Research Article Compressed Sensing MRI Reconstruction …downloads.hindawi.com/journals/jece/2015/175231.pdf · 2019-07-31 · Compressed sensing (CS) based methods have recently

Research ArticleCompressed Sensing MRI Reconstruction Algorithm Based onContourlet Transform and Alternating Direction Method

Zhenyu Hu Qiuye Wang Congcong Ming Lai Wang Yuanqing Hu and Jian Zou

School of Information and Mathematics Yangtze University Hubei 434020 China

Correspondence should be addressed to Jian Zou zoujianyangtzeueducn

Received 29 July 2015 Accepted 20 October 2015

Academic Editor Gianluca Setti

Copyright copy 2015 Zhenyu Hu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Compressed sensing (CS) based methods have recently been used to reconstruct magnetic resonance (MR) images fromundersampled measurements which is known as CS-MRI In traditional CS-MRI wavelet transform can hardly capture theinformation of image curves and edges In this paper we present a new CS-MRI reconstruction algorithm based on contourlettransform and alternating direction method (ADM) The MR images are firstly represented by contourlet transform which candescribe the imagesrsquo curves and edges fully and accuratelyThen theMR images are reconstructed by ADM which is an effective CSreconstruction method Numerical results validate the superior performance of the proposed algorithm in terms of reconstructionaccuracy and computation time

1 Introduction

CS is a new sampling and compression theory It utilizesthe sparseness of a signal in a particular domain and canreconstruct the signal from significantly fewer samples thanNyquist sampling which has been the fundamental principlein signal processing for many years [1ndash3] Due to the aboveadvantages CS has received considerable attentions in manyareas one of which is MRI reconstruction [4 5] MRI is safermore frequent and accurate for clinical diagnosis Howeverconventional MRI needs to spend much time scanning bodyregions causing the expensive cost and the nonidealizedspace resolution In themeantime the physiological propertyin the tested bodywillmake the image blurry anddistortionalTherefore under the premise of guaranteeing the image qual-ity speeding up theMRI compression and reconstruction hasbeen the powerful impetus to promote the development ofMRI techniques

For CS-MRI there are two key points that need furtherinvestigation The first one is sparse transform In MRIreconstruction the MR images themselves are not sparse buthave sparse representations in some transform domains Intraditional CS-MRI wavelet transform is commonly usedas a sparse transform [6 7] However as the limitations of

direction wavelet transform can hardly capture the infor-mation of image curves and edges fully and accuratelyIn contrast curves and edges are mainly features of MRimages Therefore more effective sparse transform should beconsidered for CS-MRI Contourlet transform also knownas Pyramid Directional Filter Bank (PDFB) is put forwardto make up for the inadequacy of the wavelet transform [8]Contourlet transform can describe the imagersquos contour anddirectional texture information fully and accurately since itrealizes any directional decomposition at each scale Further-more contourlet is constructed directly in a discrete domainand has low computing complexity Thus contourlet trans-form can be easily implemented for MR images [9 10]

The second one is the reconstruction algorithm In recentyears a number of algorithms have been put forward for thesignal reconstruction in CS for example interior-point algo-rithm [11] iterative shrinkagethresholding algorithm (ISTA)[12] fast iterative shrinkagethreshold algorithm (FISTA)[13] Sparse Reconstruction by Separable Approximation(SpaRSA) [14] But not all of these algorithms are suitable forCS-MRI since the dimensions of the MR images are hugeAlternating directionmethod (ADM) is an efficientCS recon-struction algorithm that has a faster convergence speed thansome traditional methods [15] Meanwhile ADM is able to

Hindawi Publishing CorporationJournal of Electrical and Computer EngineeringVolume 2015 Article ID 175231 5 pageshttpdxdoiorg1011552015175231

2 Journal of Electrical and Computer Engineering

solve large-scale CS problem since all the iterations of ADMonly contain the first-order information of the objectivefunction which have low computing complexity

In this paper we present a new CS-MRI reconstructionalgorithm based on contourlet transform and ADM Theproposed algorithm can recover the curves and edges of aMRimage more precisely and suit large-scale MRI reconstruc-tion

The organization of the rest of this paper is as followsin Section 2 we first introduce CS-MRI model briefly andthen present our new algorithm Numerical results willdemonstrate the effectiveness of the proposed algorithm inSection 3 Finally we conclude the paper in Section 4

2 ADM for Contourlet-Based CS-MRI

21 Contourlet-Based CS-MRI Model The basic problem ofCS is to recover a signal x from underdetermined linear mea-surement y = Φx where y isin R119898 x isin R119899 andΦ isin R119898times119899 119898 lt

119899 This underdetermined linear system has infinite solutionswhen seen from the aspect of algebra However according tothe CS theory under the assumption that x is sparse x can bereconstructed by the following optimization problem

minx x

0

st y = Φx(1)

where x0is the ℓ

0normwhichmeans the nonzero numbers

of xProblem (1) is difficult to solve since it is NP-hard It can

be relaxed as the following convex problem

minx x

1

st y = Φx(2)

where x1= sum119894|119909119894| is the sum of absolute values of x

Problem (2) is a convex optimization problem and can besolved by many algorithms

Problems (1) and (2) are under the assumption that xis sparse However in many applications the signal itself isnot sparse but has a sparse representation in some transformdomains For example x = Ψ120579 where x is the original signalwhich is not sparse and 120579 is the sparse coefficient with respectto the sparse transform matrix Ψ In this case CS modelshould be y = Φ120579 = ΦΨ

lowastx = Ax where Ψlowast denotes theinverse of Ψ The optimization problem should be changedas follows

minx

1003817100381710038171003817

Ψlowastx10038171003817100381710038171

st y = Ax(3)

22 ADM for Contourlet-Based CS-MRI In this subsectionwe solve (3) by ADMWe first introduce an auxiliary variabler and transform (3) into an equivalent problem

minxr

1003817100381710038171003817

Ψlowastx10038171003817100381710038171 Ax + r = y r le 120575 (4)

The augmented Lagrangian function of problem (4) isgiven by

minxr

119891 (x r) = 1003817100381710038171003817

Ψlowastx10038171003817100381710038171 minus 120582

T(Ax + r minus y)

+

120573

2

1003817100381710038171003817

Ax + r minus y1003817100381710038171003817

2 r le 120575

(5)

where 120582 isin R119898 is a Lagrangian multiplier and 120573 gt 0 is apenalty parameter

If we fix x = xk 120582 = 120582k that is

119891 (xk r) = 10038171003817100381710038171003817

Ψlowastxk100381710038171003817

100381710038171 minus (120582

k)

T(Axk + r minus y)

+

120573

2

10038171003817100381710038171003817

Axk + r minus y10038171003817100381710038171003817

2

(6)

The objective function 119891(xk r) is only connected with r then(5) is equivalent to

minr119891 (xk r) r le 120575 (7)

For problem (7) since (119889119889r)119891(xk r) = minus(120582k)T+120573(Axk +r minus y)T = 0 we have r = 120582k120573 minus (Axk minus y) The minimizationof (7) with respect to r is shown by

rk+1 = Ρ119861120575(

120582k

120573

minus (Axk minus y)) (8)

where Ρ119861120575

is the projection onto the set B120575 r isin R119898 r le

120575Then we fix r = rk+1 120582 = 120582120581 that is

119891 (x rk+1) = 1003817100381710038171003817

Ψlowastx10038171003817100381710038171 minus (120582

k)

T(Ax + rk+1 minus y)

+

120573

2

10038171003817100381710038171003817

Ax + rk+1 minus y10038171003817100381710038171003817

2

(9)

The objective function 119891(x rk+1) is only relevant to xthen problem (5) is equivalent to

minx119891 (x rk+1) (10)

Equation (10) is equivalent approximately to

minx1003817100381710038171003817

Ψlowastx10038171003817100381710038171 +

120573

2

100381710038171003817100381710038171003817100381710038171003817

Ax + rk+1 minus y minus 120582k

120573

100381710038171003817100381710038171003817100381710038171003817

2

(11)

Journal of Electrical and Computer Engineering 3

Require A y r0 x01205820 120573 gt 0 120574 gt 0 Γ gt 0Ensure xwhile ldquostopping criterion is not metrdquo do

rk+1 = Ρ119861120575(

120582k

120573

minus (Axk minus y))

xk+1 = Shrink(xk minus Γgk Γ120573

Ψlowast

)

120582k+1

= 120582kminus 120574120573 (Axk+1 + rk+1 minus y)

end while

Algorithm 1

Let ℎ(x) = (1205732)Ax + rk+1 minus y minus 120582k1205732 then

ℎ (x) =120573

2

[

100381710038171003817100381710038171003817100381710038171003817

Axk + rk+1 minus y minus 120582k

120573

100381710038171003817100381710038171003817100381710038171003817

2

+ (2AT(Axk + rk+1 minus y minus 120582

k

120573

))

T

+

1

2

(x minus xk)T2ATA (x minus xk)]

= 120573 [(gk)T(x minus xk) + 1

2

10038171003817100381710038171003817

A (x minus xk)10038171003817100381710038171003817

2]

= 120573 [(gk)T(x minus xk) + 1

2Γ

10038171003817100381710038171003817

x minus xk10038171003817100381710038171003817

2]

(12)

where gkΔAT(Axk + rk+1 minus y minus 120582

k120573) is the gradient of

quadratic items about (12) Γ gt 0 is a parameter and the firstequation of (12) is based on the Taylor expansionThen (11) isequivalent to

minx

1003817100381710038171003817

Ψlowastx10038171003817100381710038171 + 120573((g

k)

T(x minus xk) + 1

2Γ

10038171003817100381710038171003817

x minus xk10038171003817100381710038171003817

2) (13)

Problem (13) is equivalent approximately to

minxΨlowast

(x1 +120573

2ΓΨlowast

10038171003817100381710038171003817

x minus (xk minus Γgk)10038171003817100381710038171003817

) (14)

which has a close form solution by shrinkage (or softthresholding) formula


Ψlowast

) (15)

Finally we update the multiplier 120582 through

120582k+1

= 120582kminus 120574120573 (Axk+1 + rk+1 minus y) (16)

where 120574 gt 0 is a constantNow we present alternating direction method for prob-

lem (3) as Algorithm 1

Table 1 Comparisons of different algorithms

PSNR (dB) CPU time (s)Algorithm 1 with Contourlet 4828 452SparseMRI 3671 1617ICOTA 4579 1546FICOTA 4693 779SpaRSA 3846 1452Algorithm 1 with wavelet 4227 421

3 Numerical Experiments

In this section we present numerical results to illustrate theperformance of the proposed algorithm for MRI reconstruc-tion All experiments are made by using MATLAB 780 onthe PC with Intel Core 34GHz and 4G memory

We compare the proposed Algorithm 1 with the state-of-the-art MRI reconstruction algorithms SparseMRI [5]ICOTA [9] FICOTA [10] and SpaRSA [14] SparseMRI isbased on conjugate gradient (CG) ICOTA is based on con-tourlet transform and ISTA FICOTA is based on contourlettransform and FISTA and SpaRSA is an effective algorithmto solve the CS problemWe also compare the reconstructionperformance of contourlet and wavelet We quantify thereconstruction performance by peak signal to noise ratio(PSNR) and CPU time PSNR is defined as

PSNR = 10log10(

255

2

MSE) (17)

where MSE = (1119898119899)sum

119898minus1

119894=0sum

119899minus1

119895=0(119868ori(119894 119895) minus 119868rec(119894 119895))

2 119868oriand 119868rec are the original image and reconstructed imagerespectively and119898 119899 are the size of the images

In the following experiments we choose the measure-ment matrix Φ as a partial Fourier transform matrix with 119898rows and 119899 columns and define the sampling ratio as119898119899TheMR scanning time is less if the sampling ratio is lower Weuse the variable density sampling strategy just as the previousworks in papers [4 5 9 10] which randomly choose moreFourier coefficients from low frequencies and less coefficientsfrom high frequencies

In the first experiment we use the MR image asFigure 1(a) shows and the variable density sampling patternas Figure 1(b) shows For wavelet transformwe useDaubech-ies wavelet with 4 vanishing moments and contourlet trans-formwith decomposition [5443] just the same as paper [9]

Figure 1 shows the reconstructed images using differentalgorithms Table 1 summarizes the comparisons of differentalgorithms From Table 1 we can see that Algorithm 1 withcontourlet transform outperforms Algorithm 1 with wavelettransform in terms of PSNR although its running timeis slightly slower and Algorithm 1 outperforms other algo-rithms in terms of both PSNR and CPU time

In the second experiment we illustrate the reconstructionperformance of each algorithm as the sampling rate variesfrom 01 to 09

Figure 2 shows the variations of PSNR and CPU time ofCS-MRI reconstruction versus sampling rates for different


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

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

Figure 1 Comparison of CS-MRI reconstruction results obtained from different algorithms (a) Original MR image (b) Variable densitysampling pattern with sampling rate 02 ((c)ndash(h)) Reconstructed result from Algorithm 1 with contourlet transform SparseMRI ICOTAFICOTA SpaRSA and Algorithm 1 with wavelet transform

01 02 03 04 05 06 07 08 0935

40

45

50

55

60

65

70

Sampling rate

PSN

R

Algorithm 1 with contourletSparseMRIICOTA

FICOTASpaRSAAlgorithm 1 with wavelet

(a)

01 02 03 04 05 06 07 08 09Sampling rate



0

5

10

15

20

25

CPU

tim

e

(b)

Figure 2 Comparison of CS-MRI reconstruction results obtained from different algorithms with different sampling rates The results areaverage of 50 runs (a) PSNR versus sampling rate (b) CPU time versus sampling rate


algorithms Figure 2(a) shows that PSNR of Algorithm 1 withcontourlet is better than other algorithms with samplinggrowing Furthermore Figure 2(a) also shows that the advan-tage of contourlet becomes less obvious with the increaseof sampling rate Figure 2(b) indicates that Algorithm 1 withcontourlet is slightly slower than Algorithm 1 with waveletbut is much faster than other algorithms

4 Conclusion

In this paper we propose a novel algorithm based oncontourlet transform and the classic alternating directionmethod to solve CS-MRI problem The proposed algorithmhas low computational complexity and is suitable for largescale problem Our numerical results show that the proposedalgorithm compares favorably with these algorithms referredto in terms of PSNR and CPU time

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Undergraduate Training Programof Yangtze University for Innovation and Entrepreneurship(104892014034) and Research Funding of Yangtze Universityfor Basic Subject (2013cjp21)

References

[1] D L Donoho ldquoCompressed sensingrdquo IEEE Transactions onInformation Theory vol 52 no 4 pp 1289ndash1306 2006

[2] Y C Eldar and G Kutyniok Compressed Sensing Theory andApplications Cambridge University Press 2012

[3] E J Candes and M B Wakin ldquoAn introduction to compressivesamplingrdquo IEEE Signal Processing Magazine vol 25 no 2 pp21ndash30 2008

[4] M Lustig D L Donoho J M Santos and J M PaulyldquoCompressed sensing MRI a look at how CS can improve oncurrent imaging techniquesrdquo IEEE Signal Processing Magazinevol 25 no 2 pp 72ndash82 2008

[5] M Lustig D Donoho and J M Pauly ldquoSparse MRI theapplication of compressed sensing for rapid MR imagingrdquoMagnetic Resonance in Medicine vol 58 no 6 pp 1182ndash11952007

[6] D Zhao H Du Y Han and W Mei ldquoCompressed sensingMR image reconstruction exploiting TGV andwavelet sparsityrdquoComputational and Mathematical Methods in Medicine vol2014 Article ID 958671 11 pages 2014

[7] C Deng S Wang W Tian Z Wu and S Hu ldquoApproximatesparsity and nonlocal total variation based compressive MRimage reconstructionrdquo Mathematical Problems in Engineeringvol 2014 Article ID 137616 13 pages 2014

[8] 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

[9] X Qu W Zhang D Guo C Cai S Cai and Z Chen ldquoIterativethresholding compressed sensing MRI based on contourlettransformrdquo Inverse Problems in Science and Engineering vol 18no 6 pp 737ndash758 2010

[10] W Hao J Li X Qu and Z Dong ldquoFast iterative contourletthresholding for compressed sensing MRIrdquo Electronics Lettersvol 49 no 19 pp 1206ndash1208 2013

[11] S-J Kim K Koh M Lustig S Boyd and D GorinevskyldquoAn interior-point method for large-scale 119897

1-regularized least

squarerdquo IEEE Journal on Selected Topics in Signal Processing vol1 no 4 pp 606ndash617 2007

[12] I Daubechies M Defrise and C De Mol ldquoAn iterative thresh-olding algorithm for linear inverse problems with a sparsityconstraintrdquo Communications on Pure and Applied Mathematicsvol 57 no 11 pp 1413ndash1457 2004

[13] A Beck and M Teboulle ldquoA fast iterative shrinkage-thresholding algorithm for linear inverse problemsrdquo SIAMJournal on Imaging Sciences vol 2 no 1 pp 183ndash202 2009

[14] S J Wright R D Nowak and M A Figueiredo ldquoSparsereconstruction by separable approximationrdquo IEEE Transactionson Signal Processing vol 57 no 7 pp 2479ndash2493 2009

[15] J Yang and Y Zhang ldquoAlternating direction algorithms for 1198971-

problems in compressive sensingrdquo SIAM Journal on ScientificComputing vol 33 no 1 pp 250ndash278 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



solve large-scale CS problem since all the iterations of ADMonly contain the first-order information of the objectivefunction which have low computing complexity

In this paper we present a new CS-MRI reconstructionalgorithm based on contourlet transform and ADM Theproposed algorithm can recover the curves and edges of aMRimage more precisely and suit large-scale MRI reconstruc-tion

The organization of the rest of this paper is as followsin Section 2 we first introduce CS-MRI model briefly andthen present our new algorithm Numerical results willdemonstrate the effectiveness of the proposed algorithm inSection 3 Finally we conclude the paper in Section 4

2 ADM for Contourlet-Based CS-MRI

21 Contourlet-Based CS-MRI Model The basic problem ofCS is to recover a signal x from underdetermined linear mea-surement y = Φx where y isin R119898 x isin R119899 andΦ isin R119898times119899 119898 lt

119899 This underdetermined linear system has infinite solutionswhen seen from the aspect of algebra However according tothe CS theory under the assumption that x is sparse x can bereconstructed by the following optimization problem

minx x

0

st y = Φx(1)

where x0is the ℓ

0normwhichmeans the nonzero numbers

of xProblem (1) is difficult to solve since it is NP-hard It can

be relaxed as the following convex problem

minx x

1

st y = Φx(2)

where x1= sum119894|119909119894| is the sum of absolute values of x

Problem (2) is a convex optimization problem and can besolved by many algorithms

Problems (1) and (2) are under the assumption that xis sparse However in many applications the signal itself isnot sparse but has a sparse representation in some transformdomains For example x = Ψ120579 where x is the original signalwhich is not sparse and 120579 is the sparse coefficient with respectto the sparse transform matrix Ψ In this case CS modelshould be y = Φ120579 = ΦΨ

lowastx = Ax where Ψlowast denotes theinverse of Ψ The optimization problem should be changedas follows

minx

1003817100381710038171003817

Ψlowastx10038171003817100381710038171

st y = Ax(3)

22 ADM for Contourlet-Based CS-MRI In this subsectionwe solve (3) by ADMWe first introduce an auxiliary variabler and transform (3) into an equivalent problem

minxr

1003817100381710038171003817

Ψlowastx10038171003817100381710038171 Ax + r = y r le 120575 (4)

The augmented Lagrangian function of problem (4) isgiven by

minxr

119891 (x r) = 1003817100381710038171003817

Ψlowastx10038171003817100381710038171 minus 120582

T(Ax + r minus y)

+

120573

2

1003817100381710038171003817

Ax + r minus y1003817100381710038171003817

2 r le 120575

(5)

where 120582 isin R119898 is a Lagrangian multiplier and 120573 gt 0 is apenalty parameter

If we fix x = xk 120582 = 120582k that is

119891 (xk r) = 10038171003817100381710038171003817

Ψlowastxk100381710038171003817

100381710038171 minus (120582

k)

T(Axk + r minus y)

+

120573

2

10038171003817100381710038171003817

Axk + r minus y10038171003817100381710038171003817

2

(6)

The objective function 119891(xk r) is only connected with r then(5) is equivalent to

minr119891 (xk r) r le 120575 (7)

For problem (7) since (119889119889r)119891(xk r) = minus(120582k)T+120573(Axk +r minus y)T = 0 we have r = 120582k120573 minus (Axk minus y) The minimizationof (7) with respect to r is shown by

rk+1 = Ρ119861120575(

120582k

120573

minus (Axk minus y)) (8)

where Ρ119861120575

is the projection onto the set B120575 r isin R119898 r le

120575Then we fix r = rk+1 120582 = 120582120581 that is

119891 (x rk+1) = 1003817100381710038171003817

Ψlowastx10038171003817100381710038171 minus (120582

k)

T(Ax + rk+1 minus y)

+

120573

2

10038171003817100381710038171003817

Ax + rk+1 minus y10038171003817100381710038171003817

2

(9)

The objective function 119891(x rk+1) is only relevant to xthen problem (5) is equivalent to

minx119891 (x rk+1) (10)

Equation (10) is equivalent approximately to

minx1003817100381710038171003817

Ψlowastx10038171003817100381710038171 +

120573

2

100381710038171003817100381710038171003817100381710038171003817

Ax + rk+1 minus y minus 120582k

120573

100381710038171003817100381710038171003817100381710038171003817

2

(11)



rk+1 = Ρ119861120575(

120582k

120573



Ψlowast

)

120582k+1


end while

Algorithm 1


ℎ (x) =120573

2

[

100381710038171003817100381710038171003817100381710038171003817


120573

100381710038171003817100381710038171003817100381710038171003817

2


k

120573

))

T

+

1

2


= 120573 [(gk)T(x minus xk) + 1

2

10038171003817100381710038171003817

A (x minus xk)10038171003817100381710038171003817

2]

= 120573 [(gk)T(x minus xk) + 1

2Γ

10038171003817100381710038171003817

x minus xk10038171003817100381710038171003817

2]

(12)




minx

1003817100381710038171003817

Ψlowastx10038171003817100381710038171 + 120573((g

k)

T(x minus xk) + 1

2Γ

10038171003817100381710038171003817

x minus xk10038171003817100381710038171003817

2) (13)


minxΨlowast

(x1 +120573

2ΓΨlowast

10038171003817100381710038171003817


) (14)



Ψlowast

) (15)


120582k+1









PSNR = 10log10(

255

2

MSE) (17)

where MSE = (1119898119899)sum

119898minus1

119894=0sum

119899minus1

119895=0(119868ori(119894 119895) minus 119868rec(119894 119895))








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

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


01 02 03 04 05 06 07 08 0935

40

45

50

55

60

65

70

Sampling rate

PSN

R



(a)

01 02 03 04 05 06 07 08 09Sampling rate



0

5

10

15

20

25

CPU

tim

e

(b)




4 Conclusion




Acknowledgments


References












1-regularized least









RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











rk+1 = Ρ119861120575(

120582k

120573



Ψlowast

)

120582k+1


end while

Algorithm 1


ℎ (x) =120573

2

[

100381710038171003817100381710038171003817100381710038171003817


120573

100381710038171003817100381710038171003817100381710038171003817

2


k

120573

))

T

+

1

2


= 120573 [(gk)T(x minus xk) + 1

2

10038171003817100381710038171003817

A (x minus xk)10038171003817100381710038171003817

2]

= 120573 [(gk)T(x minus xk) + 1

2Γ

10038171003817100381710038171003817

x minus xk10038171003817100381710038171003817

2]

(12)




minx

1003817100381710038171003817

Ψlowastx10038171003817100381710038171 + 120573((g

k)

T(x minus xk) + 1

2Γ

10038171003817100381710038171003817

x minus xk10038171003817100381710038171003817

2) (13)


minxΨlowast

(x1 +120573

2ΓΨlowast

10038171003817100381710038171003817


) (14)



Ψlowast

) (15)


120582k+1









PSNR = 10log10(

255

2

MSE) (17)

where MSE = (1119898119899)sum

119898minus1

119894=0sum

119899minus1

119895=0(119868ori(119894 119895) minus 119868rec(119894 119895))








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

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


01 02 03 04 05 06 07 08 0935

40

45

50

55

60

65

70

Sampling rate

PSN

R



(a)

01 02 03 04 05 06 07 08 09Sampling rate



0

5

10

15

20

25

CPU

tim

e

(b)




4 Conclusion




Acknowledgments


References












1-regularized least









RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










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

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


01 02 03 04 05 06 07 08 0935

40

45

50

55

60

65

70

Sampling rate

PSN

R



(a)

01 02 03 04 05 06 07 08 09Sampling rate



0

5

10

15

20

25

CPU

tim

e

(b)




4 Conclusion




Acknowledgments


References












1-regularized least









RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











4 Conclusion




Acknowledgments


References












1-regularized least









RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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