research article a fast high-order total variation

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 834035 13 pageshttpdxdoiorg1011552013834035

Research ArticleA Fast High-Order Total Variation Minimization Method forMultiplicative Noise Removal

Xiao-Guang Lv1 Jiang Le12 Jin Huang2 and Liu Jun2

1 School of Science Huaihai Institute of Technology Lianyungang Jiangsu 222005 China2 School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu Sichuan 610054 China

Correspondence should be addressed to Xiao-Guang Lv xiaoguanglv126com

Received 17 October 2012 Revised 10 March 2013 Accepted 13 March 2013

Academic Editor Fatih Yaman

Copyright copy 2013 Xiao-Guang Lv et alThis 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

Multiplicative noise removal problem has received considerable attention in recent yearsThe total variation regularizationmethodfor the solution of the noise removal problem can preserve edges well but has the sometimes undesirable staircase effect In thispaper we propose a fast high-order total variation minimization method to restore multiplicative noisy images The proposedmethod is able to preserve edges and at the same time avoid the staircase effect in the smooth regions An alternating minimizationalgorithm is employed to solve the proposed high-order total variation minimization problem We discuss the convergence of thealternatingminimization algorithm Some numerical results show that the proposedmethod gives restored images of higher qualitythan some existing multiplicative noise removal methods

1 Introduction

Image denoising problem has been widely studied in theareas of image processing The problem includes additivenoise removal and multiplicative noise removal Most of theliterature deals with the additive noise model [1ndash4] Givena noisy image 119892 = 119906 + V where 119906 is the original imageand V is the noise the denoising problem is to recover 119906from the observed image 119892 The additive noise model hasbeen extensively studied We refer the reader to [5ndash11] andreferences therein for a review of various methods

In this paper we consider the problem of seeking theoriginal image from a multiplicative noisy image In themultiplicative noise model the recorded image 119892 is themultiplication of the original image 119906 and the noise V

119892 = 119906V (1)

This problemarises in ultrasound imaging synthetic apertureradar (SAR) and sonar (SAS) laser imaging and magneticfield inhomogeneity inMRI [12 13] In this work we concen-trate on the assumption that V follows a Gamma distributionwhich commonly occurs in SAR With respect to a givenresolution cell of the imaging device the complex SAR image

is computed after the SAR system receives the coherent sumof reflected monochromatic microwaves The complex SARimage at one point is usually modeled by a complex zero-mean circular Gaussian density (ie the real and imaginaryparts are independent Gaussian variable with a commonvariance) The intensity image is defined as the square ofthe magnitude of the corresponding complex SAR imageits variables follow exponentially distribution independently[14 15] Since the complex observation is acquired in highlycoherent system the intensity image occurs to have peculiargranular appearance (known as speckle) The conventionalSAR system reduces the speckle effect of the intensity imageby the multilooking process (119871-look in the case of 119871-looks)that is averaging the observed intensity images from slightlydifferent angles of the same resolution cell [16]

Due to the coherent nature of these image acquisitionprocesses nearly all the information of the original imagemay vanish when it is distorted by the multiplicative noise Ingeneral the standard additive denoisingmethod so prevalentin image processing is inadequate Therefore it is necessaryto devise efficient and reliable algorithms for recovering thetrue image from the observed multiplicative noisy imageUntil the past decade a few variational approaches devoted to

2 Mathematical Problems in Engineering

multiplicative noise removal have been proposed The earlyvariational approach for multiplicative noise removal is theone by Rudin et al [17] According to the statistical propertiesof the multiplicative noise V the recovery of the image 119906was based on solving the following constrained optimizationproblem [17]

min119906

intΩ

|119863119906| 119889119909 119889119910

subject to intΩ

119892

119906119889119909 119889119910 = 1

intΩ

(119892

119906minus 1)

2

119889119909 119889119910 = 1205902

(2)

where intΩ

|119863119906| 119889119909 119889119910 is denoted by the seminorm onboundary variation (BV) space that coincides withintΩradic1199062

119909+ 1199062119910119889119909 119889119910 when 119906 is smooth The two constraints

state that the mean of the noise is equal to 1 and the varianceis equal to 120590

2 In (2) only basic statistical properties themean and the variance of the noise V are considered whichsomehow limits the restored results

By using a maximum a posteriori (MAP) estimatorAubert and Aujol [18] proposed a function whose minimizercorresponds to the denoised image to be recovered Thisfunction is

min119906

intΩ

(log 119906 +119892

119906) 119889119909 119889119910 + 120572int

Ω

|119863119906| 119889119909 119889119910 (3)

where the total variation of 119906 is utilized as the regular-ization term and 120572 is the regularization parameter whichcontrols the trade-off between a good fit of 119892 and a smooth-ness requirement due to the total variation regularizationAlthough the functional they proposed is not convex theystill proved the existence of a minimizer and showed thecapability of their model on some numerical examples

As a result of the drawback of the objective function (3)that is not convex for all 119906 the obtained solution is likely notthe global optimal solution of (3) To overcome this problemHuang et al [19] proposed and studied a strictly convexobjective function for multiplicative noise removal problemIn [19] the authors introduced an auxiliary variable 119911 = log 119906in (3) With the new variable the proposed unconstrainedtotal variation denoising problem is described as follows

min119911119908

intΩ

(119911 + 119892119890minus119911

) 119889119909 119889119910 + 1205721119911 minus 119908

2

1198712

+ 1205722intΩ

|119863119908| 119889119909 119889119910

(4)

where 1205721and 120572

2are positive regularization parameters

The parameter 1205721measures the trade-off between an image

obtained by a maximum likelihood estimation from the firstterm and a total variation denoised image 119908 The parameter1205722measures the amount of regularization to a denoised image

119908 The main advantage of the proposed method is that thetotal variation norm can be used in the noise removal processin an efficientmannerTherefore themethod proposed in [19]has the ability to preserve edges very well in the denoisedimage

Recently Bioucas-Dias and Figueiredo [20] proposedan efficient multiplicative noise removal method by usingvariable splitting and constrained optimization After con-verting the multiplicative noise model into an additive oneby taking logarithms they used variable splitting to obtainan equivalent constrained problem and then solved thisoptimization problem by using the augmented Lagrangianmethod A set of experiments has shown that the proposedmethod which they named MIDAL (multiplicative imagedenoising by augmented Lagrangian) yields state-of-the-artresults both in terms of speed and denoising performanceIn [21] Steidl and Teuber considered a variational restorationmodel consisting of the 119868-divergence as data fitting termand the total variation seminorm or nonlocal means asregularizer for removing multiplicative noise They proposedto compute theminimizers of their restoration functionals byusingDouglas-Rachford splitting techniques For a particularsplitting they presented a semi-implicit scheme to solve theinvolved nonlinear systems of equations and proved its 119876-linear convergence

It is known that the total variation denoising methodis a PDE-based technique that preserves edges well buthas the sometimes undesirable staircase effect namely thetransformation of smooth regions (ramps) into piecewiseconstant regions (stairs) [22ndash26] Therefore the approachesinvolving the traditional total variation often cause stair-case effect since they favor solutions that are piecewiseconstant Staircase solutions fail to satisfy the evaluation ofvisual quality and they can develop false edges that do notexist in the true image To attenuate the staircase effectthere is a growing interest in the literature for replacingthe total variation norm by the high-order total variationnorm The motivation behind such attempt is to restorepotentially a wider of images which comprise more thanmerely piecewise constant regions The majority of thehigh-order norms involve second-order differential operatorsbecause piecewise-vanishing second-order derivatives leadto piecewise-linear solutions that better fit smooth intensitychanges see [27] for more details There are two classes ofsecond-order regularization methods for image restorationproblems The first class employs a second-order regularizerin a standalone way For example in [28] the authors con-sidered a fourth-order partial differential equations (PDE)model for noise removal and employed the dual algorithmof Chambolle for solving the high-order problems In [29]Hu and Jacob applied higher degree total variation (HDTV)regularization for image recovery The second class is tocombine the TV norm with a second-order regularizer Forexample a technique in [30 31] combining theTVfilterwith afourth-order PDEfilter was proposed to preserve edges and atthe same time avoid the staircase effect in smooth regions fornoise removal In [32] Papafitsoros and Schonlieb considereda high-ordermodel involving convex functions of the TV andthe TV of the first derivatives for image restoration problemsand used the split Bregman method to solve numerically thecorresponding discretized problem

In this paper we present a fast high-order total varia-tion minimization method for multiplicative noise removalproblem This technique substantially reduces the staircase

Mathematical Problems in Engineering 3

effect while preserving sharp jump discontinuities (edges)An alternating minimization algorithm is employed to solvethe proposed high-order total variationminimizationmodelWe discuss the convergence of the proposed alternatingminimization algorithm Some numerical results show thatthe proposed method gives restored images of higher qualitythan some existing multiplicative noise removal methods

The organization of this paper is outlined as followsIn the next section we present the high-order total varia-tion minimization method for multiplicative noise removalproblem In Section 3 an alternating minimization algo-rithm is employed to find the minimizer of the proposedminimization problem We analyze the convergence of theproposed alternating minimization algorithm in Section 4Some numerical experiments are given to illustrate the per-formance of the proposed algorithm in Section 5 Concludingremarks are given in the last section

2 A High-Order Total Variation MultiplicativeDenoising Model

In this section our aim is to propose a strictly convexobjective function for restoring images distorted by themultiplicative noise The most important is that we apply ahigh-order total variation to the objective function to achievesharp edges and staircase reduction efficiently We introduceour multiplicative denoising model from the statistical per-spective using the Bayesian formula

Let 119892 119906 and V denote samples of instances of somerandom variables 119865 119880 and 119881 Moreover we assume thatthe random variable V on each pixel is mutually independentand identically distributed (iid) Moreover the randomvariable V in each pixel follows Gamma distribution that isits probability density function is

119875119881(V 119871119872) =

119872119871V119871minus1

Γ (119871)119890minus119872V for V gt 0 (5)

where Γ(sdot) is the usual Gamma-function and 119871 and 119872

denote the shape scale and inverse parameters in the Gammadistribution respectivelyWe note that themean of a gamma-distributed variable is 119871119872 and the variance is 1198711198722 As themultiplicative noise in this work we assume that the mean ofV equals 1 and then we have 119871 = 119872

According to the posteriori estimation the restoredimage 119906 can be determined by

119906 = arg max119906

119875119880|119866

(119906 | 119892) (6)

From the Bayes rule we have 119875119880|119866

(119906 | 119892) = 119875119866|119880

(119892 |

119906)119875119880(119906)119875119866(119892) It then follows that

119906 = arg max119906

119875119866|119880

(119892 | 119906) 119875119880(119906)

119875119866(119892)

(7)

Using the proposition in [18] 119875119881(119892119906)(1119906) = 119875

119866|119880(119892 | 119906)

we obtain

119875119866|119880

(119892 | 119906) =119871119871

119892119871minus1

119906119871Γ (119871)119890minus119871119892119906

(8)

Taking the logarithm transformation into account we assumethat the image prior 119875

119880(119906) is given by the following

119875119880(119906) = 119875

119880|119882(119906 | 119908) 119875

119882(119908) (9)

with

119875119880|119882

(119906 | 119908) prop 119890minus1205721 log 119906minus1199082

2 (10)

119875119882(119908) = 119890

minus1205722119908HTV (11)

where1205721and1205722are two positive scalars and 119908HTV is a high-

order total variation of119908 Here we assume that the differencebetween log 119906 and 119908 follows a Gaussian distribution and 119908follows a high-order total variation prior Thus maximizing119875119880|119866

(119906 | 119892) amounts to minimizing the log-likelihood

minus log (119875119880|119866

(119906 | 119892)) = minus log (119875119866|119880

(119892 | 119906))

minus log (119875119880|119882

(119906 | 119908)) minus log (119875119882(119908))

+ log (119875119866(119892))

(12)

Since 119875119866(119892) is constant (6) can be rewritten as the following

optimization problem

119906 = arg min119906

(minus log (119875119866|119880

(119892 | 119906)) minus log (119875119880|119882

(119906 | 119908))

minus log (119875119882(119908)))

(13)

The previous computation leads us to propose the followinghigh-order total variation model for restoring images cor-rupted by the Gamma noise by introducing a new variable119911 = log 119906

min119911119908

J (119911 119908) ≐ min119911119908

intΩ

(119911 + 119892119890minus119911

) 119889119909 119889119910 + 1205721119911 minus 119908

2

1198712

+ 1205722intΩ

100381610038161003816100381610038161198632

11990810038161003816100381610038161003816119889119909 119889119910

(14)

where 1205721and 120572


In the above model 1198632119908 denotes the Hessian of 119908 and|1198632

119908| = radic1199082

119909119909+ 1199082119909119910+ 1199082119910119909+ 1199082119910119910

is just the Frobenius normof the Hessian 119863

2

119908 The parameter 1205721measures the trade-

off between an image obtained by a maximum likelihoodestimation from the first term and a high-order total variationdenoised image 119908 The parameter 120572

2measures the amount

of regularization to a denoised image 119908 The main advantageof the proposed method is that the high-order total variationnorm can be used in the noise removal process in an efficientmanner Therefore the proposed method has the ability topreserve edges very well and reduce staircase effect in thedenoised image

Moreover it is obvious that when 119906 includes an edge 119911also contains an edge at the same point that is the logarithmtransformation preserves image edges see [19] for more


details Based on this idea we can view 119911 as an image in thelogarithm domain We can apply a high-order total variationregularization to 119911 to recover its edges and reduce staircaseeffect It is interesting to note that 119911 in the proposed modelcan be positive zero or negative and the corresponding 119906 =119890119911 is still positive although 119906 in the objective function (3)should be positive Especially one can find that when 119911 inthe proposed model is a large negative value 119906 is just closeto zero

3 The Alternating Minimization Algorithm

In this section an alternating minimization algorithm isemployed to solve the proposed high-order total variationminimization problem efficiently To solve (14) we need toconsider the following two minimization subproblems

(i) For 119908 fixed determine the solution of

min119911

intΩ

(119911 + 119892119890minus119911

) 119889119909 119889119910 + 1205721119911 minus 119908

2

1198712

(15)

(ii) For 119911 fixed determine the solution of

min119908

1205721119911 minus 119908

2

1198712

+ 1205722intΩ

100381610038161003816100381610038161198632

11990810038161003816100381610038161003816119889119909 119889119910 (16)

In the discrete setting minimizing the first problem amountsto solve the following optimization problem

min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894) + 1205721119911 minus 119908

2

2 (17)

Straightforward computations can show that the minimizerof the above problem is the solution of the following nonlin-ear systems

1 minus 119892119890minus119911119894 + 2120572

1(119911119894minus 119908119894) = 0 for 119894 = 1 2 1198992 (18)

There are 1198992 decoupled nonlinear equations to be solvedThesecond derivative with respect to 119911 of the first term in (17)is equal to 119892119890minus119911 which is always greater than zero Here weassume that 119892 gt 0 in the multiplicative noise model (1)Therefore the first term of the objective function is strictlyconvex for all 119911 Hence we know that the objective function(17) is also strictly convexTherefore the corresponding non-linear equation (18) has a unique solution and the solution canbe determined by using the Newton method very efficiently

Subproblem (16) is a high-order total variation regulariza-tion process for image denoising Some effective algorithmsfor the minimizer of the high-order total variation problemhave been proposed in the literature For example a lotof PDE-based methods are widely used to preserve edgesand reduce the staircase effect for the high-order totalvariation problem see [6 22 23 30] for more details Inparticular inspired by the work from Chambolle [5] a dualalgorithm and its convergence analysis for minimizationof the high-order model are presented in [24 28] Thealgorithm overcomes the numerical difficulties related to the

nondifferentiability of the high-ordermodel Somenumericalresults show that the convergence of the dual algorithm isfaster than the gradient descent time-marching algorithm

In the present paper we employ the dual algorithmproposed by [24 28] for the high-order total variationdenoising problem (16) The discrete version of problem (16)is represented as

min119908

1205721

1205722

119911 minus 1199082

2+ 119908HTV (19)

In the dual algorithm we need to solve the followingminimization problem

min119901

10038171003817100381710038171003817119911 minus 120573 div211990110038171003817100381710038171003817

subject to 10038161003816100381610038161003816119901119895119896

10038161003816100381610038161003816le 1 forall1 le 119895 119896 le 119899

(20)

where 120573 = 120572221205721and 119901 = (

1199011111990112

1199012111990122 ) with 119901ℎ119896 isin 119877

119899times119899 forℎ 119896 = 1 2 This problem can be solved by a semi-implicitfixed point iteration

119901(119894+1)

119895119896=

119901(119894)

119895119896minus 120591119860(119894)

119895119896

1 + 120591100381610038161003816100381610038161003816119860(119894)

119895119896

100381610038161003816100381610038161003816

(21)

where 1199010 = 0 119860(119894)119895119896

= (nabla2

(div2119901(119894) minus 119911120573))119895119896 and 120591 le

164 is the step size The operator nabla2 is defined as nabla2119908 =

(119908119909119909119908119909119910

119908119910119909119908119910119910) Here div2119901(119894)

119895119896= 119863119909119909((11990111

)(119894)

119895119896) + 119863

minus

119910119909((11990112

)(119894)

119895119896) +

119863+

119909119910((11990121

)(119894)

119895119896) + 119863

119910119910((11990122

)(119894)

119895119896) see [24 28] for more details

After the minimizer 119901lowast of the constrained optimization

problem in (20) is determined the denoised image 119908 can becomputed by 119908 = 119911 minus 120573div2119901lowast In addition it is shown thatif 120591 le 164 then div2119901(119894) is convergent and 119901(119894) with the dualmethod is the solution of the minimization problem (20) as119894 rarr infin

Starting from an initial guess 119908(0) the alternating mini-mization method computes a sequence of iterates

119911(1)

119908(1)

119911(2)

119908(2)

119911(119896)

119908(119896)

(22)

such that

119911(119896)

≐ Sℎ(119908(119896minus1)

) = arg min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894)

+ 1205721

10038171003817100381710038171003817119911 minus 119908

(119896minus1)10038171003817100381710038171003817

2

2

119908(119896)

≐ S119901(119911(119896)

) = arg min119908

1205721

1205722

10038171003817100381710038171003817119911(119896)

minus 11990810038171003817100381710038171003817

2

2

+ 119908HTV

(23)

for 119896 = 1 2 We are now in a position to describe the alternating

minimization algorithm for the multiplicative noise removalproblem


Algorithm 1 The alternating minimization algorithm formultiplicative noise removal

Choose an initial guess 119908(0) and 119896 = 1

(1) For119908(119896minus1) fixed employ the Newton iterative methodto compute

119911(119896)

= arg min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894) + 1205721

10038171003817100381710038171003817119911 minus 119908

(119896minus1)10038171003817100381710038171003817

2

2

(24)

(2) For 119911(119896) fixed use a fixed point iteration to compute

119901lowast

= arg min119901

10038171003817100381710038171003817119911(119896)

minus 120573 div211990110038171003817100381710038171003817 (25)

and obtain

119908(119896)

= 119911(119896)

minus 120573div2119901lowast (26)

(3) Check the stopping criteria If a stopping criterionis satisfied then exit with an approximate restoredimage 119906(119896) = 119890

119911(119896)

otherwise let 119896 = 119896 + 1 and goto step (1)

4 Convergence Analysis

The main aim of this section is to analyze the convergenceof the proposed alternating minimization algorithmWe firstremark thatJ(119911 119908) in (14) is strictly convex as the sum of aconvex function and of two strictly convex functions see [33]for more details Hence it suffices to show that there existsa unique minimizer of the objective function J(119911 119908) Wedenote byJlowast theminimumvalue ofJ(119911 119908) In the followingwe show that our alternating minimization algorithm givesasymptotically the solution of the discrete problem associatedwith (14) We have the following theorem

Theorem 2 For any initial guess 119908(0) isin R1198992

(119911(119896) 119908(119896))generated by Algorithm 1 converges to a minimizer of theobjective functionJ(119911 119908) as 119896 rarr infin

Proof It follows from the alternating iterative process inAlgorithm 1 that

J (119911(119896)

119908(119896)

) ge J (119911(119896+1)

119908(119896)

) ge J (119911(119896+1)

119908(119896+1)

) (27)

It is obvious that the sequenceJ(119911(119896) 119908(119896)) is nonincreas-ing We note that it is bounded from below by the minimumvalue Jlowast Let 120579

1(119911) = sum

1198992

1(119911119894+ 119892119890minus119911119894) 1205792(119908) = 119908HTV and

119863Φ(119911 119908) = 120572

1119911 minus 119908

2

2 Since 120579

1and 120579

2are convex Φ is

convex and differentiable and 119863Φis the associated Bregman

distance we know from [34 35] thatJ(119911 119908) = 1205791(119911)+120579

2(119908)+

119863Φ(119911 119908) has the five-point property Hence J(119911(119896) 119908(119896))

thus converges in R toJlowast So we have

lim119896rarrinfin

J (119911(119896)

119908(119896)

) = Jlowast

(28)

We note that if the objective function J(119911 119908) is coercivethe sequence (119911

(119896)

119908(119896)

) is bounded since the sequenceJ(119911(119896) 119908(119896)) converges

Now we need to show that the objective functionJ(119891 119906 V) is coercive Let 119863

119909119909 119863119909119910 119863119910119909 and 119863

119910119910be the

second order difference matrices in the 119909119909 119909119910 119910119909 and 119910119910direction respectively and

119878 = (

119863119909119909

119863119909119910

119863119910119909

119863119910119910

) (29)

It is not difficult to obtain the lower bound of the discretehigh-order total variation as follows

119908HTV

= sum

1le119895 119896le119899

radic((119908119909119909)119895119896)2

+ ((119908119909119910)119895119896)2

+ ((119908119910119909)119895119896)2

+ ((119908119910119910)119895119896)2

ge1

2sum

1le119895 119896le119899

100381610038161003816100381610038161003816(119908119909119909)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119909119910)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119910119909)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119910119910)119895119896

100381610038161003816100381610038161003816

=1

21198781199081

(30)

Let J1(119911 119908) = 120572

1119911 minus 119908

2

2and J

2(119911 119908) = (120572

22)119878119908

1

Denote

Ω = (119911 119908)J1(119911 119908) +J

2(119911 119908) = 0 (31)

If (119911 119908) isin Ω we obtain 119911 = 119908 As (119911 119908)2rarr infin it is easy

to show that

J (119911 119908) =

1198992

sum

119894

(119911119894+ 119892119890minus119911119894) 997888rarr infin (32)

On the other hand we obtain

1198992

sum

119894

(119911119894+ 119892119890minus119911119894) ge

1198992

sum

119894

(log119892119894+ 1) (33)

since it is a strictly convex function By using the aboveinequality we have

J (119911 119908) ge

1198992

sum

119894

(log119892119894+ 1) +J

1(119911 119908) +J

2(119911 119908) (34)

Hence if (119911 119908) notin Ω with (119911 119908)2rarr infin it is not difficult

to obtain that J1(119911 119908) + J

2(119911 119908) rarr infin Thus we get that

J(119911 119908) also tends to infinity It follows from the definition ofcoercive thatJ(119911 119908) is coercive

Since the sequence (119911(119896)

119908(119896)

) is bounded we canthus extract a convergent subsequence (119911(119896119894) 119908(119896119894)) from(119911(119896)

119908(119896)

) such that lim119894rarrinfin

(119911(119896119894)

119908(119896119894)

) = ( 119908) Then


we obtain 119898 = J( 119908) Moreover we have for all 119896119894isin 119873

and all 119911

119869 (119911 119908(119896119894)

) ge J (119911(119896119894+1)

119911(119896119894)

) (35)

and for all 119896119894isin N and all 119908

J (119911(119896119894+1)

119908) ge J (119911(119896119894+1)

119908(119896119894+1)

) (36)

Let us denote by a cluster point of 119911(119896119894+1) We mayimmediately obtain from (27) that

Jlowast

= J ( 119908) = J ( 119908) (37)

Thus we have that sum1198992

119894(119894+ 119892119890minus119894) + 120572

1 minus 119908

2

2= sum1198992

119894(119894+

119892119890minus119894) + 120572

1 minus 119908

2

2 We conclude that = since and

are the minimizers of the strictly convex function Sℎ(119908)

defined in (23) Hence lim119896119894rarrinfin

119911(119896119894+1)

= Following asimilar analysis we can show that lim

119896119894rarrinfin

119908(119896119894+1)

= 119908 Bypassing to the limit in (35) and (36) we have

J (119911 119908) ge J ( 119908) forall119911

J ( 119908) ge J ( 119908) forall119908

(38)

We know that (38) can be rewritten as follows

J ( 119908) = min119911

J (119911 119908)

J ( 119908) = min119908

J ( 119908)

(39)

From the definition of J(119911 119908) (39) is equivalent to thefollowing

0 isin 120595 ( 119908)

0 isin 120601 ( 119908)

(40)

where 120595(119911 119908) = 1 minus 119892119890minus119911 + 21205721(119911 minus119908) and 120601(119911 119908) = 2120572

1(119911 minus

119908) + 1205722120597(119908HTV) The subdifferential of J(119911 119908) at ( 119908) is

given by

120597J ( 119908) = (120595 ( 119908)

120601 ( 119908)) (41)

Therefore we have

(0

0) isin 120597J ( 119908) (42)

which implies that ( 119908) is the minimizer ofJ(119911 119908) Hencethe whole sequence J(119911(119896) 119908(119896)) converges towards Jlowast theunique minimum of J(119911 119908) We deduce that the sequence(119911(119896)

119908(119896)

) converges to ( 119908) the minimizer ofJ(119911 119908) as 119896tends to infinity

5 Numerical Experiments

In this section we provide numerical results frommultiplica-tive noise removal problem to demonstrate the performance

of the proposed regularization method All computations ofthe present paper were carried out in Matlab 710 The resultswere obtained by running the Matlab codes on an Intel Corei3CPU (227GHz 227GHz) computerwithRAMof 2048MThe initial guess is chosen to be the noisy image in all tests

The quality of the restoration results with different meth-ods is compared quantitatively by using the Peak-Signal-to-Noise Ratio (PSNR) the relative error (ReErr) and StructuralSIMilarity index (SSIM) They are defined as follows

PSNR = 20 log10(

2551198991003817100381710038171003817119906(119896) minus 119906true

10038171003817100381710038172

)

ReErr =10038171003817100381710038171003817119906(119896)

minus 119906true100381710038171003817100381710038172

1003817100381710038171003817119906true10038171003817100381710038172

(43)

where 119906true and 119906(119896) are the ideal image and the restored

image of order 119899 respectively and

SSIM =(2120583119906120583+ 1198621) (2120590119906+ 1198622)

(1205832119906+ 1205832+ 1198621) (1205902119906+ 1205902+ 1198622) (44)

where 120583119906and 120583

are averages of 119906 and respectively 120590

119906

and 120590are the variance of 119906 and respectively 120590

119906is the

covariance of119906 and Thepositive constants1198621and119862

2can be

thought of as stabilizing constants for near-zero denominatorvalues In general a high PSNR-value or a small ReErr-valueindicates that the restoration is more accurate The SSIM isa well-known quality metric used to measure the similaritybetween two images This method developed by Wang etal [36] is based on three specific statistical measures thatare much closer to how the human eye perceives differencesbetween imagesThe whiter SSIMmap the closer two imagesareTherefore we use the SSIMmap to reveal areas of high orlow similarity between two images in this work

We note that a straightforward high-order total variationregularization scheme for multiplicative noise removal is tosolve the following optimization problem

min119906

intΩ

(log 119906 +119892

119906) 119889119909 119889119910 + 120572int

Ω

100381610038161003816100381610038161198632

11990610038161003816100381610038161003816119889119909 119889119910 (45)

where 120572 is a positive regularization parameter which mea-sures the trade-off between a good fit and a regularized solu-tionThe Euler-Lagrange equation for minimization problem(45) amounts to solve (omitting the boundary conditions)

119906 minus 119892

1205721199062+ (

119906119909119909

100381610038161003816100381611986321199061003816100381610038161003816

)

119909119909

+ (119906119909119910

100381610038161003816100381611986321199061003816100381610038161003816

)

119910119909

+ (119906119910119909

100381610038161003816100381611986321199061003816100381610038161003816

)

119909119910

+ (119906119910119910

100381610038161003816100381611986321199061003816100381610038161003816

)

119910119910

= 0

(46)

Hence the time-marching (TM) algorithm solving the high-order total variation model is as follows

119906119905=119892 minus 119906

1205721199062minus (

119906119909119909

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119909119909

minus (119906119909119910

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119910119909

minus (119906119910119909

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119909119910

minus (119906119910119910

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119910119910

(47)


(a) (b) (c)

Figure 1 The related images in Test 1 (a) Original image (b) Noisy image with 119871 = 10 (c) Noisy image with 119871 = 20

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

Figure 2 Restored results for Test 1 (a) Restored image by the TM algorithm for 119871 = 10 (b) Restored image by the high-order method for119871 = 10 (c) Restored image by the TM algorithm for 119871 = 20 (d) Restored image by the high-order method for 119871 = 20

where the parameter 120598 = 10minus5 is introduced to avoid divisionsby zero

In the first test we compare ourmethodwith the straight-forward high-order total variation regularization scheme Inthe time-marching algorithm for solving the straightforwardhigh-order total variation regularization problem we setthe step size as 01 in order to obtain a stable iterativeprocedure The algorithm is stopped when the maximumnumber of iterations is reached In addition we carry outmany experiments with different 120572-values in the model (45)the ones with the best results are presented in this workFor the proposed high-order total variationmethod (HighTVmethod) we terminate the iterations for the method andaccept 119906(119896) as the computed approximation of the ideal imageas soon as the maximum number of allowed outer iterationshas been carried out or the relative differences betweenconsecutive iterates 119906(1) 119906(2) 119906(3) satisfy

10038171003817100381710038171003817119906(119896+1)

minus 119906(119896)100381710038171003817100381710038172

1003817100381710038171003817119906(119896+1)

10038171003817100381710038172

lt 10minus4

(48)

In the proposed HighTV regularization method there aretwo regularization parameters We know that the quality ofthe restored image is highly depended on the regularizationparameters Similar to [19] we fix 120572

1= 19 in the proposed

method in order to reduce the computational time in searchfor good regularization parameters We determine the bestvalue of120572

2such that the relative error of such a restored image

with respect to such an ideal image is the smallest

In the first experiment we consider the ldquoCameramanrdquoimage of size 256 times 256 The original image in Figure 1(a)is distorted by the Gamma noise with 119871 = 10 and 119871 = 20The noisy images are shown in Figures 1(b) and 1(c) It is seenfromFigures 1(b) and 1(c) that the smaller 119871 is themore noisythe observed images are When 119871 = 10 the relative error(ReErr) between the noisy image and the original image is03150 When 119871 = 20 the ReErr between the noisy image andthe original image is 02234

In Figures 2(a)ndash2(d) we show the restoration resultsof the two different methods for 119871 = 10 and 119871 = 20According to the figures we see that the proposed algorithmcan provide better visual quality of restored images than thoseby the time-marching algorithm In Figures 3(a) and 3(b) wecompare the speed of convergence of the two algorithms for119871 = 10 and 119871 = 20 In the figures the 119909-axis is the numberof iterations and the 119910-axis is the the relative error betweenthe restored image and the original image We see that thespeed of convergence of the proposed algorithm is faster andthe relative error of the proposed algorithm is smaller InTable 1 we compare the results of the computational timeand the the relative error when the best restored images areobtained From Table 1 we observe that the computationaltime required by our method is lesser than that of the time-marching algorithm

In the following we compare the proposed HighTVmethod with the one proposed in [18] (AA method) the oneproposed in [19] (HNW method) and the one proposed in[20] (MIDALmethod) For the AAmethod we use the time-marching algorithm with 120598 = 10minus4 to solve the minimization


0 50 100 150 200 250 300 350 40001

015

02

025

03

035

Iteration

TM methodOur method

ReEr

r

(a)

008

01

012

014

016

018

02

022

024

0 50 100 150 200 250 300 350 400Iteration

TM methodOur method

ReEr

r(b)

Figure 3 Comparison of the TM algorithm and the proposed method for Test 1 (a) 119871 = 10 (b) 119871 = 20

(a) (b) (c)


Table 1 Restoration results with TM andHigh TVmethods for Test1

Method Parameter CPU time ReErr

119871 = 10TM 120582 = 240 121623 01150

HighTV 1205722= 0005 59285 01039

119871 = 20TM 120572 = 800 120281 00921

HighTV 1205722= 0003 46362 00871

models as proposed in [18] Similarly we set the step size as 01in order to obtain a stable iterative procedure The algorithmis stopped when the maximum number of iterations isreached In addition we carry out many experiments withdifferent 120572-values in the model (3) the ones with the bestresults are presented in this work For the HNW andMIDALmethods we use the same stopping rule as the HighTVmethod

For the MIDAL method the authors have verified exper-imentally that setting two parameter be equal yields goodresults Therefore we only select 120572 by searching for the valueleading to the smallest relative error of such a restored imagewith respect to the ideal image In the HNW method thereare two regularization parameters In [19] the authors fix1205721= 19 in all tests in order to reduce the computational time

in search for good regularization parameters We use the bestvalue of120572


with respect to such an ideal image is the smallest In this waywe have a fair comparison since we compare about the bestrestorations for four different methods

The aim of the second test is to give evidence of theeffectiveness of employing the propose algorithm over theother three algorithms for multiplicative noise removal prob-lems Especially we show that the proposed method canreduce the staircase effect In this test we consider the well-known ldquoLenardquo image of size 256 times 256 The original image in


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

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

Figure 5 Restored results by different methods for Test 2 (a) Restored image by the AA method for 119871 = 3 (b) Restored image by the HNWmethod for 119871 = 3 (c) Restored image by the MIDAL method for 119871 = 3 (d) Restored image by our method for 119871 = 3 (e) Restored image bythe AA method for 119871 = 13 (f) Restored image by the HNW method for 119871 = 13 (g) Restored image by the MIDAL method for 119871 = 13 (h)Restored image by our method for 119871 = 13

Table 2 Restoration results with four different methods for Test 2and Test 3

Image Method Parameter PSNR ReErr

Lena (119871 = 3)

AA 120582 = 15 2296 01625HNW 120572

2= 002 2339 01543

MIDAL 120572 = 2 2385 01451HighTV 120572

2= 0012 2411 01425

Lena (119871 = 13)

AA 120582 = 300 2697 01022HNW 120572

2= 0008 2714 00992

MIDAL 120572 = 4 2731 00979HighTV 120572

2= 0004 2762 00951

Barche (119871 = 6)

AA 120582 = 80 2322 01271HNW 120572

2= 0012 2369 01223

MIDAL 120572 = 3 2370 01221HighTV 120572

2= 0008 2388 01187

Barche (119871 = 16)

AA 120582 = 15 2514 01024HNW 120572

2= 0007 2567 00973

MIDAL 120572 = 45 2566 00974HighTV 120572

2= 00035 2601 00939

Figure 4(a) is distorted by the Gamma noise with 119871 = 3 and119871 = 13 The noisy images are shown in Figures 4(b) and 4(c)When119871 = 3 the PSNR and the relative error (ReErr) betweenthe noisy image and the original image are 1195 dB and05768 When 119871 = 13 the PSNR and the ReErr between thenoisy image and the original image are 1837 dB and 02765

The restored images by the AA method the HNWmethod the MIDAL method and the proposed method areshown in Figures 5(a)ndash5(h) From these figures compared

with the AA method the HNW method and the MIDALmethod the proposed approach yields better results in imagerestoration since it avoids the staircase effect of the generaltotal variation methods while at same time preserving edgesas well as the total variation methods In Figure 6 we haveenlarged some details of the eight restored images It isclear that the models involving the general total variationtransform smooth regions into piecewise constant regionsAs it is seen in the zoomed parts the proposed method out-performs another three methods since the proposed modelprocess smooth regions better than another three modelsFor the comparison of the performance quantitatively and thecomputational efficiency in Table 2 we give their restorationresults in SNRs and ReErrs We observe from Table 2 thatboth the SNR and ReErr values of the restored images by theproposed method are better than those by the AA methodHNWmethod and the MIDAL method

In the third test the ldquobarcherdquo image is considered Theoriginal image in Figure 7(a) is distorted by the Gamma noisewith 119871 = 6 and 119871 = 16 see Figures 7(b) and 7(c) When119871 = 6 the PSNR and the ReErr between the noisy image andthe original image are 1320 dB and 04090When 119871 = 16 thePSNRand theReErr between the noisy image and the originalimage are 1744 dB and 02514

The information of restored images by the AA methodthe HNW method the MIDAL method and the proposedmethod are displayed in Figures 8(a)ndash8(h) From the visualquality of restored images the proposed regularizationmethod is quite competitive with the other three algorithmsIn addition in Figures 9(a)ndash9(h) we show the SSIM mapsof the restored images It is not difficult to observe that theSSIM maps by the proposed method are whiter than the


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

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

(i) (j) (k)

Figure 6 A small portion of Lena image is emphasized to show the difference of the four methods for Test 2 (a) Original image (b) Noisyimage with 119871 = 3 (c) Noisy image with 119871 = 13 (d) Restored image by the AA method for 119871 = 3 (e) Restored image by the HNW methodfor 119871 = 3 (f) Restored image by the MIDAL method for L = 3 (g) Restored image by our method for 119871 = 3 (h) Restored image by the AAmethod for 119871 = 13 (i) Restored image by the HNWmethod for 119871 = 13 (j) Restored image by the MIDAL method for 119871 = 13 (k) Restoredimage by our method for 119871 = 13

(a) (b) (c)


AA method the HNW method and the MIDAL methodwhich means that the proposed method behaves slightlybetter The restoration results of these four different methodsin SNRs ReErrs are also presented in Table 2 It is shown

from Table 2 that both the SNR and ReErr values of therestored images by the proposed method are better thanthose by the AA method the HNWmethod and the MIDALmethod


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

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


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

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

Figure 9 Comparison of the SSIMmaps for Test 3 (a)The SSIMmap by the AAmethod for 119871 = 6 (b)The SSIMmap by the HNWmethodfor 119871 = 6 (c) The SSIM map by the MIDAL method for 119871 = 6 (d) The SSIM map by our method for 119871 = 6 (e) The SSIM map by the AAmethod for 119871 = 16 (f) The SSIM map by the HNWmethod for 119871 = 16 (g) The SSIM map by the MIDAL method for 119871 = 16 (h) The SSIMmap by our method for 119871 = 16

6 Conclusions

In this paper we investigate a fast and efficient way torestore blurred and noisy images with a high-order totalvariation minimization technique An alternating minimiza-tion algorithm is employed to solve the high-order total

variation model The edges in the restored image can bepreserved quite well and the staircase effect is reducedeffectively in the proposed algorithm Our experimentalresults show that the performance of the proposed methodis superior to that of some existing total variation restorationmethods


Acknowledgments

This research is supported by NSFC (10871034 61170311)Sichuan Province Sci amp Tech Research Project (2011JY000212ZC1802) Jiangsu Province Research Project (12KJD110001)and Research Grant KQ12013 from HuaiHai Institute ofTechnology

References

[1] L I Rudin S Osher and E Fatemi ldquoNonlinear total variationbased noise removal algorithmsrdquo Physica D vol 60 no 1ndash4 pp259ndash268 1992

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

[3] M Bertero and P Boccacci Introduction to Inverse Problems inImaging Institute of Physics Publishing London UK 1998

[4] T F Chan and J Shen Image Processing and Analysis Varia-tional PDE Wavelet and Stochastic Methods SIAM Philadel-phia Pa USA 2005

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

[6] T F Chan S Esedoglu and F E Park ldquoImage decompositioncombining staircase reduction and texture extractionrdquo Journalof Visual Communication and Image Representation vol 18 no6 pp 464ndash486 2007

[7] S Hu Z Liao andW Chen ldquoImage denoising based on dilatedsingularity priorrdquo Mathematical Problems in Engineering vol2012 Article ID 767613 15 pages 2012

[8] S Chountasis V N Katsikis and D Pappas ldquoDigital imagereconstruction in the spectral domain utilizing the Moore-Penrose inverserdquo Mathematical Problems in Engineering vol2010 Article ID 750352 14 pages 2010

[9] S Becker J Bobin and E J Candes ldquoNESTA a fast andaccurate first-order method for sparse recoveryrdquo SIAM Journalon Imaging Sciences vol 4 no 1 pp 1ndash39 2011

[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] Y L Wang J F Yang W T Yin and Y Zhang ldquoA newalternating minimization algorithm for total variation imagereconstructionrdquo SIAM Journal on Imaging Sciences vol 1 no3 pp 248ndash272 2008

[12] L Xiao L L Huang and Z HWei ldquoAweberized total variationregularization-based image multiplicative noise removal algo-rithmrdquo Eurasip Journal on Advances in Signal Processing vol2010 Article ID 490384 2010

[13] S Durand J Fadili and M Nikolova ldquoMultiplicative noiseremoval using L1 fidelity on frame coefficientsrdquo Journal ofMathematical Imaging and Vision vol 36 no 3 pp 201ndash2262010

[14] P J Green ldquoReversible jump Markov chain Monte Carlocomputation and Bayesian model determinationrdquo Biometrikavol 82 no 4 pp 711ndash732 1995

[15] C Oliver and S Quegan Understanding Synthetic ApertureRadar Images SciTech Publishing 2004

[16] J Goodman ldquoSome fundamental properties of specklerdquo Journalof the Optical Society of America vol 66 pp 1145ndash1150 1976

[17] L Rudin P-L Lions and S Osher ldquoMultiplicative denoisingand deblurring theory and algorithmsrdquo in Geometric Level Setsin Imaging Vision and Graphics S Osher and N Paragios Edspp 103ndash119 Springer New York NY USA 2003

[18] G Aubert and J-F Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008

[19] Y-M Huang M K Ng and Y-W Wen ldquoA new total variationmethod for multiplicative noise removalrdquo SIAM Journal onImaging Sciences vol 2 no 1 pp 20ndash40 2009

[20] J M Bioucas-Dias and M A T Figueiredo ldquoMultiplicativenoise removal using variable splitting and constrained opti-mizationrdquo IEEE Transactions on Image Processing vol 19 no7 pp 1720ndash1730 2010

[21] G Steidl and T Teuber ldquoRemoving multiplicative noise byDouglas-Rachford splitting methodsrdquo Journal of MathematicalImaging and Vision vol 36 no 2 pp 168ndash184 2010

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

[23] M Lysaker A Lundervold and X C Tai ldquoNoise removal usingfourth-order partial differential equation with applications tomedical magnetic resonance images in space and timerdquo IEEETransactions on Image Processing vol 12 no 12 pp 1579ndash15902003

[24] G Steidl ldquoA note on the dual treatment of higher-orderregularization functionalsrdquo Computing vol 76 no 1-2 pp 135ndash148 2006

[25] B Chen J-L Cai W-S Chen and Y Li ldquoA multiplicative noiseremoval approach based on partial differential equationmodelrdquoMathematical Problems in Engineering vol 2012 Article ID242043 14 pages 2012

[26] J Zhang Z H Wei and L Xiao ldquoAdaptive fractional-ordermulti-scale method for image denoisingrdquo Journal of Mathemat-ical Imaging and Vision vol 43 no 1 pp 39ndash49 2012

[27] S Lefkimmiatis A Bourquard and M Unser ldquoHessian-basednorm regularization for image restoration with biomedicalapplicationsrdquo IEEE Transactions on Image Processing vol 21 no3 pp 983ndash995 2012

[28] H-Z Chen J-P Song and X-C Tai ldquoA dual algorithm forminimization of the LLT modelrdquo Advances in ComputationalMathematics vol 31 no 1ndash3 pp 115ndash130 2009

[29] Y Hu and M Jacob ldquoHigher degree total variation (HDTV)regularization for image recoveryrdquo IEEE Transactions on ImageProcessing vol 21 no 5 pp 2559ndash2571 2012

[30] F Li C M Shen J S Fan and C L Shen ldquoImage restorationcombining a total variational filter and a fourth-order filterrdquoJournal of Visual Communication and Image Representation vol18 no 4 pp 322ndash330 2007

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

[32] K Papafitsoros and C B Schonlieb ldquoA combined first andsecond ordervariational approach for image reconstructionrdquohttparxivorgabs12026341

[33] D P Bertsekas A Nedic and E Ozdaglar Convex Analysis andOptimization Athena Scientific Belmont Mass USA 2003

[34] C L Byne ldquoAlternating minimization as sequential uncon-strained minimization a surveyrdquo Journal of Optimization The-ory and Applications vol 156 no 3 pp 554ndash566 2012


[35] H H Bauschke P L Combettes and D Noll ldquoJoint mini-mizationwith alternatingBregmanproximity operatorsrdquoPacificJournal of Optimization vol 2 no 3 pp 401ndash424 2006

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

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



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Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


multiplicative noise removal have been proposed The earlyvariational approach for multiplicative noise removal is theone by Rudin et al [17] According to the statistical propertiesof the multiplicative noise V the recovery of the image 119906was based on solving the following constrained optimizationproblem [17]

min119906

intΩ

|119863119906| 119889119909 119889119910

subject to intΩ

119892

119906119889119909 119889119910 = 1

intΩ

(119892

119906minus 1)

2

119889119909 119889119910 = 1205902

(2)

where intΩ

|119863119906| 119889119909 119889119910 is denoted by the seminorm onboundary variation (BV) space that coincides withintΩradic1199062

119909+ 1199062119910119889119909 119889119910 when 119906 is smooth The two constraints

state that the mean of the noise is equal to 1 and the varianceis equal to 120590

2 In (2) only basic statistical properties themean and the variance of the noise V are considered whichsomehow limits the restored results

By using a maximum a posteriori (MAP) estimatorAubert and Aujol [18] proposed a function whose minimizercorresponds to the denoised image to be recovered Thisfunction is

min119906

intΩ

(log 119906 +119892

119906) 119889119909 119889119910 + 120572int

Ω

|119863119906| 119889119909 119889119910 (3)

where the total variation of 119906 is utilized as the regular-ization term and 120572 is the regularization parameter whichcontrols the trade-off between a good fit of 119892 and a smooth-ness requirement due to the total variation regularizationAlthough the functional they proposed is not convex theystill proved the existence of a minimizer and showed thecapability of their model on some numerical examples

As a result of the drawback of the objective function (3)that is not convex for all 119906 the obtained solution is likely notthe global optimal solution of (3) To overcome this problemHuang et al [19] proposed and studied a strictly convexobjective function for multiplicative noise removal problemIn [19] the authors introduced an auxiliary variable 119911 = log 119906in (3) With the new variable the proposed unconstrainedtotal variation denoising problem is described as follows

min119911119908

intΩ

(119911 + 119892119890minus119911

) 119889119909 119889119910 + 1205721119911 minus 119908

2

1198712

+ 1205722intΩ

|119863119908| 119889119909 119889119910

(4)

where 1205721and 120572


The parameter 1205721measures the trade-off between an image

obtained by a maximum likelihood estimation from the firstterm and a total variation denoised image 119908 The parameter1205722measures the amount of regularization to a denoised image

119908 The main advantage of the proposed method is that thetotal variation norm can be used in the noise removal processin an efficientmannerTherefore themethod proposed in [19]has the ability to preserve edges very well in the denoisedimage

Recently Bioucas-Dias and Figueiredo [20] proposedan efficient multiplicative noise removal method by usingvariable splitting and constrained optimization After con-verting the multiplicative noise model into an additive oneby taking logarithms they used variable splitting to obtainan equivalent constrained problem and then solved thisoptimization problem by using the augmented Lagrangianmethod A set of experiments has shown that the proposedmethod which they named MIDAL (multiplicative imagedenoising by augmented Lagrangian) yields state-of-the-artresults both in terms of speed and denoising performanceIn [21] Steidl and Teuber considered a variational restorationmodel consisting of the 119868-divergence as data fitting termand the total variation seminorm or nonlocal means asregularizer for removing multiplicative noise They proposedto compute theminimizers of their restoration functionals byusingDouglas-Rachford splitting techniques For a particularsplitting they presented a semi-implicit scheme to solve theinvolved nonlinear systems of equations and proved its 119876-linear convergence

It is known that the total variation denoising methodis a PDE-based technique that preserves edges well buthas the sometimes undesirable staircase effect namely thetransformation of smooth regions (ramps) into piecewiseconstant regions (stairs) [22ndash26] Therefore the approachesinvolving the traditional total variation often cause stair-case effect since they favor solutions that are piecewiseconstant Staircase solutions fail to satisfy the evaluation ofvisual quality and they can develop false edges that do notexist in the true image To attenuate the staircase effectthere is a growing interest in the literature for replacingthe total variation norm by the high-order total variationnorm The motivation behind such attempt is to restorepotentially a wider of images which comprise more thanmerely piecewise constant regions The majority of thehigh-order norms involve second-order differential operatorsbecause piecewise-vanishing second-order derivatives leadto piecewise-linear solutions that better fit smooth intensitychanges see [27] for more details There are two classes ofsecond-order regularization methods for image restorationproblems The first class employs a second-order regularizerin a standalone way For example in [28] the authors con-sidered a fourth-order partial differential equations (PDE)model for noise removal and employed the dual algorithmof Chambolle for solving the high-order problems In [29]Hu and Jacob applied higher degree total variation (HDTV)regularization for image recovery The second class is tocombine the TV norm with a second-order regularizer Forexample a technique in [30 31] combining theTVfilterwith afourth-order PDEfilter was proposed to preserve edges and atthe same time avoid the staircase effect in smooth regions fornoise removal In [32] Papafitsoros and Schonlieb considereda high-ordermodel involving convex functions of the TV andthe TV of the first derivatives for image restoration problemsand used the split Bregman method to solve numerically thecorresponding discretized problem

In this paper we present a fast high-order total varia-tion minimization method for multiplicative noise removalproblem This technique substantially reduces the staircase







119875119881(V 119871119872) =

119872119871V119871minus1

Γ (119871)119890minus119872V for V gt 0 (5)




119906 = arg max119906

119875119880|119866

(119906 | 119892) (6)


(119906 | 119892) = 119875119866|119880

(119892 |


119906 = arg max119906

119875119866|119880

(119892 | 119906) 119875119880(119906)

119875119866(119892)

(7)


119866|119880(119892 | 119906)

we obtain

119875119866|119880

(119892 | 119906) =119871119871

119892119871minus1

119906119871Γ (119871)119890minus119871119892119906

(8)



119875119880(119906) = 119875

119880|119882(119906 | 119908) 119875

119882(119908) (9)

with

119875119880|119882


2 (10)

119875119882(119908) = 119890

minus1205722119908HTV (11)




minus log (119875119880|119866

(119906 | 119892)) = minus log (119875119866|119880

(119892 | 119906))

minus log (119875119880|119882

(119906 | 119908)) minus log (119875119882(119908))

+ log (119875119866(119892))

(12)



119906 = arg min119906

(minus log (119875119866|119880

(119892 | 119906)) minus log (119875119880|119882

(119906 | 119908))

minus log (119875119882(119908)))

(13)


min119911119908

J (119911 119908) ≐ min119911119908

intΩ

(119911 + 119892119890minus119911

) 119889119909 119889119910 + 1205721119911 minus 119908

2

1198712

+ 1205722intΩ

100381610038161003816100381610038161198632

11990810038161003816100381610038161003816119889119909 119889119910

(14)

where 1205721and 120572



119908| = radic1199082

119909119909+ 1199082119909119910+ 1199082119910119909+ 1199082119910119910


2











min119911

intΩ

(119911 + 119892119890minus119911

) 119889119909 119889119910 + 1205721119911 minus 119908

2

1198712

(15)


min119908

1205721119911 minus 119908

2

1198712

+ 1205722intΩ

100381610038161003816100381610038161198632

11990810038161003816100381610038161003816119889119909 119889119910 (16)


min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894) + 1205721119911 minus 119908

2

2 (17)


1 minus 119892119890minus119911119894 + 2120572

1(119911119894minus 119908119894) = 0 for 119894 = 1 2 1198992 (18)





min119908

1205721

1205722

119911 minus 1199082

2+ 119908HTV (19)


min119901

10038171003817100381710038171003817119911 minus 120573 div211990110038171003817100381710038171003817

subject to 10038161003816100381610038161003816119901119895119896

10038161003816100381610038161003816le 1 forall1 le 119895 119896 le 119899

(20)

where 120573 = 120572221205721and 119901 = (

1199011111990112

1199012111990122 ) with 119901ℎ119896 isin 119877


119901(119894+1)

119895119896=

119901(119894)

119895119896minus 120591119860(119894)

119895119896

1 + 120591100381610038161003816100381610038161003816119860(119894)

119895119896

100381610038161003816100381610038161003816

(21)

where 1199010 = 0 119860(119894)119895119896

= (nabla2

(div2119901(119894) minus 119911120573))119895119896 and 120591 le


(119908119909119909119908119909119910

119908119910119909119908119910119910) Here div2119901(119894)

119895119896= 119863119909119909((11990111

)(119894)

119895119896) + 119863

minus

119910119909((11990112

)(119894)

119895119896) +

119863+

119909119910((11990121

)(119894)

119895119896) + 119863

119910119910((11990122

)(119894)





119911(1)

119908(1)

119911(2)

119908(2)

119911(119896)

119908(119896)

(22)

such that

119911(119896)

≐ Sℎ(119908(119896minus1)

) = arg min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894)

+ 1205721

10038171003817100381710038171003817119911 minus 119908

(119896minus1)10038171003817100381710038171003817

2

2

119908(119896)

≐ S119901(119911(119896)

) = arg min119908

1205721

1205722

10038171003817100381710038171003817119911(119896)

minus 11990810038171003817100381710038171003817

2

2

+ 119908HTV

(23)







119911(119896)

= arg min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894) + 1205721

10038171003817100381710038171003817119911 minus 119908

(119896minus1)10038171003817100381710038171003817

2

2

(24)


119901lowast

= arg min119901

10038171003817100381710038171003817119911(119896)

minus 120573 div211990110038171003817100381710038171003817 (25)

and obtain

119908(119896)

= 119911(119896)



119911(119896)







J (119911(119896)

119908(119896)

) ge J (119911(119896+1)

119908(119896)

) ge J (119911(119896+1)

119908(119896+1)

) (27)


1(119911) = sum

1198992

1(119911119894+ 119892119890minus119911119894) 1205792(119908) = 119908HTV and

119863Φ(119911 119908) = 120572

1119911 minus 119908

2

2 Since 120579

1and 120579

2are convex Φ is



2(119908)+



lim119896rarrinfin

J (119911(119896)

119908(119896)

) = Jlowast

(28)


(119896)

119908(119896)



119909119909 119863119909119910 119863119910119909 and 119863

119910119910be the


119878 = (

119863119909119909

119863119909119910

119863119910119909

119863119910119910

) (29)


119908HTV

= sum

1le119895 119896le119899

radic((119908119909119909)119895119896)2

+ ((119908119909119910)119895119896)2

+ ((119908119910119909)119895119896)2

+ ((119908119910119910)119895119896)2

ge1

2sum

1le119895 119896le119899

100381610038161003816100381610038161003816(119908119909119909)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119909119910)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119910119909)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119910119910)119895119896

100381610038161003816100381610038161003816

=1

21198781199081

(30)

Let J1(119911 119908) = 120572

1119911 minus 119908

2

2and J

2(119911 119908) = (120572

22)119878119908

1

Denote

Ω = (119911 119908)J1(119911 119908) +J

2(119911 119908) = 0 (31)


to show that

J (119911 119908) =

1198992

sum

119894

(119911119894+ 119892119890minus119911119894) 997888rarr infin (32)


1198992

sum

119894

(119911119894+ 119892119890minus119911119894) ge

1198992

sum

119894

(log119892119894+ 1) (33)


J (119911 119908) ge

1198992

sum

119894

(log119892119894+ 1) +J

1(119911 119908) +J

2(119911 119908) (34)


to obtain that J1(119911 119908) + J




119908(119896)


119908(119896)


(119911(119896119894)

119908(119896119894)

) = ( 119908) Then



and all 119911

119869 (119911 119908(119896119894)

) ge J (119911(119896119894+1)

119911(119896119894)

) (35)


J (119911(119896119894+1)

119908) ge J (119911(119896119894+1)

119908(119896119894+1)

) (36)


Jlowast

= J ( 119908) = J ( 119908) (37)


119894(119894+ 119892119890minus119894) + 120572

1 minus 119908

2

2= sum1198992

119894(119894+

119892119890minus119894) + 120572

1 minus 119908

2




119911(119896119894+1)



119908(119896119894+1)


J (119911 119908) ge J ( 119908) forall119911

J ( 119908) ge J ( 119908) forall119908

(38)


J ( 119908) = min119911

J (119911 119908)

J ( 119908) = min119908

J ( 119908)

(39)


0 isin 120595 ( 119908)

0 isin 120601 ( 119908)

(40)


1(119911 minus


given by

120597J ( 119908) = (120595 ( 119908)

120601 ( 119908)) (41)

Therefore we have

(0

0) isin 120597J ( 119908) (42)


119908(119896)






PSNR = 20 log10(

2551198991003817100381710038171003817119906(119896) minus 119906true

10038171003817100381710038172

)

ReErr =10038171003817100381710038171003817119906(119896)

minus 119906true100381710038171003817100381710038172

1003817100381710038171003817119906true10038171003817100381710038172

(43)



SSIM =(2120583119906120583+ 1198621) (2120590119906+ 1198622)

(1205832119906+ 1205832+ 1198621) (1205902119906+ 1205902+ 1198622) (44)

where 120583119906and 120583


119906


119906is the


2can be



min119906

intΩ

(log 119906 +119892

119906) 119889119909 119889119910 + 120572int

Ω

100381610038161003816100381610038161198632

11990610038161003816100381610038161003816119889119909 119889119910 (45)


119906 minus 119892

1205721199062+ (

119906119909119909

100381610038161003816100381611986321199061003816100381610038161003816

)

119909119909

+ (119906119909119910

100381610038161003816100381611986321199061003816100381610038161003816

)

119910119909

+ (119906119910119909

100381610038161003816100381611986321199061003816100381610038161003816

)

119909119910

+ (119906119910119910

100381610038161003816100381611986321199061003816100381610038161003816

)

119910119910

= 0

(46)


119906119905=119892 minus 119906

1205721199062minus (

119906119909119909

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119909119909

minus (119906119909119910

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119910119909

minus (119906119910119909

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119909119910

minus (119906119910119910

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119910119910

(47)


(a) (b) (c)


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




10038171003817100381710038171003817119906(119896+1)

minus 119906(119896)100381710038171003817100381710038172

1003817100381710038171003817119906(119896+1)

10038171003817100381710038172

lt 10minus4

(48)










0 50 100 150 200 250 300 350 40001

015

02

025

03

035

Iteration

TM methodOur method

ReEr

r

(a)

008

01

012

014

016

018

02

022

024

0 50 100 150 200 250 300 350 400Iteration

TM methodOur method

ReEr

r(b)


(a) (b) (c)




119871 = 10TM 120582 = 240 121623 01150

HighTV 1205722= 0005 59285 01039

119871 = 20TM 120572 = 800 120281 00921

HighTV 1205722= 0003 46362 00871








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

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




Lena (119871 = 3)

AA 120582 = 15 2296 01625HNW 120572

2= 002 2339 01543

MIDAL 120572 = 2 2385 01451HighTV 120572

2= 0012 2411 01425

Lena (119871 = 13)

AA 120582 = 300 2697 01022HNW 120572

2= 0008 2714 00992

MIDAL 120572 = 4 2731 00979HighTV 120572

2= 0004 2762 00951

Barche (119871 = 6)

AA 120582 = 80 2322 01271HNW 120572

2= 0012 2369 01223

MIDAL 120572 = 3 2370 01221HighTV 120572

2= 0008 2388 01187

Barche (119871 = 16)

AA 120582 = 15 2514 01024HNW 120572

2= 0007 2567 00973

MIDAL 120572 = 45 2566 00974HighTV 120572

2= 00035 2601 00939







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

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

(i) (j) (k)


(a) (b) (c)





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

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


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

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


6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119875119881(V 119871119872) =

119872119871V119871minus1

Γ (119871)119890minus119872V for V gt 0 (5)




119906 = arg max119906

119875119880|119866

(119906 | 119892) (6)


(119906 | 119892) = 119875119866|119880

(119892 |


119906 = arg max119906

119875119866|119880

(119892 | 119906) 119875119880(119906)

119875119866(119892)

(7)


119866|119880(119892 | 119906)

we obtain

119875119866|119880

(119892 | 119906) =119871119871

119892119871minus1

119906119871Γ (119871)119890minus119871119892119906

(8)



119875119880(119906) = 119875

119880|119882(119906 | 119908) 119875

119882(119908) (9)

with

119875119880|119882


2 (10)

119875119882(119908) = 119890

minus1205722119908HTV (11)




minus log (119875119880|119866

(119906 | 119892)) = minus log (119875119866|119880

(119892 | 119906))

minus log (119875119880|119882

(119906 | 119908)) minus log (119875119882(119908))

+ log (119875119866(119892))

(12)



119906 = arg min119906

(minus log (119875119866|119880

(119892 | 119906)) minus log (119875119880|119882

(119906 | 119908))

minus log (119875119882(119908)))

(13)


min119911119908

J (119911 119908) ≐ min119911119908

intΩ

(119911 + 119892119890minus119911

) 119889119909 119889119910 + 1205721119911 minus 119908

2

1198712

+ 1205722intΩ

100381610038161003816100381610038161198632

11990810038161003816100381610038161003816119889119909 119889119910

(14)

where 1205721and 120572



119908| = radic1199082

119909119909+ 1199082119909119910+ 1199082119910119909+ 1199082119910119910


2











min119911

intΩ

(119911 + 119892119890minus119911

) 119889119909 119889119910 + 1205721119911 minus 119908

2

1198712

(15)


min119908

1205721119911 minus 119908

2

1198712

+ 1205722intΩ

100381610038161003816100381610038161198632

11990810038161003816100381610038161003816119889119909 119889119910 (16)


min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894) + 1205721119911 minus 119908

2

2 (17)


1 minus 119892119890minus119911119894 + 2120572

1(119911119894minus 119908119894) = 0 for 119894 = 1 2 1198992 (18)





min119908

1205721

1205722

119911 minus 1199082

2+ 119908HTV (19)


min119901

10038171003817100381710038171003817119911 minus 120573 div211990110038171003817100381710038171003817

subject to 10038161003816100381610038161003816119901119895119896

10038161003816100381610038161003816le 1 forall1 le 119895 119896 le 119899

(20)

where 120573 = 120572221205721and 119901 = (

1199011111990112

1199012111990122 ) with 119901ℎ119896 isin 119877


119901(119894+1)

119895119896=

119901(119894)

119895119896minus 120591119860(119894)

119895119896

1 + 120591100381610038161003816100381610038161003816119860(119894)

119895119896

100381610038161003816100381610038161003816

(21)

where 1199010 = 0 119860(119894)119895119896

= (nabla2

(div2119901(119894) minus 119911120573))119895119896 and 120591 le


(119908119909119909119908119909119910

119908119910119909119908119910119910) Here div2119901(119894)

119895119896= 119863119909119909((11990111

)(119894)

119895119896) + 119863

minus

119910119909((11990112

)(119894)

119895119896) +

119863+

119909119910((11990121

)(119894)

119895119896) + 119863

119910119910((11990122

)(119894)





119911(1)

119908(1)

119911(2)

119908(2)

119911(119896)

119908(119896)

(22)

such that

119911(119896)

≐ Sℎ(119908(119896minus1)

) = arg min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894)

+ 1205721

10038171003817100381710038171003817119911 minus 119908

(119896minus1)10038171003817100381710038171003817

2

2

119908(119896)

≐ S119901(119911(119896)

) = arg min119908

1205721

1205722

10038171003817100381710038171003817119911(119896)

minus 11990810038171003817100381710038171003817

2

2

+ 119908HTV

(23)







119911(119896)

= arg min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894) + 1205721

10038171003817100381710038171003817119911 minus 119908

(119896minus1)10038171003817100381710038171003817

2

2

(24)


119901lowast

= arg min119901

10038171003817100381710038171003817119911(119896)

minus 120573 div211990110038171003817100381710038171003817 (25)

and obtain

119908(119896)

= 119911(119896)



119911(119896)







J (119911(119896)

119908(119896)

) ge J (119911(119896+1)

119908(119896)

) ge J (119911(119896+1)

119908(119896+1)

) (27)


1(119911) = sum

1198992

1(119911119894+ 119892119890minus119911119894) 1205792(119908) = 119908HTV and

119863Φ(119911 119908) = 120572

1119911 minus 119908

2

2 Since 120579

1and 120579

2are convex Φ is



2(119908)+



lim119896rarrinfin

J (119911(119896)

119908(119896)

) = Jlowast

(28)


(119896)

119908(119896)



119909119909 119863119909119910 119863119910119909 and 119863

119910119910be the


119878 = (

119863119909119909

119863119909119910

119863119910119909

119863119910119910

) (29)


119908HTV

= sum

1le119895 119896le119899

radic((119908119909119909)119895119896)2

+ ((119908119909119910)119895119896)2

+ ((119908119910119909)119895119896)2

+ ((119908119910119910)119895119896)2

ge1

2sum

1le119895 119896le119899

100381610038161003816100381610038161003816(119908119909119909)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119909119910)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119910119909)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119910119910)119895119896

100381610038161003816100381610038161003816

=1

21198781199081

(30)

Let J1(119911 119908) = 120572

1119911 minus 119908

2

2and J

2(119911 119908) = (120572

22)119878119908

1

Denote

Ω = (119911 119908)J1(119911 119908) +J

2(119911 119908) = 0 (31)


to show that

J (119911 119908) =

1198992

sum

119894

(119911119894+ 119892119890minus119911119894) 997888rarr infin (32)


1198992

sum

119894

(119911119894+ 119892119890minus119911119894) ge

1198992

sum

119894

(log119892119894+ 1) (33)


J (119911 119908) ge

1198992

sum

119894

(log119892119894+ 1) +J

1(119911 119908) +J

2(119911 119908) (34)


to obtain that J1(119911 119908) + J




119908(119896)


119908(119896)


(119911(119896119894)

119908(119896119894)

) = ( 119908) Then



and all 119911

119869 (119911 119908(119896119894)

) ge J (119911(119896119894+1)

119911(119896119894)

) (35)


J (119911(119896119894+1)

119908) ge J (119911(119896119894+1)

119908(119896119894+1)

) (36)


Jlowast

= J ( 119908) = J ( 119908) (37)


119894(119894+ 119892119890minus119894) + 120572

1 minus 119908

2

2= sum1198992

119894(119894+

119892119890minus119894) + 120572

1 minus 119908

2




119911(119896119894+1)



119908(119896119894+1)


J (119911 119908) ge J ( 119908) forall119911

J ( 119908) ge J ( 119908) forall119908

(38)


J ( 119908) = min119911

J (119911 119908)

J ( 119908) = min119908

J ( 119908)

(39)


0 isin 120595 ( 119908)

0 isin 120601 ( 119908)

(40)


1(119911 minus


given by

120597J ( 119908) = (120595 ( 119908)

120601 ( 119908)) (41)

Therefore we have

(0

0) isin 120597J ( 119908) (42)


119908(119896)






PSNR = 20 log10(

2551198991003817100381710038171003817119906(119896) minus 119906true

10038171003817100381710038172

)

ReErr =10038171003817100381710038171003817119906(119896)

minus 119906true100381710038171003817100381710038172

1003817100381710038171003817119906true10038171003817100381710038172

(43)



SSIM =(2120583119906120583+ 1198621) (2120590119906+ 1198622)

(1205832119906+ 1205832+ 1198621) (1205902119906+ 1205902+ 1198622) (44)

where 120583119906and 120583


119906


119906is the


2can be



min119906

intΩ

(log 119906 +119892

119906) 119889119909 119889119910 + 120572int

Ω

100381610038161003816100381610038161198632

11990610038161003816100381610038161003816119889119909 119889119910 (45)


119906 minus 119892

1205721199062+ (

119906119909119909

100381610038161003816100381611986321199061003816100381610038161003816

)

119909119909

+ (119906119909119910

100381610038161003816100381611986321199061003816100381610038161003816

)

119910119909

+ (119906119910119909

100381610038161003816100381611986321199061003816100381610038161003816

)

119909119910

+ (119906119910119910

100381610038161003816100381611986321199061003816100381610038161003816

)

119910119910

= 0

(46)


119906119905=119892 minus 119906

1205721199062minus (

119906119909119909

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119909119909

minus (119906119909119910

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119910119909

minus (119906119910119909

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119909119910

minus (119906119910119910

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119910119910

(47)


(a) (b) (c)


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




10038171003817100381710038171003817119906(119896+1)

minus 119906(119896)100381710038171003817100381710038172

1003817100381710038171003817119906(119896+1)

10038171003817100381710038172

lt 10minus4

(48)










0 50 100 150 200 250 300 350 40001

015

02

025

03

035

Iteration

TM methodOur method

ReEr

r

(a)

008

01

012

014

016

018

02

022

024

0 50 100 150 200 250 300 350 400Iteration

TM methodOur method

ReEr

r(b)


(a) (b) (c)




119871 = 10TM 120582 = 240 121623 01150

HighTV 1205722= 0005 59285 01039

119871 = 20TM 120572 = 800 120281 00921

HighTV 1205722= 0003 46362 00871








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

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




Lena (119871 = 3)

AA 120582 = 15 2296 01625HNW 120572

2= 002 2339 01543

MIDAL 120572 = 2 2385 01451HighTV 120572

2= 0012 2411 01425

Lena (119871 = 13)

AA 120582 = 300 2697 01022HNW 120572

2= 0008 2714 00992

MIDAL 120572 = 4 2731 00979HighTV 120572

2= 0004 2762 00951

Barche (119871 = 6)

AA 120582 = 80 2322 01271HNW 120572

2= 0012 2369 01223

MIDAL 120572 = 3 2370 01221HighTV 120572

2= 0008 2388 01187

Barche (119871 = 16)

AA 120582 = 15 2514 01024HNW 120572

2= 0007 2567 00973

MIDAL 120572 = 45 2566 00974HighTV 120572

2= 00035 2601 00939







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

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

(i) (j) (k)


(a) (b) (c)





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

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


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

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


6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














min119911

intΩ

(119911 + 119892119890minus119911

) 119889119909 119889119910 + 1205721119911 minus 119908

2

1198712

(15)


min119908

1205721119911 minus 119908

2

1198712

+ 1205722intΩ

100381610038161003816100381610038161198632

11990810038161003816100381610038161003816119889119909 119889119910 (16)


min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894) + 1205721119911 minus 119908

2

2 (17)


1 minus 119892119890minus119911119894 + 2120572

1(119911119894minus 119908119894) = 0 for 119894 = 1 2 1198992 (18)





min119908

1205721

1205722

119911 minus 1199082

2+ 119908HTV (19)


min119901

10038171003817100381710038171003817119911 minus 120573 div211990110038171003817100381710038171003817

subject to 10038161003816100381610038161003816119901119895119896

10038161003816100381610038161003816le 1 forall1 le 119895 119896 le 119899

(20)

where 120573 = 120572221205721and 119901 = (

1199011111990112

1199012111990122 ) with 119901ℎ119896 isin 119877


119901(119894+1)

119895119896=

119901(119894)

119895119896minus 120591119860(119894)

119895119896

1 + 120591100381610038161003816100381610038161003816119860(119894)

119895119896

100381610038161003816100381610038161003816

(21)

where 1199010 = 0 119860(119894)119895119896

= (nabla2

(div2119901(119894) minus 119911120573))119895119896 and 120591 le


(119908119909119909119908119909119910

119908119910119909119908119910119910) Here div2119901(119894)

119895119896= 119863119909119909((11990111

)(119894)

119895119896) + 119863

minus

119910119909((11990112

)(119894)

119895119896) +

119863+

119909119910((11990121

)(119894)

119895119896) + 119863

119910119910((11990122

)(119894)





119911(1)

119908(1)

119911(2)

119908(2)

119911(119896)

119908(119896)

(22)

such that

119911(119896)

≐ Sℎ(119908(119896minus1)

) = arg min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894)

+ 1205721

10038171003817100381710038171003817119911 minus 119908

(119896minus1)10038171003817100381710038171003817

2

2

119908(119896)

≐ S119901(119911(119896)

) = arg min119908

1205721

1205722

10038171003817100381710038171003817119911(119896)

minus 11990810038171003817100381710038171003817

2

2

+ 119908HTV

(23)







119911(119896)

= arg min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894) + 1205721

10038171003817100381710038171003817119911 minus 119908

(119896minus1)10038171003817100381710038171003817

2

2

(24)


119901lowast

= arg min119901

10038171003817100381710038171003817119911(119896)

minus 120573 div211990110038171003817100381710038171003817 (25)

and obtain

119908(119896)

= 119911(119896)



119911(119896)







J (119911(119896)

119908(119896)

) ge J (119911(119896+1)

119908(119896)

) ge J (119911(119896+1)

119908(119896+1)

) (27)


1(119911) = sum

1198992

1(119911119894+ 119892119890minus119911119894) 1205792(119908) = 119908HTV and

119863Φ(119911 119908) = 120572

1119911 minus 119908

2

2 Since 120579

1and 120579

2are convex Φ is



2(119908)+



lim119896rarrinfin

J (119911(119896)

119908(119896)

) = Jlowast

(28)


(119896)

119908(119896)



119909119909 119863119909119910 119863119910119909 and 119863

119910119910be the


119878 = (

119863119909119909

119863119909119910

119863119910119909

119863119910119910

) (29)


119908HTV

= sum

1le119895 119896le119899

radic((119908119909119909)119895119896)2

+ ((119908119909119910)119895119896)2

+ ((119908119910119909)119895119896)2

+ ((119908119910119910)119895119896)2

ge1

2sum

1le119895 119896le119899

100381610038161003816100381610038161003816(119908119909119909)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119909119910)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119910119909)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119910119910)119895119896

100381610038161003816100381610038161003816

=1

21198781199081

(30)

Let J1(119911 119908) = 120572

1119911 minus 119908

2

2and J

2(119911 119908) = (120572

22)119878119908

1

Denote

Ω = (119911 119908)J1(119911 119908) +J

2(119911 119908) = 0 (31)


to show that

J (119911 119908) =

1198992

sum

119894

(119911119894+ 119892119890minus119911119894) 997888rarr infin (32)


1198992

sum

119894

(119911119894+ 119892119890minus119911119894) ge

1198992

sum

119894

(log119892119894+ 1) (33)


J (119911 119908) ge

1198992

sum

119894

(log119892119894+ 1) +J

1(119911 119908) +J

2(119911 119908) (34)


to obtain that J1(119911 119908) + J




119908(119896)


119908(119896)


(119911(119896119894)

119908(119896119894)

) = ( 119908) Then



and all 119911

119869 (119911 119908(119896119894)

) ge J (119911(119896119894+1)

119911(119896119894)

) (35)


J (119911(119896119894+1)

119908) ge J (119911(119896119894+1)

119908(119896119894+1)

) (36)


Jlowast

= J ( 119908) = J ( 119908) (37)


119894(119894+ 119892119890minus119894) + 120572

1 minus 119908

2

2= sum1198992

119894(119894+

119892119890minus119894) + 120572

1 minus 119908

2




119911(119896119894+1)



119908(119896119894+1)


J (119911 119908) ge J ( 119908) forall119911

J ( 119908) ge J ( 119908) forall119908

(38)


J ( 119908) = min119911

J (119911 119908)

J ( 119908) = min119908

J ( 119908)

(39)


0 isin 120595 ( 119908)

0 isin 120601 ( 119908)

(40)


1(119911 minus


given by

120597J ( 119908) = (120595 ( 119908)

120601 ( 119908)) (41)

Therefore we have

(0

0) isin 120597J ( 119908) (42)


119908(119896)






PSNR = 20 log10(

2551198991003817100381710038171003817119906(119896) minus 119906true

10038171003817100381710038172

)

ReErr =10038171003817100381710038171003817119906(119896)

minus 119906true100381710038171003817100381710038172

1003817100381710038171003817119906true10038171003817100381710038172

(43)



SSIM =(2120583119906120583+ 1198621) (2120590119906+ 1198622)

(1205832119906+ 1205832+ 1198621) (1205902119906+ 1205902+ 1198622) (44)

where 120583119906and 120583


119906


119906is the


2can be



min119906

intΩ

(log 119906 +119892

119906) 119889119909 119889119910 + 120572int

Ω

100381610038161003816100381610038161198632

11990610038161003816100381610038161003816119889119909 119889119910 (45)


119906 minus 119892

1205721199062+ (

119906119909119909

100381610038161003816100381611986321199061003816100381610038161003816

)

119909119909

+ (119906119909119910

100381610038161003816100381611986321199061003816100381610038161003816

)

119910119909

+ (119906119910119909

100381610038161003816100381611986321199061003816100381610038161003816

)

119909119910

+ (119906119910119910

100381610038161003816100381611986321199061003816100381610038161003816

)

119910119910

= 0

(46)


119906119905=119892 minus 119906

1205721199062minus (

119906119909119909

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119909119909

minus (119906119909119910

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119910119909

minus (119906119910119909

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119909119910

minus (119906119910119910

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119910119910

(47)


(a) (b) (c)


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




10038171003817100381710038171003817119906(119896+1)

minus 119906(119896)100381710038171003817100381710038172

1003817100381710038171003817119906(119896+1)

10038171003817100381710038172

lt 10minus4

(48)










0 50 100 150 200 250 300 350 40001

015

02

025

03

035

Iteration

TM methodOur method

ReEr

r

(a)

008

01

012

014

016

018

02

022

024

0 50 100 150 200 250 300 350 400Iteration

TM methodOur method

ReEr

r(b)


(a) (b) (c)




119871 = 10TM 120582 = 240 121623 01150

HighTV 1205722= 0005 59285 01039

119871 = 20TM 120572 = 800 120281 00921

HighTV 1205722= 0003 46362 00871








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

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




Lena (119871 = 3)

AA 120582 = 15 2296 01625HNW 120572

2= 002 2339 01543

MIDAL 120572 = 2 2385 01451HighTV 120572

2= 0012 2411 01425

Lena (119871 = 13)

AA 120582 = 300 2697 01022HNW 120572

2= 0008 2714 00992

MIDAL 120572 = 4 2731 00979HighTV 120572

2= 0004 2762 00951

Barche (119871 = 6)

AA 120582 = 80 2322 01271HNW 120572

2= 0012 2369 01223

MIDAL 120572 = 3 2370 01221HighTV 120572

2= 0008 2388 01187

Barche (119871 = 16)

AA 120582 = 15 2514 01024HNW 120572

2= 0007 2567 00973

MIDAL 120572 = 45 2566 00974HighTV 120572

2= 00035 2601 00939







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

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

(i) (j) (k)


(a) (b) (c)





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

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


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

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


6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119911(119896)

= arg min119911

1198992

sum

1

(119911119894+ 119892119890minus119911119894) + 1205721

10038171003817100381710038171003817119911 minus 119908

(119896minus1)10038171003817100381710038171003817

2

2

(24)


119901lowast

= arg min119901

10038171003817100381710038171003817119911(119896)

minus 120573 div211990110038171003817100381710038171003817 (25)

and obtain

119908(119896)

= 119911(119896)



119911(119896)







J (119911(119896)

119908(119896)

) ge J (119911(119896+1)

119908(119896)

) ge J (119911(119896+1)

119908(119896+1)

) (27)


1(119911) = sum

1198992

1(119911119894+ 119892119890minus119911119894) 1205792(119908) = 119908HTV and

119863Φ(119911 119908) = 120572

1119911 minus 119908

2

2 Since 120579

1and 120579

2are convex Φ is



2(119908)+



lim119896rarrinfin

J (119911(119896)

119908(119896)

) = Jlowast

(28)


(119896)

119908(119896)



119909119909 119863119909119910 119863119910119909 and 119863

119910119910be the


119878 = (

119863119909119909

119863119909119910

119863119910119909

119863119910119910

) (29)


119908HTV

= sum

1le119895 119896le119899

radic((119908119909119909)119895119896)2

+ ((119908119909119910)119895119896)2

+ ((119908119910119909)119895119896)2

+ ((119908119910119910)119895119896)2

ge1

2sum

1le119895 119896le119899

100381610038161003816100381610038161003816(119908119909119909)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119909119910)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119910119909)119895119896

100381610038161003816100381610038161003816+100381610038161003816100381610038161003816(119908119910119910)119895119896

100381610038161003816100381610038161003816

=1

21198781199081

(30)

Let J1(119911 119908) = 120572

1119911 minus 119908

2

2and J

2(119911 119908) = (120572

22)119878119908

1

Denote

Ω = (119911 119908)J1(119911 119908) +J

2(119911 119908) = 0 (31)


to show that

J (119911 119908) =

1198992

sum

119894

(119911119894+ 119892119890minus119911119894) 997888rarr infin (32)


1198992

sum

119894

(119911119894+ 119892119890minus119911119894) ge

1198992

sum

119894

(log119892119894+ 1) (33)


J (119911 119908) ge

1198992

sum

119894

(log119892119894+ 1) +J

1(119911 119908) +J

2(119911 119908) (34)


to obtain that J1(119911 119908) + J




119908(119896)


119908(119896)


(119911(119896119894)

119908(119896119894)

) = ( 119908) Then



and all 119911

119869 (119911 119908(119896119894)

) ge J (119911(119896119894+1)

119911(119896119894)

) (35)


J (119911(119896119894+1)

119908) ge J (119911(119896119894+1)

119908(119896119894+1)

) (36)


Jlowast

= J ( 119908) = J ( 119908) (37)


119894(119894+ 119892119890minus119894) + 120572

1 minus 119908

2

2= sum1198992

119894(119894+

119892119890minus119894) + 120572

1 minus 119908

2




119911(119896119894+1)



119908(119896119894+1)


J (119911 119908) ge J ( 119908) forall119911

J ( 119908) ge J ( 119908) forall119908

(38)


J ( 119908) = min119911

J (119911 119908)

J ( 119908) = min119908

J ( 119908)

(39)


0 isin 120595 ( 119908)

0 isin 120601 ( 119908)

(40)


1(119911 minus


given by

120597J ( 119908) = (120595 ( 119908)

120601 ( 119908)) (41)

Therefore we have

(0

0) isin 120597J ( 119908) (42)


119908(119896)






PSNR = 20 log10(

2551198991003817100381710038171003817119906(119896) minus 119906true

10038171003817100381710038172

)

ReErr =10038171003817100381710038171003817119906(119896)

minus 119906true100381710038171003817100381710038172

1003817100381710038171003817119906true10038171003817100381710038172

(43)



SSIM =(2120583119906120583+ 1198621) (2120590119906+ 1198622)

(1205832119906+ 1205832+ 1198621) (1205902119906+ 1205902+ 1198622) (44)

where 120583119906and 120583


119906


119906is the


2can be



min119906

intΩ

(log 119906 +119892

119906) 119889119909 119889119910 + 120572int

Ω

100381610038161003816100381610038161198632

11990610038161003816100381610038161003816119889119909 119889119910 (45)


119906 minus 119892

1205721199062+ (

119906119909119909

100381610038161003816100381611986321199061003816100381610038161003816

)

119909119909

+ (119906119909119910

100381610038161003816100381611986321199061003816100381610038161003816

)

119910119909

+ (119906119910119909

100381610038161003816100381611986321199061003816100381610038161003816

)

119909119910

+ (119906119910119910

100381610038161003816100381611986321199061003816100381610038161003816

)

119910119910

= 0

(46)


119906119905=119892 minus 119906

1205721199062minus (

119906119909119909

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119909119909

minus (119906119909119910

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119910119909

minus (119906119910119909

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119909119910

minus (119906119910119910

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119910119910

(47)


(a) (b) (c)


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




10038171003817100381710038171003817119906(119896+1)

minus 119906(119896)100381710038171003817100381710038172

1003817100381710038171003817119906(119896+1)

10038171003817100381710038172

lt 10minus4

(48)










0 50 100 150 200 250 300 350 40001

015

02

025

03

035

Iteration

TM methodOur method

ReEr

r

(a)

008

01

012

014

016

018

02

022

024

0 50 100 150 200 250 300 350 400Iteration

TM methodOur method

ReEr

r(b)


(a) (b) (c)




119871 = 10TM 120582 = 240 121623 01150

HighTV 1205722= 0005 59285 01039

119871 = 20TM 120572 = 800 120281 00921

HighTV 1205722= 0003 46362 00871








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

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




Lena (119871 = 3)

AA 120582 = 15 2296 01625HNW 120572

2= 002 2339 01543

MIDAL 120572 = 2 2385 01451HighTV 120572

2= 0012 2411 01425

Lena (119871 = 13)

AA 120582 = 300 2697 01022HNW 120572

2= 0008 2714 00992

MIDAL 120572 = 4 2731 00979HighTV 120572

2= 0004 2762 00951

Barche (119871 = 6)

AA 120582 = 80 2322 01271HNW 120572

2= 0012 2369 01223

MIDAL 120572 = 3 2370 01221HighTV 120572

2= 0008 2388 01187

Barche (119871 = 16)

AA 120582 = 15 2514 01024HNW 120572

2= 0007 2567 00973

MIDAL 120572 = 45 2566 00974HighTV 120572

2= 00035 2601 00939







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

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

(i) (j) (k)


(a) (b) (c)





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

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


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

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


6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











and all 119911

119869 (119911 119908(119896119894)

) ge J (119911(119896119894+1)

119911(119896119894)

) (35)


J (119911(119896119894+1)

119908) ge J (119911(119896119894+1)

119908(119896119894+1)

) (36)


Jlowast

= J ( 119908) = J ( 119908) (37)


119894(119894+ 119892119890minus119894) + 120572

1 minus 119908

2

2= sum1198992

119894(119894+

119892119890minus119894) + 120572

1 minus 119908

2




119911(119896119894+1)



119908(119896119894+1)


J (119911 119908) ge J ( 119908) forall119911

J ( 119908) ge J ( 119908) forall119908

(38)


J ( 119908) = min119911

J (119911 119908)

J ( 119908) = min119908

J ( 119908)

(39)


0 isin 120595 ( 119908)

0 isin 120601 ( 119908)

(40)


1(119911 minus


given by

120597J ( 119908) = (120595 ( 119908)

120601 ( 119908)) (41)

Therefore we have

(0

0) isin 120597J ( 119908) (42)


119908(119896)






PSNR = 20 log10(

2551198991003817100381710038171003817119906(119896) minus 119906true

10038171003817100381710038172

)

ReErr =10038171003817100381710038171003817119906(119896)

minus 119906true100381710038171003817100381710038172

1003817100381710038171003817119906true10038171003817100381710038172

(43)



SSIM =(2120583119906120583+ 1198621) (2120590119906+ 1198622)

(1205832119906+ 1205832+ 1198621) (1205902119906+ 1205902+ 1198622) (44)

where 120583119906and 120583


119906


119906is the


2can be



min119906

intΩ

(log 119906 +119892

119906) 119889119909 119889119910 + 120572int

Ω

100381610038161003816100381610038161198632

11990610038161003816100381610038161003816119889119909 119889119910 (45)


119906 minus 119892

1205721199062+ (

119906119909119909

100381610038161003816100381611986321199061003816100381610038161003816

)

119909119909

+ (119906119909119910

100381610038161003816100381611986321199061003816100381610038161003816

)

119910119909

+ (119906119910119909

100381610038161003816100381611986321199061003816100381610038161003816

)

119909119910

+ (119906119910119910

100381610038161003816100381611986321199061003816100381610038161003816

)

119910119910

= 0

(46)


119906119905=119892 minus 119906

1205721199062minus (

119906119909119909

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119909119909

minus (119906119909119910

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119910119909

minus (119906119910119909

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119909119910

minus (119906119910119910

100381610038161003816100381611986321199061003816100381610038161003816 + 120598

)

119910119910

(47)


(a) (b) (c)


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




10038171003817100381710038171003817119906(119896+1)

minus 119906(119896)100381710038171003817100381710038172

1003817100381710038171003817119906(119896+1)

10038171003817100381710038172

lt 10minus4

(48)










0 50 100 150 200 250 300 350 40001

015

02

025

03

035

Iteration

TM methodOur method

ReEr

r

(a)

008

01

012

014

016

018

02

022

024

0 50 100 150 200 250 300 350 400Iteration

TM methodOur method

ReEr

r(b)


(a) (b) (c)




119871 = 10TM 120582 = 240 121623 01150

HighTV 1205722= 0005 59285 01039

119871 = 20TM 120572 = 800 120281 00921

HighTV 1205722= 0003 46362 00871








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

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




Lena (119871 = 3)

AA 120582 = 15 2296 01625HNW 120572

2= 002 2339 01543

MIDAL 120572 = 2 2385 01451HighTV 120572

2= 0012 2411 01425

Lena (119871 = 13)

AA 120582 = 300 2697 01022HNW 120572

2= 0008 2714 00992

MIDAL 120572 = 4 2731 00979HighTV 120572

2= 0004 2762 00951

Barche (119871 = 6)

AA 120582 = 80 2322 01271HNW 120572

2= 0012 2369 01223

MIDAL 120572 = 3 2370 01221HighTV 120572

2= 0008 2388 01187

Barche (119871 = 16)

AA 120582 = 15 2514 01024HNW 120572

2= 0007 2567 00973

MIDAL 120572 = 45 2566 00974HighTV 120572

2= 00035 2601 00939







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

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

(i) (j) (k)


(a) (b) (c)





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

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


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

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


6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)


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




10038171003817100381710038171003817119906(119896+1)

minus 119906(119896)100381710038171003817100381710038172

1003817100381710038171003817119906(119896+1)

10038171003817100381710038172

lt 10minus4

(48)










0 50 100 150 200 250 300 350 40001

015

02

025

03

035

Iteration

TM methodOur method

ReEr

r

(a)

008

01

012

014

016

018

02

022

024

0 50 100 150 200 250 300 350 400Iteration

TM methodOur method

ReEr

r(b)


(a) (b) (c)




119871 = 10TM 120582 = 240 121623 01150

HighTV 1205722= 0005 59285 01039

119871 = 20TM 120572 = 800 120281 00921

HighTV 1205722= 0003 46362 00871








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

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




Lena (119871 = 3)

AA 120582 = 15 2296 01625HNW 120572

2= 002 2339 01543

MIDAL 120572 = 2 2385 01451HighTV 120572

2= 0012 2411 01425

Lena (119871 = 13)

AA 120582 = 300 2697 01022HNW 120572

2= 0008 2714 00992

MIDAL 120572 = 4 2731 00979HighTV 120572

2= 0004 2762 00951

Barche (119871 = 6)

AA 120582 = 80 2322 01271HNW 120572

2= 0012 2369 01223

MIDAL 120572 = 3 2370 01221HighTV 120572

2= 0008 2388 01187

Barche (119871 = 16)

AA 120582 = 15 2514 01024HNW 120572

2= 0007 2567 00973

MIDAL 120572 = 45 2566 00974HighTV 120572

2= 00035 2601 00939







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

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

(i) (j) (k)


(a) (b) (c)





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

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


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

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


6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 40001

015

02

025

03

035

Iteration

TM methodOur method

ReEr

r

(a)

008

01

012

014

016

018

02

022

024

0 50 100 150 200 250 300 350 400Iteration

TM methodOur method

ReEr

r(b)


(a) (b) (c)




119871 = 10TM 120582 = 240 121623 01150

HighTV 1205722= 0005 59285 01039

119871 = 20TM 120572 = 800 120281 00921

HighTV 1205722= 0003 46362 00871








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

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




Lena (119871 = 3)

AA 120582 = 15 2296 01625HNW 120572

2= 002 2339 01543

MIDAL 120572 = 2 2385 01451HighTV 120572

2= 0012 2411 01425

Lena (119871 = 13)

AA 120582 = 300 2697 01022HNW 120572

2= 0008 2714 00992

MIDAL 120572 = 4 2731 00979HighTV 120572

2= 0004 2762 00951

Barche (119871 = 6)

AA 120582 = 80 2322 01271HNW 120572

2= 0012 2369 01223

MIDAL 120572 = 3 2370 01221HighTV 120572

2= 0008 2388 01187

Barche (119871 = 16)

AA 120582 = 15 2514 01024HNW 120572

2= 0007 2567 00973

MIDAL 120572 = 45 2566 00974HighTV 120572

2= 00035 2601 00939







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

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

(i) (j) (k)


(a) (b) (c)





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

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


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

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


6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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

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




Lena (119871 = 3)

AA 120582 = 15 2296 01625HNW 120572

2= 002 2339 01543

MIDAL 120572 = 2 2385 01451HighTV 120572

2= 0012 2411 01425

Lena (119871 = 13)

AA 120582 = 300 2697 01022HNW 120572

2= 0008 2714 00992

MIDAL 120572 = 4 2731 00979HighTV 120572

2= 0004 2762 00951

Barche (119871 = 6)

AA 120582 = 80 2322 01271HNW 120572

2= 0012 2369 01223

MIDAL 120572 = 3 2370 01221HighTV 120572

2= 0008 2388 01187

Barche (119871 = 16)

AA 120582 = 15 2514 01024HNW 120572

2= 0007 2567 00973

MIDAL 120572 = 45 2566 00974HighTV 120572

2= 00035 2601 00939







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

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

(i) (j) (k)


(a) (b) (c)





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

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


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

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


6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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

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

(i) (j) (k)


(a) (b) (c)





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

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


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

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


6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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

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


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

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


6 Conclusions




Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a fast high-order total variation

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