research article a nonmonotone gradient algorithm for...

Research ArticleA Nonmonotone Gradient Algorithm for TotalVariation Image Denoising Problems

Peng Wang12 Shifang Yuan1 Xiangyun Xie1 and Shengwu Xiong2

1School of Mathematics and Computational Science Wuyi University Jiangmen 529020 China2School of Computer Science and Technology Wuhan University of Technology Wuhan 430070 China

Correspondence should be addressed to Peng Wang p wong126com

Received 3 June 2016 Accepted 1 August 2016

Academic Editor Raffaele Solimene

Copyright copy 2016 Peng Wang 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

The total variation (TV)model has been studied extensively because it is able to preserve sharp attributes and capture some sparselycritical information in images However TV denoising problem is usually ill-conditioned that the classical monotone projectedgradient method cannot solve the problem efficiently Therefore a new strategy based on nonmonotone approach is digged out asaccelerated spectral project gradient (ASPG) for solving TV Furthermore traditional TV is handled by vectorizing which makesthe scheme farmore complicated for designing algorithms In order to simplify the computing process a new technique is developedin view of matrix rather than traditional vector Numerical results proved that our ASPG algorithm is better than some state-of-the-art algorithms in both accuracy and convergence speed

1 Introduction

Blur and noise are unavoidable during the acquisition andtransmission of images and this may be introduced by thepoor lighting conditions faulty camera or unperfect trans-mission channels and so on It is interesting and challengingto restore the ideal image from its degraded version Severaleffective mathematical models and algorithms have emergedsuch as partial differential equations [1] sparse-land models[2 3] and regularization [2]

This paper mainly elaborated on one of the most popularand effective tools for image denoising problem named TV[3ndash5] as it has been shown to preserve sharp edges bothexperimentally and theoretically In view of the good perfor-mance in image processing TV has been extensively studiedin many image processing fields such as denoising [3 4]deblurring [6] and inpainting [5 7] Meanwhile variousalgorithms [3 5ndash8] for solving TV are also proposed Eventhough there are so many algorithms most of them aremerely related to vectorization [3 6] As images are two-dimensional signals it is more intuitive and suitable to proc-ess by matrix than by vector Furthermore it is more efficientand simpler for image processing in a matrix view In this

work some universal and useful properties of TV are given inmatrix version while it also applies to other algorithms evenwithout any modification

Over the last few decades there have been many differentvariations of the projected gradient method (PG) to solveTV denoising problems [9ndash11] Generally the classical PGmethod has complexity result of 119874(1119896) as one of its draw-backs [12] where 119896 is the iteration counter Many authorsdevoted themselves into enhancing PG Birgin et al [13 14]imposed a nonmonotone technique to classical PG and devel-oped the spectral projected gradient (SPG)method Nesterov[15] gave a concise and efficient strategy which improved theconvergence speed of PG to 119874(1119896

2) Recently similar tech-

niques were tactfully managed to different traditional algo-rithms [8 10 11 16 17] and corresponding optimal complexityresults were obtained Beck and Teboulle [10] proposed afast duality-based gradient projection methods to solve TVChambolle and Pock [11] and Goldfarb et al [16] acceleratedthe classical primal-dual algorithm and alternating directionaugmented Lagrangian methods respectively more recentlyOuyang et al [17] adopted the same technique to enhance theconvergence of alternating direction method of multipliers(ADMM) Though so many studies have been done based

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4310612 9 pageshttpdxdoiorg10115520164310612

2 Mathematical Problems in Engineering

on Nesterovrsquos theory to the best of our knowledge no workexplores the possibility between Nesterovrsquos scheme [15] andthe spectral projected gradient algorithm Although the SPGapproach is interesting the strong disturbance always occursbefore its convergence and thus casts a shadow over theapproachmeanwhile the key thought inNesterovrsquos work [15]is that a combination of past iterates is used to compute thenext iterateThis work is purposed to reduce the SPGrsquos distur-bance byNesterovrsquos method and propose an accelerated spec-tral projected gradient (ASPG) for TV denoising problem

2 Primal-Dual Framework ofTotal Variation Denoising

In this section we consider the TV denoising problem whichis formulated as

min119906isinΩ

119906 minus 1198872+ 2120582 119906TV (1)

where 119906 and 119887 denote an ideal and the noised imagerespectively the box region Ω is practically selected as[0 1] or [0 255] to constrain the value of 119906 and 119906TVis called TV seminorm usually divided into isotropic andanisotropic types which are respectively defined as the paperof Chambolle [9] It is worth mentioning that our approachcan be applied to TV

119868and TV

119860 However for the sake of

compactness we will put emphasis on TV119868

21 Notations and Outline To smooth the process somemain notations are introduced here The standard innerproduct in Euclidean space 119877

119898times119899 is defined as ⟨119860 119861⟩ =

tr(119860119879119861) Denote the right linear space in Euclidean space119877(119898minus1)times119899

times 119877119898times(119899minus1)

= [119860 119861] | 119860 isin 119877(119898minus1)times119899

119861 isin 119877119898times(119899minus1)

For convenience we writeC = 119877

(119898minus1)times119899times 119877119898times(119899minus1) Thus we

can define the inner product as

⟨[1198601 1198611] [1198602 1198611]⟩ = tr (119860119879

21198601) + tr (119861119879

21198611)

forall [119860119894 119861119894] isin 119877(119898minus1)times119899


(119894 = 1 2)

(2)

The rest of this paper is organized as follows In the follow-ing subsections some important operators in matrix versionfor solving TV followed by the algorithm for TV-based imagedenoising are explained In Section 3 the spectral projectedgradient algorithm is first recapped and then the acceleratedversion ASPG is derived by it To show the effectiveness ofthe proposed approach numerical comparisons with existingstate-of-the-art methods are carried out in Section 4 FinallySection 5 contains the conclusion

22 Some Preliminaries for TV Denoising In this section wewill define some useful operators which are necessary forour algorithm The operators are defined in matrix version

instead of the traditional vector version [3 6] which makesTV more understandable and concise

(a) Subset 119878 is a subset in the inner product space C asfollows

119878 = (1198751 1198752) | (1198751 1198752) isin C 119901

2

1(119894119895)+ 1199012

2(119894119895)

le 1 (for 119894 = 1 (119898 minus 1) 119895 = 1 (119899 minus 1)) 1199011(119894119899)

le 1 (for 119894 = 1 (119898 minus 1)) 1199012(119898119895)

le 1 (for 119895 = 1 (119899 minus 1))

(3)

where 119901119896(119894119895)

is the entry of 119875119896(for 119896 = 1 2)

(b) Projected Operator In this paper two kinds of projectionoperators 119875

Ω(119906) and 119875

119878(P) are necessary An image 119906 pro-

jected onto a box setΩ equiv [1198871 1198872] is easily computed

(119875Ω119906)119894119895=

1198871

if 119906119894119895lt 1198871

119906119894119895

if 1198871le 119906119894119895le 1198872

1198872

if 119906119894119895gt 1198872

(4)

Taking P = (1198751 1198752) isin C for TV

119868problem (119875

119878P)119894119895

isin 119878 isgiven by

(1198751198781198751)119894119895

=

(1198751)119894119895

max 1 10038161003816100381610038161003816(1198751)11989411989910038161003816100381610038161003816

if 119894 = 1 (119898 minus 1)

(1198751)119894119895

max 1radic(1198751)2

119894119895+ (1198752)2

119894119895

otherwise

(1198751198781198752)119894119895

=

(1198752)119894119895

max 1 10038161003816100381610038161003816(1198752)11989411989910038161003816100381610038161003816

if 119894 = 1 (119898 minus 1)

(1198752)119894119895


119894119895+ (1198752)2

119894119895

otherwise

(5)

while for TV119860problem the projected operator is defined as

(1198751198781198751)119894119895=

(1198751)119894119895

max 1100381610038161003816100381610038161003816(1198751)119894119895

100381610038161003816100381610038161003816

(1198751198781198752)119894119895=

(1198752)119894119895

max 1100381610038161003816100381610038161003816(1198752)119894119895

100381610038161003816100381610038161003816

(6)

Mathematical Problems in Engineering 3

(c) Gradient Operator If an image 119906 isin 119877119898times119899 the gradient is

an element inDu isin C defined as Du = (119863119909119906119863119910119906) with

(119863119909119906)119894119895= 119906119894119895minus 119906119894+1119895

119894 = 1 (119898 minus 1) 119895 = 1 119899

(119863119910119906)119894119895= 119906119894119895minus 119906119894119895+1

119894 = 1 119898 119895 = 1 (119899 minus 1)

(7)

(d) Divergence Operator The divergence of a vector P =

(1198751 1198752) isin 119878 is an array in 119877

119898times119899 computed by

(div P)119894119895= 1199011(119894119895)

minus 1199011(119894minus1119895)

+ 1199012(119894119895)

minus 1199012(119894119895minus1)

(8)

which implies that 1199011(0119895)

= 1199011(119898119895)

= 1199012(1198940)

= 1199012(119894119899)

equiv 0 for119894 = 1 119898 119895 = 1 119899

It should bementioned that the operators (b) (c) and (d)are defined with the assumption of periodic boundary whichis adopted in the rest of this paper

23 The Primal Dual Solution of TV As the denoising prob-lem (1) is not differentiable the traditional optimal methodsare not applicable Though there are so many techniques toovercome the difficulty we employ the scheme of Chambolle[9 18] for its effectiveness and simpleness

Now we give the TV-based image denoising algorithm inprimal-dual view which mainly contains two steps

Algorithm 1 (TV-based image denoising (TVID))Input119887 observed image120582 regularization parameterStep 0 Take P

0= (0(119898minus1)times119899

0119898times(119899minus1)

)Step 1 119906 = 119875

Ω(119887 minus 120582 div P) Compute

Step 2 Plowast = argminPisin119878119887 minus 120582 div P2119865minus 119906lowastminus 119887 +

120582 div P2119865

Output 119906lowast = 119875Ω(119887 minus 120582 div Plowast)

Theorem 2 The TVID algorithm can obtain an analyticsolution of the TV image denoising problem (1)

Proof According to the primal-dual approach [9 18] the TVseminorm can be written as follows

119906TV = supPisin119878

⟨119906 div P⟩ (9)

Substituting (9) into (1) then we get

min119906isinΩ

maxPisin119878

119906 minus 1198872+ 2120582 ⟨119906 div P⟩ (10)

By the min-max theorem the above equation is equivalent to

maxPisin119878

min119906isinΩ

119906 minus 1198872+ 2120582 ⟨119906 div P⟩ (11)

Obviously the inner part of the above equation

min119906isinΩ

119906 minus 1198872+ 2120582 ⟨119906 div P⟩ (12)

has an analytic solution

119906lowast= 119875Ω(119887 minus 120582 div P) (13)

where 119875Ω(120577) denotes the projection of 120577 onto the convex set

Ω

3 Accelerated Spectral ProjectedGradient Based TV Denoising

In this section first there will be a brief on spectral projectedgradient method [14] Next an accelerated strategy will beused to enhance SPG

31 Introduction to Spectral Projected Gradient Method Bycombining Barzilai-Borwein nonmonotone [19] ideas withtraditional projected gradient algorithm Birgin et al [13 14]developed a new gradient algorithm that is SPGwhich is suit-able for convex optimization problemswith projection easy tobe computed on feasible set By incorporating a spectral steplength and anonmonotone globalization strategy SPG speedsup the traditional PG and maintains it to be simple and easyto code

The SPG method aims to minimize 119891 on a closed andconvex setΩ that is

min119909isinΩ

119891 (119909) (14)

As the traditional projection method the SPG method hasthe form

119909119896+1

= 119909119896+ 120572119896119889119896 (15)

where 120572119896is the step length and the search direction 119889

119896has

been defined as [14]

119889119896= 119875Ω[119909119896minus 120582119896nabla119891 (119909

119896)] minus 119909

119896 (16)

In the classical SPG method the nonmonotone decreasecriterion is realized by an integer parameter 119872 ge 1 whichguarantees the object function decrease every 119872 iterationsIn the line search stage a sufficient decrease parameter 120574 isin

(0 1) is used to satisfy Armijo criterion Besides some otherparameters are also required safeguarding parameters 0 lt

120572min lt 120572max lt infin for the spectral step length and safeguard-ing parameters 0 lt 120590

1lt 1205902lt 1 for the quadratic interpola-

tion For later reference we now recall the theorem by Birginet al [14] in which convergence of classical SPG is shown

Theorem 3 (see [14]) The spectral projected algorithm is welldefined and any accumulation point of the sequence 119909

119896 that

it generates is a constrained stationary point of (14)

32 Accelerated SPG for TVID In this section we willelaborate our algorithm on TV denoising problem (1) Fromalgorithm TVID problem (1) can be solved if and only ifStep 2 is solved Thus we proposed an accelerated spectralprojected gradient (ASPG) for solving Step 2 The proposedASPGmainly consists of two parts a nonmonotone projected


gradient for finding the feasible solutions and a combinationof the former two iterations for reducing the turbulence of thealgorithm and accelerating the convergence speed In order tosolve Step 2 in algorithmTVID substituting (13) into (11) andsimplifying yield

minPisin119878

119866 (P) = 119887 minus 120582 div P2119865minus1003817100381710038171003817119906lowastminus 119887 + 120582 div P1003817100381710038171003817

2

119865 (17)

And the gradient of (17) is

nabla119866 (P) = minus2120582119863 (119875Ω(119887 minus 120582 div P)) (18)

where 119863(sdot) can be computed by the gradient operator inSection 22 Substituting (18) into (16) and simplifying themtogether with (13) we get the search direction of ASPG

119889119896= 119875119878(P119896+ 2120582120572

119896119863(119906119896)) minus P

119896 (19)

where119875119878(sdot) is defined as the projected operator in Section 22

The nonmonotone linear search and the spectral steplength are updated just as the classical SPG [14] Followingnonmonotone line search ASPG adopts Nesterovrsquos [15] tech-nique to improve the performance of classical SPG Finally itshould be mentioned that the ASPG algorithm is used for thedual step in TVID algorithm that is the variation P in ASPGis an element of the inner product spaceC Consequently theinner product involving ASPG should be computed as (2)

Algorithm4 (accelerated spectral projected gradient (ASPG))

InputQ0isin C integer119872 ge 1 120572

0isin [120572min 120572max]

the regularization parameter 120582an integer Iter and the observed image 119887

Step 0 Take P1= Q0= (0(119898minus1)times119899

0119898times(119899minus1)

) 1199051= 1

Step 1 (119896 = 1 Iter) Compute

Step 2 Iterate

119906119896= 119875Ω(119887 minus 120582 div P

119896)

119889119896= 119875119878(P119896+ 2120582120572

119896119863(119906119896)) minus P

119896

Step 21 Nonmonotone Line SearchCompute119866max = max119866(P

119896minus119895) | 0 le 119895 le min(119896119872 minus 1)

and set 120591 larr 1

Step 211

If 119866(P119896+ 120591119889119896) le 119866max minus 2120574120591120582⟨119889

119896 119863(119906119896)⟩

then set 120591119896larr 120591 and stop

Step 212 Compute

120591119905= 1205821205912⟨119889119896 119863(119906119896)⟩2[119866(P

119896+ 120591119889119896) minus 119866(P

119896) +

2120582120591⟨119889119896 119863(119906119896)⟩]

If 120591119905isin [1205901 1205902120591] then set 120591 larr 120591

119905

Otherwise set 120591 larr 1205912Go to Step 211Step 22Q

119896= 119875119878(P119896+ 120572119889119896)

Step 3

119905119896+1

= (1 + radic1 + 41199052

119896)2

P119896+1

= Q119896+ ((119905119896minus 1)119905

119896+1)(Q119896minusQ119896minus1

)

Step 4 Compute the spectral step length

119878119896= P119896+1

minus P119896

119884119896= minus2120582(119863(119906

119896+1) minus 119863(119906

119896))

If ⟨119884119896 119878119896⟩ le 0 then set 120572

119896+1larr 120572max

Otherwise120572119896+1

= max120572minmin⟨119878119896 119878119896⟩⟨119884119896 119878119896⟩ 120572max

Output 119906lowast = 119875Ω(119887 minus 120582 div PIter)

33 Convergence Analysis of ASPG Since our method can beseen as an accelerated version of classical SPG in the dualspace its convergence can be deduced fromTheorem 3 Nowwe summarize the convergence of ASPG in the followingtheorem

Theorem 5 The ASPG for TVID is well defined and anyaccumulation point of the sequence P

119896 is a constrained

stationary point of (17) furthermore the output 119906lowast of ASPGis a stationary point of TVID problem (1)

Proof By (13) 119906lowast generated by ASPG is a stationary point of(1) if and only if P is a stationary of (17)

Equation (18) shows that the 119896th gradient of (17) can begotten by 119906

119896 that is nabla119866(P

119896) = minus2120582119863(119875

Ω(119906119896)) Substituting

it into (16) we get the search direction of ASPG that is (19)This means that the search direction of ASPG is the sameas that of the SPG direction except that (19) is in the dualspace C Furthermore the definition of gradient operator119863(sdot) shows that119863(119906

119896) is an element ofCTherefore the inner

product ⟨119889119896 119863(119906119896)⟩ can be computed by (2) Moreover both

the nonmonotone line search and the spectral step lengthmainly involve the inner products ⟨119889

119896 119863(119906119896)⟩ ⟨119878119896 119878119896⟩ and

⟨119878119896 119884119896⟩ Consequently the nonmonotone line search and the

spectral step length of ASPG are just different from those ofthe classical SPG in the inner product

Step 3 in ASPG is an additional step besides the classicalSPG It can be known that the step is a linear combinationof the previous two results which will not change the con-vergence of the whole algorithmTherefore fromTheorem 3it follows that P is a stationary of (17) This implies theconclusion

4 Numerical Experiments

In this section the proposed ASPG algorithm (ASPGProposed) is employed to solve the TVID problem fol-lowed by the comparison with some state-of-the-art algo-rithms the classical projected gradient given by Chambolle


(a) Barbara (b) Boats

(c) Satellite (d) Pepper

Figure 1 Original image

(PG Chambolle) [9] Birginrsquos spectral projected gradient [1314] (SPG Birgin) and the fast gradient projection byBeck andTeboulle [10] (FPG Beck) The effect of these experiments ismeasured by relative error peak signal-to-noise ratio (PSNR)and structural similarity index measurement (SSIM) Thetest images are 256-by-256 images as shown in Figure 1 (a)Barbara (b) Boats (c) Satellite and (d) Peppers All the imagepixel values are scaled to the interval [0 1]

In the following the applicability of the proposedalgorithm on image denoising problems is illustrated Thedegraded images are generated by adding Gaussian noisewith zero mean and standard deviation of size 005 to theoriginal images (Figure 1) Throughout the experiments theparameters involved in SPG and ASPG are set as follows120572min = 10

minus30 120572max = 1030 1205901= 01 120590

2= 09 120574 = 10

minus4and 119872 = 3 For the value of the regularization parameter 120582for all schemes tune itmanually and choose the one that givesthe most satisfactory results for all four images

The PSNR and the relative error of the four images withdifferent algorithms are shown in Table 1 To exclude therandom fact to the results the last column of Table 1 showsthe average metrics of the four tested images and the metricsof the best method for each test case are highlighted inboldface Similar to FPG Beck improving PG Chambolle byNesterovrsquos strategy the proposedASPG improves the classicalSPG Birgin Table 1 illustrates that ASPG and FPG Beck arebetter than SPG Birgin and PG Chambolle respectivelyThisfact shows that the accelerated scheme is valid for the ASPGas well as the FPG Beck Furthermore we can find that ASPGis superior to FPG Beck in that the former is based on thenonmonotonic line search Table 1 shows that among all thealgorithms the proposed ASPG algorithm performs the bestin both PSNR and relative error

The following goes with a deep comparison of thementioned algorithms To illustrate this more clearly inFigure 2 relative error PSNR and SSIM value are introduced


0

01

02

03

04

05

06

07

08

09

1

0 20 40 60 80 100 120

SPG_BirginASPG_Proposed

PG_Chambolle

FPG_Beck

(a) Relative error

5

10

15

20

25

30

35

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(b) PSNR

0

01

02

03

04

05

06

07

08

09

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(c) SSIM

Figure 2 The evaluation index

Table 1 Numerical comparison for images (a)ndash(d) in Figure 1

Barbara Boats Satellite Pepper AveragePSNR Error PSNR Error PSNR Error PSNR Error PSNR Error

PG Chambolle 288919 00735 291882 00654 295731 01077 294148 00662 292670 00782SPG Birgin 297996 00662 304961 00562 324344 00775 308096 00564 308849 00641FPG Beck 289254 00732 292779 00647 312876 00884 294227 00661 297284 00731ASPG 302991 00625 313927 00507 326485 00756 320317 00490 315930 00594

here which are related to the iterative times for noisedBarbara (generated from Figure 1(a)) From Figure 2 wecan see that the classical SPG has strong disturbance beforeits convergence while the ASPG reduces the disturbanceand accelerates the convergence speed Meanwhile Figure 2

also illustrates that the nonmonotone line search improvesall the metrics PSNR relative error and SSIM One morepoint to mention is that here we only show the relativeerror PSNR and SSIM value which are generated from thedegraded version of Figure 1(a) while for the other figures


(a) Barbara noised (b) Denoised SPG

(c) Denoised FPG (d) Denoised ASPG

Figure 3 Denoised by different algorithms

(ie Figures 1(b) 1(c) and 1(d)) they are just different invalue compared with that of Figure 1(a) but the perfor-mance priority of PG Chambolle SPG Birgin FPG Beckand ASPG is the same which all verify that ASPG is thebest Finally Figure 3(a) is corrupted fromFigure 1(a) Figures3(b)ndash3(d) illustrate the denoised results which processed bySPG Birgin FPG Beck and ASPG

Remote sensing images are widely used in so many fields[20ndash23] However noise is unavoided during the acquisitionand transmission of images To restore the the ideal imagefrom its degraded version is an interesting field of remotesensing image processing Here we employ the proposedASPG algorithm to denoise a remote sensing image (Fig-ure 4) The results are shown as that from Figures 4(a)ndash4(f)and a selected area is magnified to better prove them in moredetail Both the original and magnified areas are highlightedin green

5 Conclusions

In this paper an accelerated scheme for spectral project gra-dient is proposed for TV denoising problems To handle TV

in matrix space a concise version for total variation model isintroduced Numerical examples have been done to verify theperformance of algorithm which show that it is much betterthan that of some existing state-of-the-art methods

Competing Interests

There are no competing interests related to this paper

Acknowledgments

This work was supported by the National High-Tech RampDProgram of China (863 Program) (no 2015AA015403)National Natural Science Foundation (nos 11271040 and11361027) the Natural Science Foundation of GuangdongProvince (no 2014A030313625 no 2015A030313646 andno 2016A030313005) the Nature Science Foundation ofHubei Province (no 2015CFA059) Science amp TechnologyPillar Program of Hubei Province (no 2014BAA146) theTraining plan for the Outstanding Young Teachers in HigherEducation of Guangdong (no SYq2014002) the Key Project


250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(a) Ground truth

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(b) Noised

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(c) PG (PSNR = 283365)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(d) SPG (PSNR = 288468)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(e) FPG (PSNR = 282593)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(f) ASPG (PSNR = 288547)

Figure 4 Remote sensing image denoising


of Department of Education of Guangdong Province (no2014KZDXM055) Opening Project of Guangdong ProvinceKey Laboratory of Computational Science at the Sun Yat-senUniversity (no 2016007) Science and Technology Project ofJiangmen City and Science Foundation for Young Teachersof Wuyi University (no 2015zk09)

References

[1] W Tian T Ma Y Zheng et al ldquoWeighted curvature-preservingPDE image filtering methodrdquo Computers amp Mathematics withApplications vol 70 no 6 pp 1336ndash1344 2015

[2] W Dong L Zhang G Shi and X Wu ldquoImage deblurringand super-resolution by adaptive sparse domain selection andadaptive regularizationrdquo IEEETransactions on Image Processingvol 20 no 7 pp 1838ndash1857 2011

[3] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013

[4] L I Rudin S Osher and E Fatemi ldquoNonlinear total variationbased noise removal algorithmsrdquo Physica D Nonlinear Phenom-ena vol 60 no 1ndash4 pp 259ndash268 1992

[5] Q Cheng H Shen L Zhang and P Li ldquoInpainting for remotelysensed images with a multichannel nonlocal total variationmodelrdquo IEEE Transactions on Geoscience and Remote Sensingvol 52 no 1 pp 175ndash187 2014

[6] CWu J Zhang andX-C Tai ldquoAugmented Lagrangianmethodfor total variation restoration with non-quadratic fidelityrdquoInverse Problems and Imaging vol 5 no 1 pp 237ndash261 2011

[7] Y-W Wen R H Chan and A M Yip ldquoA primal-dual methodfor total-variation-based wavelet domain inpaintingrdquo IEEETransactions on Image Processing vol 21 no 1 pp 106ndash114 2012

[8] 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

[9] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004

[10] A Beck and M Teboulle ldquoFast gradient-based algorithms forconstrained total variation image denoising and deblurringproblemsrdquo IEEE Transactions on Image Processing vol 18 no11 pp 2419ndash2434 2009

[11] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011

[12] Y Nesterov Introductory Lectures on Convex Optimization ABasic Course vol 87 Springer Berlin Germany 2013

[13] E G Birgin J MMartınez andM Raydan ldquoSpectral projectedgradient methods review and perspectivesrdquo Journal of Statisti-cal Software vol 60 no 3 pp 1ndash21 2014

[14] E G Birgin J M Martınez and M Raydan ldquoNonmonotonespectral projected gradient methods on convex setsrdquo SIAMJournal on Optimization vol 10 no 4 pp 1196ndash1211 2000

[15] Y Nesterov ldquoA method of solving a convex programmingproblem with convergence rate o (11198962)rdquo Soviet MathematicsDoklady vol 27 pp 372ndash376 1983

[16] D Goldfarb S Ma and K Scheinberg ldquoFast alternatinglinearization methods for minimizing the sum of two convexfunctionsrdquo Mathematical Programming vol 141 no 1-2 pp349ndash382 2013

[17] Y Ouyang Y Chen G Lan and J Pasiliao Jr ldquoAn acceleratedlinearized alternating direction method of multipliersrdquo SIAMJournal on Imaging Sciences vol 8 no 1 pp 644ndash681 2015

[18] A Chambolle ldquoTotal variation minimization and a class ofbinary MRF modelsrdquo in Energy Minimization Methods inComputer Vision and Pattern Recognition A Rangarajan BVemuri and A L Yuille Eds vol 3757 of Lecture Notes inComputer Science pp 136ndash152 Springer New York NY USA2005

[19] J Barzilai and J M Borwein ldquoTwo-point step size gradientmethodsrdquo IMA Journal of Numerical Analysis vol 8 no 1 pp141ndash148 1988

[20] R Solimene A Cuccaro A DellrsquoAversano I Catapano andF Soldovieri ldquoGround clutter removal in GPR surveysrdquo IEEEJournal of Selected Topics in Applied Earth Observations andRemote Sensing vol 7 no 3 pp 792ndash798 2014

[21] E Izquierdo-Verdiguier R Jenssen L Gomez-Chova and GCamps-Valls ldquoSpectral clustering with the probabilistic clusterkernelrdquo Neurocomputing vol 149 no 5 pp 1299ndash1304 2015

[22] J C Jimenez-Munoz J A Sobrino and A R Gillespie ldquoSurfaceemissivity retrieval from airborne hyperspectral scanner datainsights on atmospheric correction and noise removalrdquo IEEEGeoscience and Remote Sensing Letters vol 9 no 2 pp 180ndash1842012

[23] D Tuia M Volpi L Copa M Kanevski and J Munoz-MarıldquoA survey of active learning algorithms for supervised remotesensing image classificationrdquo IEEE Journal of Selected Topics inSignal Processing vol 5 no 3 pp 606ndash617 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

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


on Nesterovrsquos theory to the best of our knowledge no workexplores the possibility between Nesterovrsquos scheme [15] andthe spectral projected gradient algorithm Although the SPGapproach is interesting the strong disturbance always occursbefore its convergence and thus casts a shadow over theapproachmeanwhile the key thought inNesterovrsquos work [15]is that a combination of past iterates is used to compute thenext iterateThis work is purposed to reduce the SPGrsquos distur-bance byNesterovrsquos method and propose an accelerated spec-tral projected gradient (ASPG) for TV denoising problem

2 Primal-Dual Framework ofTotal Variation Denoising

In this section we consider the TV denoising problem whichis formulated as

min119906isinΩ

119906 minus 1198872+ 2120582 119906TV (1)

where 119906 and 119887 denote an ideal and the noised imagerespectively the box region Ω is practically selected as[0 1] or [0 255] to constrain the value of 119906 and 119906TVis called TV seminorm usually divided into isotropic andanisotropic types which are respectively defined as the paperof Chambolle [9] It is worth mentioning that our approachcan be applied to TV

119868and TV

119860 However for the sake of

compactness we will put emphasis on TV119868

21 Notations and Outline To smooth the process somemain notations are introduced here The standard innerproduct in Euclidean space 119877

119898times119899 is defined as ⟨119860 119861⟩ =

tr(119860119879119861) Denote the right linear space in Euclidean space119877(119898minus1)times119899


= [119860 119861] | 119860 isin 119877(119898minus1)times119899

119861 isin 119877119898times(119899minus1)

For convenience we writeC = 119877

(119898minus1)times119899times 119877119898times(119899minus1) Thus we

can define the inner product as

⟨[1198601 1198611] [1198602 1198611]⟩ = tr (119860119879

21198601) + tr (119861119879

21198611)

forall [119860119894 119861119894] isin 119877(119898minus1)times119899


(119894 = 1 2)

(2)

The rest of this paper is organized as follows In the follow-ing subsections some important operators in matrix versionfor solving TV followed by the algorithm for TV-based imagedenoising are explained In Section 3 the spectral projectedgradient algorithm is first recapped and then the acceleratedversion ASPG is derived by it To show the effectiveness ofthe proposed approach numerical comparisons with existingstate-of-the-art methods are carried out in Section 4 FinallySection 5 contains the conclusion

22 Some Preliminaries for TV Denoising In this section wewill define some useful operators which are necessary forour algorithm The operators are defined in matrix version

instead of the traditional vector version [3 6] which makesTV more understandable and concise

(a) Subset 119878 is a subset in the inner product space C asfollows

119878 = (1198751 1198752) | (1198751 1198752) isin C 119901

2

1(119894119895)+ 1199012

2(119894119895)

le 1 (for 119894 = 1 (119898 minus 1) 119895 = 1 (119899 minus 1)) 1199011(119894119899)

le 1 (for 119894 = 1 (119898 minus 1)) 1199012(119898119895)

le 1 (for 119895 = 1 (119899 minus 1))

(3)

where 119901119896(119894119895)

is the entry of 119875119896(for 119896 = 1 2)

(b) Projected Operator In this paper two kinds of projectionoperators 119875

Ω(119906) and 119875

119878(P) are necessary An image 119906 pro-

jected onto a box setΩ equiv [1198871 1198872] is easily computed

(119875Ω119906)119894119895=

1198871

if 119906119894119895lt 1198871

119906119894119895

if 1198871le 119906119894119895le 1198872

1198872

if 119906119894119895gt 1198872

(4)

Taking P = (1198751 1198752) isin C for TV

119868problem (119875

119878P)119894119895

isin 119878 isgiven by

(1198751198781198751)119894119895

=

(1198751)119894119895

max 1 10038161003816100381610038161003816(1198751)11989411989910038161003816100381610038161003816

if 119894 = 1 (119898 minus 1)

(1198751)119894119895


119894119895+ (1198752)2

119894119895

otherwise

(1198751198781198752)119894119895

=

(1198752)119894119895

max 1 10038161003816100381610038161003816(1198752)11989411989910038161003816100381610038161003816

if 119894 = 1 (119898 minus 1)

(1198752)119894119895


119894119895+ (1198752)2

119894119895

otherwise

(5)

while for TV119860problem the projected operator is defined as

(1198751198781198751)119894119895=

(1198751)119894119895

max 1100381610038161003816100381610038161003816(1198751)119894119895

100381610038161003816100381610038161003816

(1198751198781198752)119894119895=

(1198752)119894119895

max 1100381610038161003816100381610038161003816(1198752)119894119895

100381610038161003816100381610038161003816

(6)




(119863119909119906)119894119895= 119906119894119895minus 119906119894+1119895

119894 = 1 (119898 minus 1) 119895 = 1 119899

(119863119910119906)119894119895= 119906119894119895minus 119906119894119895+1

119894 = 1 119898 119895 = 1 (119899 minus 1)

(7)




(div P)119894119895= 1199011(119894119895)

minus 1199011(119894minus1119895)

+ 1199012(119894119895)

minus 1199012(119894119895minus1)

(8)


= 1199011(119898119895)

= 1199012(1198940)

= 1199012(119894119899)

equiv 0 for119894 = 1 119898 119895 = 1 119899





0= (0(119898minus1)times119899

0119898times(119899minus1)

)Step 1 119906 = 119875



120582 div P2119865





⟨119906 div P⟩ (9)


min119906isinΩ

maxPisin119878

119906 minus 1198872+ 2120582 ⟨119906 div P⟩ (10)


maxPisin119878

min119906isinΩ

119906 minus 1198872+ 2120582 ⟨119906 div P⟩ (11)


min119906isinΩ

119906 minus 1198872+ 2120582 ⟨119906 div P⟩ (12)




Ω





min119909isinΩ

119891 (119909) (14)


119909119896+1

= 119909119896+ 120572119896119889119896 (15)


119896has


119889119896= 119875Ω[119909119896minus 120582119896nabla119891 (119909

119896)] minus 119909

119896 (16)







119896 that





minPisin119878


2

119865 (17)




119889119896= 119875119878(P119896+ 2120582120572

119896119863(119906119896)) minus P

119896 (19)





0isin [120572min 120572max]



0119898times(119899minus1)

) 1199051= 1


Step 2 Iterate

119906119896= 119875Ω(119887 minus 120582 div P

119896)

119889119896= 119875119878(P119896+ 2120582120572

119896119863(119906119896)) minus P

119896




Step 211

If 119866(P119896+ 120591119889119896) le 119866max minus 2120574120591120582⟨119889

119896 119863(119906119896)⟩


Step 212 Compute

120591119905= 1205821205912⟨119889119896 119863(119906119896)⟩2[119866(P

119896+ 120591119889119896) minus 119866(P

119896) +

2120582120591⟨119889119896 119863(119906119896)⟩]


119905


119896= 119875119878(P119896+ 120572119889119896)

Step 3

119905119896+1

= (1 + radic1 + 41199052

119896)2

P119896+1

= Q119896+ ((119905119896minus 1)119905

119896+1)(Q119896minusQ119896minus1

)


119878119896= P119896+1

minus P119896

119884119896= minus2120582(119863(119906

119896+1) minus 119863(119906

119896))

If ⟨119884119896 119878119896⟩ le 0 then set 120572

119896+1larr 120572max

Otherwise120572119896+1

= max120572minmin⟨119878119896 119878119896⟩⟨119884119896 119878119896⟩ 120572max









119896) = minus2120582119863(119875






119896 119863(119906119896)⟩ ⟨119878119896 119878119896⟩ and












minus30 120572max = 1030 1205901= 01 120590

2= 09 120574 = 10





0

01

02

03

04

05

06

07

08

09

1

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(a) Relative error

5

10

15

20

25

30

35

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(b) PSNR

0

01

02

03

04

05

06

07

08

09

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(c) SSIM













5 Conclusions



Competing Interests


Acknowledgments



250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(a) Ground truth

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(b) Noised

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(c) PG (PSNR = 283365)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(d) SPG (PSNR = 288468)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(e) FPG (PSNR = 282593)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(f) ASPG (PSNR = 288547)




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(119863119909119906)119894119895= 119906119894119895minus 119906119894+1119895

119894 = 1 (119898 minus 1) 119895 = 1 119899

(119863119910119906)119894119895= 119906119894119895minus 119906119894119895+1

119894 = 1 119898 119895 = 1 (119899 minus 1)

(7)




(div P)119894119895= 1199011(119894119895)

minus 1199011(119894minus1119895)

+ 1199012(119894119895)

minus 1199012(119894119895minus1)

(8)


= 1199011(119898119895)

= 1199012(1198940)

= 1199012(119894119899)

equiv 0 for119894 = 1 119898 119895 = 1 119899





0= (0(119898minus1)times119899

0119898times(119899minus1)

)Step 1 119906 = 119875



120582 div P2119865





⟨119906 div P⟩ (9)


min119906isinΩ

maxPisin119878

119906 minus 1198872+ 2120582 ⟨119906 div P⟩ (10)


maxPisin119878

min119906isinΩ

119906 minus 1198872+ 2120582 ⟨119906 div P⟩ (11)


min119906isinΩ

119906 minus 1198872+ 2120582 ⟨119906 div P⟩ (12)




Ω





min119909isinΩ

119891 (119909) (14)


119909119896+1

= 119909119896+ 120572119896119889119896 (15)


119896has


119889119896= 119875Ω[119909119896minus 120582119896nabla119891 (119909

119896)] minus 119909

119896 (16)







119896 that





minPisin119878


2

119865 (17)




119889119896= 119875119878(P119896+ 2120582120572

119896119863(119906119896)) minus P

119896 (19)





0isin [120572min 120572max]



0119898times(119899minus1)

) 1199051= 1


Step 2 Iterate

119906119896= 119875Ω(119887 minus 120582 div P

119896)

119889119896= 119875119878(P119896+ 2120582120572

119896119863(119906119896)) minus P

119896




Step 211

If 119866(P119896+ 120591119889119896) le 119866max minus 2120574120591120582⟨119889

119896 119863(119906119896)⟩


Step 212 Compute

120591119905= 1205821205912⟨119889119896 119863(119906119896)⟩2[119866(P

119896+ 120591119889119896) minus 119866(P

119896) +

2120582120591⟨119889119896 119863(119906119896)⟩]


119905


119896= 119875119878(P119896+ 120572119889119896)

Step 3

119905119896+1

= (1 + radic1 + 41199052

119896)2

P119896+1

= Q119896+ ((119905119896minus 1)119905

119896+1)(Q119896minusQ119896minus1

)


119878119896= P119896+1

minus P119896

119884119896= minus2120582(119863(119906

119896+1) minus 119863(119906

119896))

If ⟨119884119896 119878119896⟩ le 0 then set 120572

119896+1larr 120572max

Otherwise120572119896+1

= max120572minmin⟨119878119896 119878119896⟩⟨119884119896 119878119896⟩ 120572max









119896) = minus2120582119863(119875






119896 119863(119906119896)⟩ ⟨119878119896 119878119896⟩ and












minus30 120572max = 1030 1205901= 01 120590

2= 09 120574 = 10





0

01

02

03

04

05

06

07

08

09

1

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(a) Relative error

5

10

15

20

25

30

35

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(b) PSNR

0

01

02

03

04

05

06

07

08

09

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(c) SSIM













5 Conclusions



Competing Interests


Acknowledgments



250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(a) Ground truth

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(b) Noised

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(c) PG (PSNR = 283365)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(d) SPG (PSNR = 288468)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(e) FPG (PSNR = 282593)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(f) ASPG (PSNR = 288547)




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minPisin119878


2

119865 (17)




119889119896= 119875119878(P119896+ 2120582120572

119896119863(119906119896)) minus P

119896 (19)





0isin [120572min 120572max]



0119898times(119899minus1)

) 1199051= 1


Step 2 Iterate

119906119896= 119875Ω(119887 minus 120582 div P

119896)

119889119896= 119875119878(P119896+ 2120582120572

119896119863(119906119896)) minus P

119896




Step 211

If 119866(P119896+ 120591119889119896) le 119866max minus 2120574120591120582⟨119889

119896 119863(119906119896)⟩


Step 212 Compute

120591119905= 1205821205912⟨119889119896 119863(119906119896)⟩2[119866(P

119896+ 120591119889119896) minus 119866(P

119896) +

2120582120591⟨119889119896 119863(119906119896)⟩]


119905


119896= 119875119878(P119896+ 120572119889119896)

Step 3

119905119896+1

= (1 + radic1 + 41199052

119896)2

P119896+1

= Q119896+ ((119905119896minus 1)119905

119896+1)(Q119896minusQ119896minus1

)


119878119896= P119896+1

minus P119896

119884119896= minus2120582(119863(119906

119896+1) minus 119863(119906

119896))

If ⟨119884119896 119878119896⟩ le 0 then set 120572

119896+1larr 120572max

Otherwise120572119896+1

= max120572minmin⟨119878119896 119878119896⟩⟨119884119896 119878119896⟩ 120572max









119896) = minus2120582119863(119875






119896 119863(119906119896)⟩ ⟨119878119896 119878119896⟩ and












minus30 120572max = 1030 1205901= 01 120590

2= 09 120574 = 10





0

01

02

03

04

05

06

07

08

09

1

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(a) Relative error

5

10

15

20

25

30

35

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(b) PSNR

0

01

02

03

04

05

06

07

08

09

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(c) SSIM













5 Conclusions



Competing Interests


Acknowledgments



250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(a) Ground truth

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(b) Noised

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(c) PG (PSNR = 283365)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(d) SPG (PSNR = 288468)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(e) FPG (PSNR = 282593)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(f) ASPG (PSNR = 288547)




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















minus30 120572max = 1030 1205901= 01 120590

2= 09 120574 = 10





0

01

02

03

04

05

06

07

08

09

1

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(a) Relative error

5

10

15

20

25

30

35

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(b) PSNR

0

01

02

03

04

05

06

07

08

09

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(c) SSIM













5 Conclusions



Competing Interests


Acknowledgments



250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(a) Ground truth

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(b) Noised

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(c) PG (PSNR = 283365)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(d) SPG (PSNR = 288468)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(e) FPG (PSNR = 282593)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(f) ASPG (PSNR = 288547)




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

01

02

03

04

05

06

07

08

09

1

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(a) Relative error

5

10

15

20

25

30

35

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(b) PSNR

0

01

02

03

04

05

06

07

08

09

0 20 40 60 80 100 120


PG_Chambolle

FPG_Beck

(c) SSIM













5 Conclusions



Competing Interests


Acknowledgments



250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(a) Ground truth

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(b) Noised

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(c) PG (PSNR = 283365)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(d) SPG (PSNR = 288468)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(e) FPG (PSNR = 282593)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(f) ASPG (PSNR = 288547)




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















5 Conclusions



Competing Interests


Acknowledgments



250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(a) Ground truth

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(b) Noised

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(c) PG (PSNR = 283365)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(d) SPG (PSNR = 288468)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(e) FPG (PSNR = 282593)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(f) ASPG (PSNR = 288547)




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(a) Ground truth

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(b) Noised

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(c) PG (PSNR = 283365)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(d) SPG (PSNR = 288468)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(e) FPG (PSNR = 282593)

250

200

150

100

50

150 200 25050 100

60

70

80110 120 130

(f) ASPG (PSNR = 288547)




References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a nonmonotone gradient algorithm for...

Documents