is therefore not as grossly limitedas the choice of numerical values. The other parameter π isdynamically obtained and is therefore not manually tuned.With all this the number of parameters is reduced to basicallyonly one. Our method approximates TV model, for highervalues of the thresholding parameterπΎ. Experimental results,presented here, demonstrate the effectiveness of our modelover the classical models of TV, PM, and D-πΌ-PM.
The structure of this paper is as follows. In Section 2, wepresent the proposed model (6), its properties, justificationsof the model, and the associated evolution equation. InSection 3, we give certain preliminary definitions we relyon from time to time in this paper; we also prove theexistence and uniqueness of the solution to the minimizationproblem (6). In Section 4, we discuss the associated evolutionequation to theminimization problem (6). Also, we define theweak solution to the evolution problem (12)β(14), derive theestimates for the solution of an approximating problem (32),prove the existence and uniqueness of the weak solution ofthe evolution problem (12)β(14), and discuss the asymptoticbehaviour of the weak solution as π‘ β β. In Section 5,we give the numerical schemes and experimental results todemonstrate the strength and effectiveness of our method.We have also presented a brief comparative discussion ofour results with the PM, original TV methods, and D-πΌ-PMmethods. A brief summary concludes the paper in Section 6.
Abstract and Applied Analysis 3
2. Proposed Model
In this section, motivated by the methods of [11, 21, 24],we propose a modified version of total variation denoisingmodel. The model is based on minimization of a totalvariation functional with a logarithm-based strictly convexmodification. First, we present the model and certain impor-tant properties of it.
2.1. The New Energy Functional. The new strictly convexenergy functional is given as follows:
minπ’βBV(Ξ©)β©πΏ2(Ξ©)
{πΌ (π’) = β«
Ξ©
Ξ¦ (|βπ’|) ππ₯ +
π
2
β«
Ξ©
(π’ β π)2
ππ₯} ,
(6)
where
Ξ¦ (π ) = π β
1
πΎ
ln (1 + πΎπ ) , (7)
whereπΎ is a positive parameter, and π is the noise image.Calculating directly, we can obtain the following propo-
sition.
Proposition 1. The function Ξ¦(π ) satisfies the following prop-erties:
(i) Ξ¦(0) = 0, Ξ¦(π ) = πΎπ /(1 + πΎπ ), and therefore Ξ¦(π ) isa strictly nondecreasing function for π > 0;
(ii) Ξ¦(π ) = πΎ/(1 + πΎπ )2 > 0, and thereforeΞ¦(π ) is strictlyconvex;
(iii) Ξ¦(π ) = (πΎ/2)π 2 + π(π 2) as π β 0;(iv) limsβ+β(Ξ¦(π )/π ) = 1, πΌ|π | β€ Ξ¦(|π |) < |π |, 0 < πΌ < 1,
and therefore Ξ¦(π ) is the linear growth.
Compared with the work in [31], the new model satisfiesthe linear growth condition, which is much more natural.And the existence result for the flow associated with theminimization problem in [31] is only in one and two dimen-sions because themethods employed there use general resultson maximal monotone operators and evolution operators inHilbert spaces.
Remark 2. In this method,Ξ¦ satisfies someminimal hypoth-eses; namely:
(H1) Ξ¦ is a strictly convex, nondecreasing function fromR+ to R+, with Ξ¦(0) = 0;
(H2) because in the homogeneous regions weak gradientsshould be smoothed, there should be weak penaliza-tion in these region. Thus as |βπ’| = π β 0, Ξ¦(π ) βπΎπ
2; this signals isotropic diffusion;(H3) because edges, signaled by regions of strong gradients,
should be protected, regularization near the edgeswillbe penalized strongly. Thus, as |βπ’| = π β +β,πΌ|π | < Ξ¦(|π |) < Ξ|π |, 0 < πΌ β€ Ξ; this not onlydemonstrates the linear growth nature of the modelbut also signals the TV edge preservation behaviourof the model.
In [27], the authors obtained
limπ½β0
π½π½(π’) = π½
0(π’) , (8)
where π½π½(π’) = β«
Ξ©
βπ½ + |βπ’|2
ππ₯. Actually, the perturba-tion π½ is always used to the eliminate singularity of theterm div(π·π’/|π·π’|) in the numerical experiments. With theinclusion or addition of π½, what is then implemented isnot actually the TV formulation, but an approximation ofit. And, so, because of the approximation, the time step inthe discrete scheme becomes limited only to smaller values,as π½ diminishes [32]. The proposed model beats all thesechallenges and therefore has an easier design for numericalimplementation without the need for lifting parameters.Moreover, it is noticed that, for π β R,
limπΎβ+β
Ξ¦ (|π |) = |π | , (9)
which is the form of the TV model. This means that anymerits accruing from the TV model could be still obtainedas πΎ becomes larger and larger.
Remark 3. (1) Since TV potential (i.e., Ξ¦(π ) = π ) is onlyconvex (i.e., Ξ¦(π ) = 0), it gives a local minimum, but itcannot guarantee uniqueness of the minimum energy. Theproposed energy potential (i.e.,Ξ¦(π ) = π β (1/πΎ) ln(1 + πΎπ )),however, is strictly convex (i.e., Ξ¦(π ) = πΎ/(1 + πΎπ )2 >0). And, with additional strict convexity in the fidelity part,π»(π§) = (π§ β π)
2, the resultant energy functional is strictlyconvex in both |βπ’| and π’, thereby giving a global minimumenergy and hence guaranteeing uniqueness of the results.
(2) The proposed method seeks to reduce the number ofparameters subject to manual manipulation. In the imple-mentation,πΎ ismade to depend on time, thereby leaving time(evolution parameter) as the only parameter to be tweaked.
Other modifications of the TV model are only convex in|βπ’| and therefore do not guarantee uniqueness of solutions;hence uniqueness of results is not easy to assure. Besides, theevolution system of the models contains 1/|βπ’| component,which runs into a spike when |βπ’| = 0, in smoothregions. This then makes only the implementation of theapproximations of the corresponding formulations possible(see [13, 20, 24, 27, 29]).
2.2. The Associated Evolution Equation. The Euler-Lagrangeequation for the energy functional (6) is
0 = β div( πΎβπ’1 + πΎ |βπ’|
) + π (π’ β π) , π₯ β Ξ©, (10)
with
ππ’
π βπ
= 0, π₯ β πΞ©. (11)
To compute a solution of (10) numerically, it is embeddedinto a dynamical scheme, where π‘ is used as the evolution
4 Abstract and Applied Analysis
parameter. Hence, corresponding to the proposed minimiza-tion model (6), we have the following evolution system:
π’π‘= div( πΎβπ’
1 + πΎ |βπ’|
) β π (π’ β π) , (π₯, π‘) β Ξ© Γ (0, π) ,
(12)
π’ (π₯, 0) = π (π₯) , π₯ β Ξ©, (13)
ππ’
π βπ
= 0, (π₯, π‘) β πΞ© Γ [0, π] . (14)
Note 1. The modified formulation also does not have thedisadvantage of running into a singularity, since the denom-inator does not become zero. This particular observationmakes it more convenient to design a numerical scheme forthe model.
Furthermore, to see more clearly the action of the diffu-sion operator (kernel), we decompose the divergence term onthe premise of the local image structures. That is, we breakit into the tangential (π) and normal (π) directions to theisophote lines. Consequently we have
div( βπ’1 + πΎ |βπ’|
) =
1
(1 + πΎ |βπ’|)2π’ππ
+
1
1 + πΎ |βπ’|
π’ππ,
(15)
where
π’ππ
=
1
|βπ’|2(π’
2
π₯1
π’π₯1π₯1
+ π’2
π₯2
π’π₯2π₯2
+ 2π’π₯1
π’π₯2
π’π₯1π₯2
) ,
π’ππ
=
1
|βπ’|2(π’
2
π₯1
π’π₯2π₯2
+ π’2
π₯2
π’π₯1π₯1
β 2π’π₯1
π’π₯2
π’π₯1π₯2
) .
(16)
The divergence term is visibly a weighted sum of the direc-tional derivatives, along the normal and tangential directions.Since we are also seeking to preserve the edges and otherimportant features of the image, smoothing in the tangentialdirection should be encouragedmore than that in the normaldirection, in the regions near the boundaries (edges).Observein the above formulation that, as |βπ’| increases, the coefficientof π’
ππdiminishes faster, reducing diffusion in the normal
direction, thus preserving edges. The edges are signalled byhigher values of the magnitude of gradient of π’. However, as|βπ’| decreases, there is relatively uniform diffusion in bothπ’ππ
and π’ππ
directions, thereby achieving isotropic diffusionin homogeneous regions. The homogeneous regions aresignaled by diminishing values of the magnitude of gradientof image π’. The proposed model is, therefore, sensitive tothe local image structure. However, TV based denoisingmodel and most other modifications do not have that π’
ππ
component.This leads to a situation where the homogeneousparts are processed into piecewise constant regions, whoseboundaries reflect staircases or false edges in the image. Thissituation, for instance, affects the results of a modification byChen et al. [24] when π = 1, and the model becomes thetraditional TV model.
3. The Minimization Problem
In this section, we prove the existence and uniqueness of theminimization problem (6). But, first, we give the followingpreliminary presentations which will guide our reasoning inthe subsequent sections of this paper.
3.1. Preliminaries
Definition 4. Let Ξ© be an open subset of Rπ. A function π’ βπΏ1 has a bounded variation inΞ© if
sup {β«Ξ©
π’ div πππ₯ : π β πΆ10(Ξ©;R
π
) ,πβ€ 1} < β, (17)
where BV(Ξ©) denotes the space of such functions. Then BV-norm is given by
ββπ’βBV = β«Ξ©
|βπ’| + |π’|πΏ1(Ξ©). (18)
Definition 5 (see [33]). If π’ β BV(Ξ©), then
π·π’ = βπ’ β Lπ
+ π·π
π’ (19)
and π·π’ is a radon measure, where βπ’ is the density of theabsolutely continuous part ofπ·π’with respect to the Lebesguemeasure,Lπ, andπ·π π’ is the singular part.
Lemma 6 (see [34]). Let π : R β R+ be convex, even,nondecreasing onR+ with linear growth at infinity. Also letΞ¦βbe the recession function of Ξ¦ defined by
Ξ¦β
(π) = limπ ββ
Ξ¦ (π π)
π
. (20)
Then for π’ β π΅π(Ξ©) and setting Ξ¦(π) = Ξ¦(|π|) we have
β«
Ξ©
Ξ¦ (π·π’) = β«
Ξ©
Ξ¦ (|βπ’|) ππ₯ + Ξ¦β
(1) β«
Ξ©
π·π
π’. (21)
This implies that π’ β β«Ξ©
Ξ¦(π·π’) is lower semicontinuous forthe π΅π(Ξ©) weakβ topology.
3.2. Existence and Uniqueness of Solution to the MinimizationProblem. The linear growth condition makes it natural toconsider a solution in the space
π = {π’ β πΏ2
(Ξ©) ; βπ’ β πΏ1
(Ξ©)2
} . (22)
However, this space is not reflexive. But sequences boundedinπ are also bounded in BV(Ξ©) and are therefore compact forBV weakβ topology. Moreover, due to the fact that πΏ1-spaceis separable, weakβ topology allows us to obtain compactnessresults even if the space is not reflexive [34β37]. Hence,by denoting BV(Ξ©) weakβ topology simply as BV(Ξ©), wetherefore seek the solution to the minimization problem (6)in the space BV(Ξ©) β© πΏ2(Ξ©).
Theorem 7. The minimization problem πΌ(π’) (refer to (6)) hasa unique solution π’ β π΅π(Ξ©) β© πΏ2(Ξ©).
Abstract and Applied Analysis 5
Proof . From the linear growth property ofΞ¦(π ) we have
πΌ |π | β€ Ξ¦ (|π |) < |π | , for 0 < πΌ < 1, (23)
which implies that
limπ β+β
Ξ¦ (π ) = +β. (24)
Thus πΌ(π’) is coercive. Therefore, let π’πbe a minimizing
sequence in BV(Ξ©) β© πΏ2(Ξ©). Then
πΌ (π’π) β€ π, (25)
whereπ denotes a generic constant that may differ from lineto line. It clearly follows from above that
β«
Ξ©
Ξ¦(βπ’
π
) ππ₯ β€ πΌ (π’
π) β€ π, β«
Ξ©
π’π
2
ππ₯ β€ π. (26)
This implies that {π’π} is bounded in BV(Ξ©) β© πΏ2(Ξ©). Hence
there exists a subsequence {π’ππ
} of {π’π} and a function π’ β
BV(Ξ©) β© πΏ2(Ξ©) such that
π’ππ
β π’, strongly in πΏ1 (Ξ©) ,
π’ππ
β π’, weakly in πΏ2 (Ξ©) .(27)
And, by Lemma 6 and the weak lower semicontinuity of theπΏ2-norm, we have that
πΌ (π’) β€ lim infπββ
πΌ (π’ππ
) = minBV(Ξ©)β©πΏ2(Ξ©)
πΌ (V) . (28)
Hence, there exists a solution of the minimization problem.Uniqueness of the solution follows from the strict convexityof the functional.
4. Existence and Uniqueness forthe Evolution Equation
In this section, we define a weak solution for the evolutionequations (12)β(14). That is, if (6) has a minimum point π’,then it should formally satisfy the Euler-Lagrange equation(12), subject to the boundary conditions (13)-(14). We definea weak solution that we are seeking for the equation system(12)β(14). We then propose an approximating problem andderive certain a priori estimates to the approximating prob-lem. These will aid the process of proving the existence anduniqueness of the weak solution. Finally, we show the largetime behavior (asymptotic stability) of the weak solution.
4.1. Definition of Weak Solution. Let V β πΏ2(0, π;π»1(Ξ©)), π βBV(Ξ©), V > 0, and πV/π βπ = 0, where Ξ©
π= Ξ© Γ [0, π] and
π’ is a solution to (12)β(14). Multiplying (12) by (V β π’) andintegrating overΞ©, we have
β«
Ξ©
ππ‘π’ (V β π’) ππ₯ + β«
Ξ©
πΎβπ’
1 + πΎ |βπ’|
(βV β βπ’) ππ₯
= βπβ«
Ξ©
(π’ β π) (V β π’) ππ₯.(29)
Then applying the convexity condition of Ξ¦, that is, Ξ¦(π₯) βΞ¦(π¦) β₯ Ξ¦
(π¦)(π₯ β π¦), on both the second term on the leftside and on the right side of the above equation gives
β«
Ξ©
ππ‘π’ (V β π’) ππ₯ + β«
Ξ©
Ξ¦ (|βπ’|) ππ₯ +
π
2
β«
Ξ©
(V β π)2ππ₯
β₯ β«
Ξ©
Ξ¦ (|βπ’|) +
π
2
β«
Ξ©
(π’ β π)2
ππ₯.
(30)
Integrating (30) with respect to π‘ we get
β«
π‘
0
β«
Ξ©
ππ‘π’ (V β π’) ππ₯ ππ‘ + β«
π‘
0
πΌ (V) ππ‘ β₯ β«π‘
0
πΌ (π’) ππ‘. (31)
Now, since V β πΏ2(0, π;π»1(Ξ©)), we may choose V = π’ + ππ,where π β πΆβ
0(Ξ©). We observe that the left-hand side of (31)
has aminimum at π = 0.This shows that π’ is a solution of (12)in the distributional sense.The facts raised abovemotivate thefollowing definition for a weak solution to (12)β(14).
Definition 8. A function π’ β πΏ2(0, π;BV(Ξ©) β© πΏ2) is called aweak solution of (12)β(14) if π
π‘π’ β πΏ
2
(Ξ©π), π’(π₯, 0) = π(π₯), on
Ξ© and π’ satisfies (31) for every V β πΏ2((0, π);BV(Ξ©)β©πΏ2) andπ‘ β [0, π].
4.2. Approximating Energy Functional. Let Ξ© be an openbounded subset of Rπ and let π β BV(Ξ©) β© πΏβ. Then let usconsider the following approximating energy functional for1 < π β€ 2:
πΌπ(π’) = β«
Ξ©
Ξ¦π(βπ’) ππ₯ +
π
2
β«
Ξ©
(π’ β π)2
ππ₯, (32)
where
Ξ¦π(π ) =
1
π
(|π |π
β
1
πΎ
ln (1 + πΎ|π |π)) , (33)
with the following properties:
(1) Ξ¦π(π ) = πΎ|π |
2πβ2
π /(1+πΎ|π |π
) andΞ¦π(π ) β₯ 0 βπ β Rπ;
(2) Ξ¦π(π ) = πΎ|π |
2πβ2
[(2π β 1) + πΎ(π β
1)|π |π
]/[1 + πΎ|π |π
]
2
> 0. Hence Ξ¦(π ) is strictlyconvex in π .
Remark 9. Since π β BV(Ξ©) we can have a sequence ππ β
π1,π
(Ξ©) with
ππ β π as π β 0 in πΏ2 (Ξ©) ,
β«
Ξ©
βπ
π
β β«
Ξ©
βπ.
(34)
And as a consequence of Theorem 2.9 in [20] we have
β«
Ξ©
βπ
π
β€ πΆ. (35)
From property (iii) ofΞ¦(π ) it is easy to see that
Ξ¦(βπ) β€βπ. (36)
6 Abstract and Applied Analysis
This implies that
Ξ¦(βππ ) β€
βπ
π
β€ πΆ. (37)
Hence
β«
Ξ©
Ξ¦(βππ ) β β«
Ξ©
Ξ¦(βπ) , as π β 0. (38)
On the strength of (32) and Remark 9 let us consider theapproximate evolution problem
ππ‘π’π ,π
= div (Ξ¦π(βπ’
π ,π)) β π (π’
π ,πβ π
π ) , in Ξ© Γ [0, π] ,
(39)
ππ’π ,π
π βπ
= 0; on πΞ© Γ [0, π] , (40)
π’π ,π
(π₯, 0) = ππ , π₯ β Ξ©. (41)
Let the followingπΏβ bound also hold for the solution of (39)β(41).Then, the following lemma indicates the boundedness ofthe solutionπ’
π ,πof the above approximate evolution problem.
Lemma 10. Suppose ππ β πΏ
β
(Ξ©) β© π΅π(Ξ©) and that π’π ,π
is asolution to the problem (39)β(41); then
inf ππ β€ π’
π ,πβ€ supπ
π . (42)
Proof. By a method similar to that of Zhou in [33], let π =supπ
π and define (π’
π ,πβπ)
+such that
(π’π ,π
βπ)+
:= {
(π’π ,π
βπ) , if π’π ,π
βπ β₯ 0,
0, otherwise;(43)
then multiplying (39) and integrating overΞ© yields
β«
Ξ©
ππ‘π’π ,π(π’
π ,πβπ)
+
ππ₯ + β«
Ξ©
Ξ¦
π(βπ’
π ,π) β βπ’
π ,πππ₯
+ πβ«
Ξ©
(π’π ,π
β ππ ) (π’
π ,πβπ)
+
ππ₯ = 0.
(44)
Observe that the last two integrals in the above equation arenonnegative, based on the definition of (π’
π ,πβπ)
+
and thefact thatΦ
π(π ) β π β₯ 0. Therefore, we have
1
2
β«
Ξ©
π
ππ‘
(π’π ,π
βπ)
2
+
ππ₯ β€ 0, (45)
which indicates that π½(π‘) = (1/2) β«Ξ©
(π’π ,π
βπ)2
+
ππ₯ is adecreasing function in π‘. Since π½(0) = 0 we have that
π½ (π‘) = 0, βπ‘ β [0, π] . (46)
Hence π’π ,π
β€ π = supππ . Conversely, multiplying (39)
by (π’π ,π
β π)β
, where π = inf ππ , and employing a similar
argument, we obtain that π’π ,π
β₯ βπ, βπ‘ β [0, π]. Therefore,inf π
π β€ π’
π ,πβ€ supπ
π , βπ‘ β [0, π].
Another estimate (bound) is obtained via the followinglemma.
Lemma 11. The approximate evolution problem (39)β(41) hasa unique solution π’
π ,πβ πΏ
β
(0, π;BV(Ξ©)) with ππ‘π’π ,π
β
πΏ2
(0, π; πΏ2
(Ξ©)) such that
β«
π‘
0
β«
Ξ©
ππ‘π’π ,π
(V β π’π ,π) ππ₯ ππ‘ + β«
π‘
0
πΌπ(V) ππ‘ β₯ β«
π‘
0
πΌπ(π’
π ,π) ππ‘,
(47)
for any π‘ β [0, π] and V β πΏ2(0, π;π1,π(Ξ©)) with πV/π βπ = 0.Moreover
β«
β
0
β«
Ξ©
ππ‘π’π ,π
2
ππ₯ ππ‘
+ supπ‘β[0,π]
{β«
Ξ©
Ξ¦π(βπ’
π ,π) ππ₯ +
π
2
β«
Ξ©
(π’π ,π
β ππ )
2
ππ₯}
β€ πΌπ(βπ
π ) .
(48)
Proof. Since Ξ¦(βπ’) is a lower semicontinuous and strictlyconvex function, then, by definitions provided in [34, 38, 39],β div(Ξ¦(βπ’)), which is a subdifferential of β«
Ξ©
Ξ¦(βπ’)ππ₯, ismaximal monotone. The approximate problem (39)β(41) hasa unique solution such that π’
π ,πβ πΏ
β
(0, π;π1,π
(Ξ©)). Now,to show that π’
π ,πis the weak solution to the approximating
problem (39)β(41), as governed by (47), we multiply (39) byVβπ’
π ,πfor V β πΏ2(0, π;π1,π(Ξ©))with πV/π βπ = 0 and integrate
overΞ© and π‘ β [0, π]. Thus
β«
π‘
0
β«
Ξ©
ππ‘π’π ,π
(V β π’π ,π) ππ₯ ππ‘
= β«
π‘
0
β«
Ξ©
(div (Ξ¦π(βπ’
π ,π)) (V β π’
π ,π)) ππ₯ ππ‘
β πβ«
π‘
0
β«
Ξ©
((π’π ,π
β ππ ) (V β π’
π ,π)) ππ₯ ππ‘,
β«
π‘
0
β«
Ξ©
ππ‘π’π ,π
(V β π’π ,π) ππ₯ ππ‘
+ β«
π‘
0
β«
Ξ©
Ξ¦
π(βπ’
π ,π) (βV β βπ’
π ,π) ππ₯ ππ‘
= βπβ«
π‘
0
β«
Ξ©
(π’π ,π
β ππ ) (V β π’
π ,π) ππ₯ ππ‘.
(49)
Applying convexity condition on both sides of the aboveequation we obtain
β«
π‘
0
β«
Ξ©
ππ‘π’π ,π
(V β π’π ,π) ππ₯ ππ‘ + β«
π‘
0
β«
Ξ©
Ξ¦π(βV) ππ₯ ππ‘
β
π
2
β«
π‘
0
β«
Ξ©
(V β ππ )2
ππ₯ ππ‘
β₯ β«
π‘
0
β«
Ξ©
Ξ¦π(βπ’
π ,π) ππ₯ ππ‘ +
π
2
β«
π‘
0
β«
Ξ©
(π’π ,π
β ππ)
2
ππ₯ ππ‘,
(50)
which implies (47).
Abstract and Applied Analysis 7
Then, multiplying (39) by οΏ½ΜοΏ½π ,π
and integrating overΞ© andπ‘ β [0, π] we obtain
β«
π‘
0
β«
Ξ©
(ππ‘π’π ,π)
2
ππ₯ ππ‘ = β«
π‘
0
β«
Ξ©
div (Ξ¦π(βπ’
π ,π)) οΏ½ΜοΏ½
π ,πππ₯ ππ‘
β πβ«
π‘
0
β«
Ξ©
(π’π ,π
β ππ ) οΏ½ΜοΏ½
π ,πππ₯ ππ‘,
(51)
which yields
β«
π‘
0
β«
Ξ©
(ππ‘π’π ,π)
2
ππ₯ ππ‘ + β«
π‘
0
ππ‘(β«
Ξ©
Ξ¦π(βπ’
π ,π) ππ₯) ππ‘
+
π
2
β«
π‘
0
ππ‘(β«
Ξ©
(π’π ,π
β ππ ) ππ₯)
2
ππ‘ = 0.
(52)
From the above equation together with (41) we obtain
β«
π‘
0
β«
Ξ©
(ππ‘π’π ,π)
2
ππ₯ ππ‘ + β«
Ξ©
Ξ¦π(βπ’
π ,π)
π‘
0
ππ₯
+
π
2
β«
Ξ©
(π’π ,π
β ππ )
2
π‘
0
ππ₯ = 0,
(53)
which by (40) and (41) produces (48).
4.3. Existence and Uniqueness of the Solution of the EvolutionProblem. Next, using the a priori estimates obtained above,through the approximate problem, we proceed to prove theexistence and uniqueness of the solution of the evolutionproblem (12)β(14).
Theorem 12. Let π β π΅π(Ξ©) β© πΏβ(Ξ©). Then there exists aunique weak solution π’ β πΏβ(0,β; π΅π(Ξ©) β© πΏβ(Ξ©)), π
π‘π’ β
πΏ2
(πβ), and π’(π₯, 0) = π such that
β«
β
0
β«
Ξ©
ππ‘π’
2
ππ₯ ππ‘ + supπ‘β[0,β)
{πΌ (π’)} β€ πΌ (π) , (54)
where π‘ β [0,β), πβ:= Ξ© Γ [0,β).
Proof. From Lemmas 10 and 11 we see that
ππ‘π’π ,π
πΏ2(πβ)
+
π’π ,π
πΏβ(πβ)
+
π’π ,π
πΏβ(0,β;π
1,π(Ξ©)β©πΏ
β(Ξ©))
β€ π,
(55)
whereπ is some constant that may vary from line to line. Let{π’
π ,π} by Lemma 10 be a bounded sequence of solutions of the
problem (39)β(41). Then there exists a subsequence denoted
by {π’π ,ππ
} of {π’π ,π} and a function π’
π β πΏ
β
(Ξ©), with οΏ½ΜοΏ½π β
πΏ2
(Ξ©; [0, π]), such that, as ππβ 1,
π’π ,ππ
β π’π
strongly in πΏ1 (Ξ©) for each π‘ β [0,β) , (56)
ππ‘π’π ,ππ
β ππ‘π’π
weakly in πΏ2 (πβ) , (57)
π’π ,ππ
β
β π’π
weakβ in πΏβ (πβ) , (58)
π’π ,ππ
β π’π
in πΏ2 (Ξ©) uniformly in π‘, (59)
limπ‘β0+
β«
Ξ©
π’π ,ππ
(π₯, π‘) β ππ (π₯)
2
ππ₯ = 0. (60)
Indeed, from (55), there is a sequence {π’π ,ππ
} and a func-tion π’
π β πΏ
β
(πβ) with οΏ½ΜοΏ½
π β πΏ
2
(πβ) such that (57) and (58)
hold. Observe also that for any π β πΏ2(Ξ©), as π β β,
β«
Ξ©
(π’π ,ππ(π₯, π‘) β π
π (π₯)) π (π₯) ππ₯
= β«
π‘
0
ππ (β«
Ξ©
π’π ,ππ(π₯, π ) π (π₯) ππ₯) ππ
β β«
π‘
0
ππ (β«
Ξ©
π’π (π₯, π ) π (π₯) ππ₯) ππ
= β«
Ξ©
(π’π (π₯, π‘) β π
π (π₯)) π (π₯) ππ₯,
(61)
which indicates that for each π‘π’π ,ππ
β π’π
in πΏ2 (Ξ©) . (62)By Lemmas 10 and 11, for each π‘ β [0,β), {π’
π ,ππ
(π₯, π‘)} is abounded sequence inπ1,1(Ξ©). Combining that fact with (62)we obtain that for each π‘ as π
πβ 1
π’π ,ππ
β π’π
in πΏ1 (Ξ©) . (63)Moreover, (60) follows from the fact thatπ’π ,ππ
(β , π‘) β π’π ,ππ
(β , π‘
)
2
πΏ2(Ξ©)
β€
π‘ β π‘
β«
π‘
0
β«
Ξ©
(ππ‘π’π ,ππ
)
2
ππ₯ ππ‘.
(64)
From (64) π‘ β π’π ,ππ
(β , π‘) β πΏ2
(Ξ©) is equicontinuous, and from(55) and (58) we have that
π’π ,ππ
β π’π
in πΏ2 (Ξ©) . (65)It then follows by standard argument thatwe canhaveπ’
π ,ππ
β
π’π in πΏ2(Ξ©) uniformly in π‘, giving us (59). From Lemma 10
and (58) we obtain that π’π β πΏ
β
(0,β,BV(Ξ©)β©πΏβ(Ξ©))withοΏ½ΜοΏ½π β πΏ
2
(πβ).
Next we show that, for all V β πΏ2(0, π,π1,π(Ξ©) β© πΏ2(Ξ©)),V > 0 and πV/π βπ = 0 and for each π‘ β [0,β)
β«
π‘
0
β«
Ξ©
ππ‘π’π (V β π’
π ) ππ₯ ππ‘ + β«
π‘
0
β«
Ξ©
Ξ¦ (βV) ππ₯ ππ‘
β
π
2
β«
π‘
0
β«
Ξ©
(V β ππ )2
ππ₯ ππ‘
β₯ β«
π‘
0
β«
Ξ©
Ξ¦(βπ’π ) ππ₯ ππ‘ β
π
2
β«
π‘
0
β«
Ξ©
(π’π β π
π)2
ππ₯ ππ‘.
(66)
8 Abstract and Applied Analysis
To show this end, from Lemma 11, we obtain
β«
Ξ©
ππ‘π’π ,ππ
(V β π’π ,ππ
) ππ₯ + β«
Ξ©
Ξ¦ππ(βV) ππ₯
+
π
2
β«
Ξ©
(V β ππ )2
ππ₯
β₯ β«
Ξ©
Ξ¦ππ
(βπ’π ,ππ
) ππ₯ +
π
2
β«
Ξ©
(π’π ,ππ
β ππ )
2
ππ₯.
(67)
From (56) and (58) we deduce that there exists a subsequence{π’
π ,ππ
} such that
π’π ,ππ
β π’π
strongly in πΏ2 (Ξ©; [0, π]) . (68)
Using (57), (59), and (68) andwe letππβ 1; in (67)we obtain
β«
Ξ©
ππ‘π’π (V β π’
π ) ππ₯ + β«
Ξ©
Ξ¦ππ(βV) ππ₯ +
π
2
β«
Ξ©
(V β π)2ππ₯
β₯ lim infπββ
β«
Ξ©
Ξ¦ππ
(βπ’π ,ππ
) ππ₯ +
π
2
β«
Ξ©
(π’π β π
π )2
ππ₯.
(69)
But we define from the lower semicontinuity theorem that
β«
Ξ©
Ξ¦(βπ’π ) ππ₯ β€ lim inf
πββ
β«
Ξ©
Ξ¦ππ
(βπ’π ,ππ
) ππ₯. (70)
Using (70) in (69) and integrating over π β [0, π] we obtain
β«
π
0
β«
Ξ©
ππ‘π’π (V β π’
π ) ππ₯ ππ‘ + β«
π
0
β«
Ξ©
Ξ¦ (βV) ππ₯ ππ‘
+
π
2
β«
π
0
β«
Ξ©
(V β ππ )2
ππ₯ ππ‘
β₯ β«
π
0
β«
Ξ©
Ξ¦π (βπ’
π ) ππ₯ ππ‘ +
π
2
β«
π
0
β«
Ξ©
(π’π β π
π )2
ππ₯ ππ‘,
(71)
which confirms (66).Now, to complete the proof of the existence of solution to
(12)β(14), it remains to pass to the limit as π β 0 in (71).Replacing π by π
πin (48), letting π β β (π
πβ 1), and
using (57)β(59) and (70) we obtain
β«
β
0
β«
Ξ©
ππ‘π’π
2
ππ₯ ππ‘
+ supπ‘β[0,β)
{β«
Ξ©
Ξ¦(βπ’π ) ππ₯ +
π
2
β«
Ξ©
(π’π β π
π )2
ππ₯}
β€ β«
Ξ©
Ξ¦(βππ ) ππ₯.
(72)
From Lemma 10 and also from the above inequality we seethat π’
π is uniformly bounded in πΏβ(0,β,BV(Ξ©) β© πΏβ(Ξ©))
and ππ‘π’π is uniformly bounded in πΏ2(π
β).Thismeanswe can
extract a subsequence {π’π π
} of {π’π } such that as π β β (π
πβ
0) we have
ππ‘π’π π
β ππ‘π’ weakly in πΏ2 (π
β) ,
π’π π
β
β π’ weakβ in πΏβ (Ξ©β) ,
π’π π
β π’ strongly in πΏ1 (Ξ©, [0, π]) , for each π‘ β [0,β) ,
π’π ,ππ
β π’π
in πΏ2 (Ξ©) uniformly in π‘,
limπ‘β0+
β«
Ξ©
π’π ,ππ
(π₯, π‘) β ππ (π₯)
2
ππ₯ = 0.
(73)
Now, replacing π with π πin (71), letting π β β (π
πβ 0),
and applying the lower semicontinuity in (70) we obtain
β«
π
0
β«
Ξ©
ππ‘π’ (V β π’) ππ₯ ππ‘ + β«
Ξ©
πΌ (V) ππ‘ β₯ β«Ξ©
πΌ (π’) ππ‘, (74)
for all V β πΏ2(0, π,π1,π(Ξ©)β©πΏ2(Ξ©)), V > 0 and πV/π βπ = 0 andfor each π‘ β [0,β). Hence π’ is a weak solution to (12)β(14).Replacing π by π
πin (72), letting π β β(π
πβ 0), and using
(57)β(59) and (70) we obtain (54).
Uniqueness of the Weak Solution. With reference to the def-inition of solution inequality as given in (31), let π’
1and π’
2
be two weak solutions to the problem (12) to (14) such thatπ’1(π₯, 0) = π’
2(π₯, 0) = π. Then we have, for two solutions, two
inequalities:
β«
π
0
β«
Ξ©
ππ‘π’1(π’
2β π’
1) ππ₯ ππ‘+ β«
π
0
πΌ (π’2) ππ‘ β₯ β«
π
0
πΌ (π’1) ππ‘,
β«
π
0
β«
Ξ©
ππ‘π’2(π’
1β π’
2) ππ₯ ππ‘+ β«
π
0
πΌ (π’1) ππ‘ β₯ β«
π
0
πΌ (π’2) ππ‘.
(75)
Now, adding the two inequalities (75) we obtain a more com-pact inequality given by
β«
π
0
β«
Ξ©
(ππ‘π’2β π
π‘π’1) (π’
1β π’
2) ππ₯ ππ‘ β₯ 0. (76)
This implies that
β«
π
0
π
ππ‘
β«
Ξ©
(π’1β π’
2)2
ππ₯ ππ‘ β€ 0, (77)
which implies βπ’1βπ’
2β = 0 for a.e. (π₯, π‘) β π
β.This confirms
the uniqueness of the weak solution.
4.4. Large Time Behavior. Finally, we will assess the asymp-totic limit of the weak solution π’(β , π‘) as π‘ β β. The essenceof this part is to demonstrate that over time the solution ofthe evolution problem (12)β(14) ultimately converges to theunique minimizer of the functional (6).
Theorem 13. As π‘ β β the weak solution π’ of the evolutionequation (12)β(14) converges weakly in πΏ2(Ξ©) to a minimizer π’of the functional πΌ(π’) in (6).
Abstract and Applied Analysis 9
Proof. Let V β BV(Ξ©) β© πΏβ(Ξ©) into (31) to obtain
β
1
2
β«
Ξ©
(V (π₯) β π’)2
π
0
ππ₯ + β«
π
0
πΌ (V (π₯)) ππ‘ β₯ β«π
0
πΌ (π’) ππ‘. (78)
The equation above simplifies to
β«
Ξ©
(π’ (π₯, π ) β π) V (π₯) ππ₯
β
1
2
β«
Ξ©
(π’2
(π₯, π ) β π2
) ππ₯ + π πΌ (V (π₯))
β₯ β«
π
0
πΌ (π’) ππ‘.
(79)
By taking π’(π₯, π ) = (1/π ) β«π 0
π’(π₯, π‘)ππ‘, and have for each π ,π’(π₯, π ) β BV(Ξ©)β©πΏβ(Ξ©). Given that π’ is uniformly boundedin BV(Ξ©), we conclude that there exists sequence π’(π₯, π
π)
and its subsequence is still denoted by {π’(π₯, π π)} such that
π’(π₯, π π) β π’(π₯, π ) in πΏ1(Ξ©) and π’(π₯, π
π) β π’(π₯, π ) weakβ in
BV(Ξ©), as π πβ β, respectively. Hence dividing inequality
(79) by π and taking the limit as π πβ β we obtain
πΌ (V) β₯ πΌ (π’) . (80)
This demonstrates that indeed π’ is a weak solution to (12)β(14) and also the unique minimizer of problem (6).
5. Numerical Experiments
In this section, we present the performance of our methodin denoising images involving a Gaussian white noise. Wehave then compared our results with the ones obtained by theclassical methods of PMmethod [7], TVmethod [8], and themore recent D-πΌ-PM method [11].
5.1. Numerical Scheme. In the following two subsections, twonumerical discrete schemes, the PM scheme (PMS) and theadditive operator splitting (AOS) scheme, have been pro-posed.
5.1.1. PM Scheme. Here, we have proposed a numericalscheme similar to the original PMmethod, whereby (12)β(14)are discretized as follows:
πΆπ
ππ,π=
πΎ
1 + πΎ
βππ’π,π
, πΆπ
ππ,π=
πΎ
1 + πΎ
βππ’π,π
,
πΆπ
πΈπ,π=
πΎ
1 + πΎ
βπΈπ’π,π
, πΆπ
ππ,π=
πΎ
1 + πΎ
βππ’π,π
,
divππ,π= [πΆ
π
ππ,πβππ’π,π+ πΆ
π
ππ,πβππ’π,π+ πΆ
π
πΈπ,πβπΈπ’π,π
+ πΆπ
ππ,πβππ’π,π] ,
(81)
and π is dynamically determined according to the followingdiscretization scheme:
ππ
=
1
π2|Ξ©|
β
π,π
divππ,π(π’
π,πβ π
π,π) ,
where |Ξ©| = ππ is the size of image.
(82)
Hence from (81) and (82) we have
π’π+1
π,π= π’
π
π,π+ πdivπ
π,πβ π
π
π (π’π,πβ π
π,π) ,
π’0
π,π= π
π,π, π’
π
π,0= π’
π
π,1, π’
π
0,π= π’
π
1,π,
π’π
π,π= π’
π
πβ1,π, π’
π
π,π= π’
π
π,πβ1,
(83)
where
βππ’π,π= π’
πβ1,πβ π’
π,π, β
ππ’π,π= π’
π+1,πβ π’
π,π,
βπΈπ’π,π= π’
π,π+1β π’
π,π, β
ππ’π,π= π’
π,πβ1β π’
π,π,
(84)
for π = 0, 1, 2, . . . , π and π = 0, 1, 2, . . . ,π.
5.1.2. AOS Scheme. In this part, using a AOS scheme, theproblem (12)β(14) has been discretized as follows:
π0
= 0,
π’π+1
=
1
π
π
β
π=1
[πΌ β πππ΄π(π’
π
)]
β1
[π’π
+ ππ (π β π’π
)] ,
divπ =(π’
π+1
β π’π
)
π
,
ππ
=
1
π2ππ
(π’ β π) divπ,
π’0
π,π= π
π,π= π (πβ, πβ) , π’
π
π,0= π’
π
π,1,
π’π
0,π= π’
π
1,π, π’
π
πΌ,π= π’
π
πΌβ1,π, π’
π
π,π½= π’
π
π,π½β1,
(85)
where π΄π(π’
π
) = [ππ,π(π’
π
)],
ππ,π(π’
π
) :=
{{{{{{
{{{{{{
{
πΆπ
π+ πΆ
π
π
2β2
, [π β N (π)] ,
β β
πβN(π)
πΆπ
π+ πΆ
π
π
2β2
, (π = π) ,
0, (else) ,
πΆπ
π:=
πΎ
1 + πΎ
βπ’
π
π,π
,
(86)
where
βπ’
π
π,π
=
1
2
β
π,πβN(π)
π’π
πβ π’
π
π
2β
, (87)
whereN(π) is the set of the two neighbors of pixel π (boundarypixels have only one neighbor).
It is observed that AOS schemes with large time stepsstill reveal average grey value invariance, stability based onextremum principle, Lyapunov functionals, and convergenceto a constant steady state [10]. The AOS scheme is lessthan twice the typical effort needed for the PM scheme, arather low price for gaining absolute stability [10]. It is worthnoting that if we use div(βπ’/βπ + |βπ’|2) to approximate
10 Abstract and Applied Analysis
Table 1: Numerical results for synthetic image (300 Γ 300).
Algorithm Parameters Number of steps CPU time (sec) PSNR MAE PSNRE SSIMπ πΎ π
PM 30 12 0.25 232 12.73 34.23 2.84 25.26 0.9829TV 30 n/a 0.2 203 17.86 32.79 3.76 15.76 0.9748D-πΌ-PM 30 1 0.25 42 2.40 37.30 3.38 24.76 0.9838PMS 30 n/a 0.20 262 12.01 35.75 2.15 25.60 0.9870AOS 30 n/a 3.00 17 2.51 33.30 2.92 25.40 0.9839
Table 2: Numerical results for Lena image (300 Γ 300).
Algorithm Parameters Number of steps CPU time (sec) PSNR MAE PSNRE SSIMπ πΎ π
PM 30 12 0.25 60 2.43 27.10 7.75 21.70 0.7234TV 30 n/a 0.2 149 13.87 27.46 7.49 22.63 0.7960D-πΌ-PM 30 4 2 13 0.90 28.17 7.35 24.78 0.7880PMS 30 n/a 0.2 122 5.80 27.73 7.45 26.60 0.8060AOS 30 n/a 2.0 7 1.24 28.06 7.27 27.80 0.8509
div(βπ’/|βπ’|) (TV kernel), in numerical scheme, the AOSscheme may be unstable because of the small number π.Our approximation, however, effectively avoids instabilitiesarising from such a scenario, as there is no need for liftingthe denominator, since it cannot not be equal to zero. Thishas been an additional motivation for considering AOS forour numerical experiments.The possibility of occurrence of azero denominator in the evolution problem is a phenomenonthatmakes the numerical implementation of TVproblematic.
5.2. Comparison with Other Methods. The experiments inthis work were performed on a Compaq610 computer, havingIntel Core 2 Duo CPU T5870 each 2.00GHz, physical RAMof 4.00GB, and Professional Windows 8 64-bit OperatingSystem, on MATLAB R2013b. The image restoration per-formance was measured in terms of the peak-signal-to-noise ratio (PSNR), mean absolute deviation/error (MAE),structural similarity index measure (SSIM), the measure ofsimilarity of edges (PSNRE), and visual effects. The iterationstopping mechanism was based on the maximal PSNR or thelowestMAE.The PSNR andMAE values were obtained usingthe formula by Durand et al. [40] and are given by
PSNR = 10log10
ππmax π’
0βmin π’
0
2
π’ β π’
0
2
πΏ2
ππ΅,
MAE =π’ β π’
0
πΏ1
ππ
.
(88)
Also, taking the edge map EM(π’) from the evolution equa-tion, we have
EM (π’) = πΎ(1 + πΎ |βπ’|)
. (89)
And corresponding edge similarity measure is given, accord-ing to the formulation by Guo et al. [11], by
PSNRE = PSNR (EM (π’) ,EM (π’0)) ππ΅. (90)
The SSIM measures have been obtained according to theformula by Wang et al. [41] given by
SSIM (π’, π’0) = πΏ (π’, π’
0) β πΆ (π’, π’
0) β π (π’, π’
0) , (91)
where π’0denotes the noise-free image, π’ is the denoised
image,πΓπ is the dimension of image, and |max π’0βmin π’
0|
yields the gray scale range of the original image. In addition,πΏ(π’, π’
0) = (2π
π’ππ’0
+ π1)/(π
2
π’+ π
2
π’0
+ π1) compares the two
imagesβ mean luminances ππ’and π
π’0
. The maximal value ofπΏ(π’, π’
0) = 1, if π
π’= π
π’0
and πΆ(π’, π’0) = (2π
π’ππ’0
+ π2)/(π
2
π’+
π2
π’0
+ π2), measures the closeness of contrast of the two
images π’ and π’0. Contrast is determined in terms of standard
deviation, π. Contrast comparison measure πΆ(π’, π’0) = 1
maximally if and only if ππ’= π
π’0
, that is, when the imageshave equal contrast.
π (π’, π’0) = (π
π’π’0
+ π3)/(π
π’ππ’0
+ π3), where π
π’π’0
is covari-ance between π’ and π’
0, is a structure comparison measure
which determines the correlation between the images π’ andπ’0. It attains maximal value of 1 if structurally the two images
coincide, but its value is equal to zero when there is absolutelyno structural coincidence. The quantities π
1, π
2, and π
3are
small positive constants that avert the possibility of havingzero denominators.
The results of our method were compared to PMmethod[7], TV method [8], and the D-πΌ-PM method [11]. Tables 1and 2, respectively, give a summary of the results from theexperiments, using synthetic image in Figure 1 and Lenaimage in Figure 3.The parameters considered here are thresh-olding parameter πΎ, variance πΏ, the time step parameter π,and the convolution parameter π
1applied in the D-πΌ-PM
method. The fidelity parameter π was dynamically obtainedaccording to (82) or under the AOS scheme in Section 5.1.2,while the rest of the parameters were chosen to guaranteestability and attainment of optimal results.
For nontexture images as displayed in Figure 1 and theresults of our method using PMS scheme, as shown in
Abstract and Applied Analysis 11
(a) Original image (b) Noisy image (c) Proposed model I (PMS)
(d) Proposed model I (AOS) (e) PM model (f) TV model
(g) D-πΌ-PM model
Figure 1: Synthetic image (300 Γ 300). (a) Original image. (b) Noisy image corrupted by Gaussian noise for π = 30. (c) Our algorithm byPMS; π = 0.2 (262 steps). (d) Our algorithm by AOS scheme; π = 3 (17 steps). (e) PMmethod;πΎ = 5 and π = 0.25 (232 steps). (f) TVmethod;π = 0.2 (203 steps). (g) D-πΌ-PM method; π
1= 0.5, π = 30, π = 0.25, and πΎ = 1 (42 steps).
the fourth row of Table 1, demonstrate better performancethan those PM and TV as indicated by the higher PSNR,PSNRE, and SSIM values and lower MAE. And, in spiteof higher iterative steps, the corresponding CPU time islower compared to TV and the traditional Perona-Malik(PM) model. And although D-πΌ-PM model shows betterresults than PSNR and MAE results, it can be observed thatin terms of edge feature recovery measure (PSNRE) andgeneral structural coincidence measure (SSIM) our methodperforms better. Moreover, implementing our model by theOAS scheme revealed faster execution in terms of both the
CPU time and iteration steps (see Table 1). The MAE haslower and even better PSNR results than TV (see Table 1).Looking at visual results of our method using PM scheme(PMS) and AOS, respectively, in Figures 1(c) and 1(d), thereis manifestly better visual appeal for our method comparedto the PM method (see Figure 3(e)) which shows somespeckles, TVmethod (see Figure 1(f)) which exhibits staircaseeffects and slight loss in contrast, and the D-πΌ-method (seeFigure 3(g)) which shows slightly deformed edges.
However, for real image such as the given Lena image inFigure 3, the results as shown in Table 2 indicate that our
12 Abstract and Applied Analysis
0 50 100 150 200 2500
1
2
3
4
5
6
7
Pixel count (horizontal dimension)
Deg
ree o
f sim
ilarit
y
Original imageNew AOS
Γ104
(a) Proposed model I (AOS)
0 50 100 150 200 2500
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2
3
4
5
6
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ree o
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0 50 100 150 200 2500
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ree o
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0 50 100 150 200 2500
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Γ104
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0 50 100 150 200 2500
1
2
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5
6
7
Pixel count (horizontal dimension)
Deg
ree o
f sim
ilarit
y
Original imageD-πΌ-PM
Γ104
(e) D-πΌ-PM model
Figure 2: Synthetic image (300Γ300). Similarity graphs between the original image and results of our method (AOS and PMS), PMmethod,TV method, and D-πΌ-PM method, respectively.
Abstract and Applied Analysis 13
(a) Original image (b) Noisy image (c) Proposed model I (PMS)
(d) Proposed model I (AOS) (e) PM model (f) TV model
(g) D-πΌ-PM model
Figure 3: Lena image (300 Γ 300). (a) Original image. (b) Noisy image corrupted by Gaussian noise for π = 30. (c) Our algorithm by PMS;π = 0.2 (122 steps). (d) Our algorithm by AOS scheme; π = 2 (7 steps). (e) PM method; πΎ = 12 and π = 0.25 (60 steps). (f) TV method;π = 0.2 (149 steps). (g) D-πΌ-PM method; π
1= 0.5, π = 30, π = 2, and πΎ = 4 (13 steps).
method by AOS scheme gives superior results as shown bythe extremely lower iteration steps (7 steps), very short CPUprocessing time (1.24 sec), better PSNR (28.06), and evenlower MAE (7.27) compared to those of the TV method andPM method. Note that our model implemented using AOSperforms better than the samemodel implemented using PMscheme (PMS) for real images. And, in spite of the slightlysuperior PSNR and MAE values by the D-πΌ-PM method,our method not only gives better edge preservation as evi-denced by the higher PSNRE, but also gives closer structuralcoincidence than the other three models. Further, the visual
appeal of our method, whether using PMS (Figure 3(c)) orAOS scheme (Figure 3(d)), excels those of the traditionalPerona-Malik (PM) (see Figure 3(e)) method which showsspeckles and a bit of blur, TVmethod (see Figure 3(f)) whichintroduces staircasing effects on the denoised image, andeven the D-πΌ-PMmethod, which shows blockiness and someslight speckle effects.
In addition, from the similarity curves given in Figures 2and 4, it can be observed in the light of the marked areas,for instance, that this model performs better than its com-parisons, both for the synthetic image and real image. Note
14 Abstract and Applied Analysis
0 50 100 150 200 3002500
1
2
3
4
5
6
7
Pixel count (horizontal dimension)
Deg
ree o
f sim
ilarit
y
Γ104
Original imageNew AOS
(a) Proposed model I (AOS)
0 50 100 150 200 3002500
1
2
3
4
5
6
7
Pixel count (horizontal dimension)
Deg
ree o
f sim
ilarit
y
Original imageNew PMS
Γ104
(b) Proposed model I (PMS)
0 50 100 150 200 3002500
1
2
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7
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ree o
f sim
ilarit
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Original imagePM
Γ104
(c) PM model
0 50 100 150 200 3002500
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ree o
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Original imageTV
Γ104
(d) TV model
0 50 100 150 200 3002500
1
2
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4
5
6
7
Pixel count (horizontal dimension)
Deg
ree o
f sim
ilarit
y
Original imageD-πΌ-PM
Γ104
(e) D-πΌ-PM model
Figure 4: Lena image (300Γ 300). Similarity graphs between the original image and results of our method (AOS and PMS), PMmethod, TVmethod, and D-πΌ-PM method, respectively.
Abstract and Applied Analysis 15
also that, even though in performance metrics, especially interms of PSNR and MAE, the D-πΌ-PM method seems toperform better, the similarity curves attest that our methodgenerates restored images that more closely match the origi-nal image than the results obtained by the D-πΌ-PM method(see Figures 2(e) and 4(e)).
6. Conclusion
In this paper, we have proposed a modified total variationmodel based on the strictly convex modification for imagedenoising. The main idea was to offer a better image restora-tionmodel that is strictly convex and is, therefore, not subjectto backward diffusion which has the potential of introducingblurs and to limit the number of parameters available formanualmanipulation.This, probably, explains why, in spite ofthe fact that ourmodel tends toTVasπΎ tends to+β, its prac-tical implementation enjoys better image visual sharpness.This visual sharpness is comparably visible in the results byPM method, but it is still marred behind the oversmoothingand speckle effects in PM images. In fact, the reason why theD-πΌ-PM method tends to give images that have some blurand deformed edges is attributable to the backward diffusionthat it introduces in the course of diffusion. Indeed, it is diffi-cult to assure convergence to a solution of minimum energygiven the backward-forward motion of the diffusion processinduced by the D-πΌ-PM algorithm. At the practical level, theincreased number of parameters in the D-πΌ-PM algorithmtends to make it difficult to arrive at a definite permutation ofparameter values that would give an optimal result.
For the proposed method, we have demonstrated theexistence and uniqueness of the solution of the model.Moreover, numerical implementation of our model also doesnot suffer potential inaccuracies typical of the TV method,since we do not need to add any small perturbation constantin TV method. Our thresholding parameter πΎ dependson the evolution parameter π‘ and therefore does not haveto be constrained to smaller values as in the case of PMmodel. The resultant evolution equation has been discretizedand implemented using PM scheme and AOS scheme todemonstrate the performance of our algorithm. From thegiven experimental results, the PSNR values, MAE values,Iterative steps, the CPU processing time, PSNRE, SSIM, andvisual appeal of our denoised images all testify that ourmethod is actually a good balance of the PM, TV, and D-πΌ-PM methods and hence a better image restoration model.
However, in real life application, we observe that the suc-cess of any denoising algorithm depends on the type of imagebeing considered, the type of noise, the application intendedfor the results of the restoration, the extent of degradation,and indeed the implementation platform. And although wehave only considered additive Gaussian white noise, differentkinds of noise (whether Poisson, speckle, salt, or paper,among others) will require different formulations for thefidelity part and may even demand the application of morethan just one formulation for effectiveness. The choice ofplatform and scheme of implementation must also be appro-priately made for efficient performance of the formulation.
With respect to the use of the restoration results, thereare situations where the noise removal, generally, may becounterproductive. This usually occurs when the oscillationsdue to the noise are of comparable scale to those of thefeatures being targeted for preservation. A case like this mayrequire a combination of formulations [42, 43].
For images that are heavily degraded, it might be nec-essary to do a preconvolution, to blur the noise effects, andthen to use an effective formulation such as this one torecover semantically important features such as the edges andcontours of the image.
Conflict of Interests
The authors declare that they have no conflict of interestswhatsoever and do approve the publication of this paper.
Acknowledgments
This work is partially supported by the National ScienceFoundation of China (11271100 and 11301113), the Ph.D.Programs Foundation of Ministry of Education of China(no. 20132302120057), the class General Financial Grantfrom the China Postdoctoral Science Foundation (Grantno. 2012M510933), the Fundamental Research Funds for theCentral Universities (Grant nos. HIT. NSRIF. 2011003 andHIT. A. 201412), the Program for Innovation Research ofScience in Harbin Institute of Technology (Grant no. PIRSOF HIT A201403), and Harbin Science and TechnologyInnovative Talents Project of Special Fund (2013RFXYJ044).
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