Research Article Fast Total-Variation Image Deconvolution with … · 2019. 7. 31. · Research Article Fast Total-Variation Image Deconvolution with Adaptive Parameter Estimation

Research ArticleFast Total-Variation Image Deconvolution with AdaptiveParameter Estimation via Split Bregman Method

Chuan He1 Changhua Hu1 Wei Zhang2 Biao Shi1 and Xiaoxiang Hu1

1 Unit 302 Xirsquoan Institute of High-tech Xirsquoan 710025 China2Unit 403 Xirsquoan Institute of High-tech Xirsquoan 710025 China

Correspondence should be addressed to Chuan He hechuan8512163com

Received 16 August 2013 Accepted 27 December 2013 Published 17 February 2014

Academic Editor Yi-Hung Liu

Copyright copy 2014 Chuan He et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The total-variation (TV) regularization has been widely used in image restoration domain due to its attractive edge preservationability However the estimation of the regularization parameter which balances the TV regularization term and the data-fidelityterm is a difficult problem In this paper based on the classical split Bregman method a new fast algorithm is derived tosimultaneously estimate the regularization parameter and to restore the blurred image In each iteration the regularizationparameter is refreshed conveniently in a closed form according to Morozovrsquos discrepancy principle Numerical experiments inimage deconvolution show that the proposed algorithm outperforms some state-of-the-art methods both in accuracy and in speed

1 Introduction

Digital image restoration which aims at recovering an esti-mate of the original scene from the degraded observationis a recurrent task with many real-world applications forexample remote sensing astronomy and medical imagingDuring acquisition the observed images are often degradedby relativemotion between the camera and the original scenedefocusing of the lens system atmospheric turbulence andso forth In most cases the degradation can be modeled asa spatially linear shift invariant system where the originalimage is convolved by a spatially invariant point spreadfunction (PSF) and contaminated with Gaussian white noise[1]

Without loss of generality we assume that the digital gray-scale images used throughout this paper have an119898times119899domainand are represented by 119898119899 vectors formed by stacking upthe image matrix rows So the (119894 119895)th pixel becomes the((119894 minus 1)119899 + 119895)th entry of the vector Then in general thedegradation process can be modeled as the following discretelinear inverse problem

f = Huclean + n (1)where f and uclean are the observed image and the originalimage respectively both expressed in vectorial form H is

the convolution operator in accordance with the spatiallyinvariant PSF which is assumed to be known and n is avector of zero mean Gaussian white noise of variance 1205902 Inmost cases H is ill-conditioned so that directly estimatinguclean from f is of no possibility The solution of (1) is highlysensitive to noise in the observed image and it becomes awell-known ill-posed linear inverse problem (IPLIP) Theinverse filtering in a least square form which tries to solvethis problem directly usually results in an estimation of nousability

If we get some prior knowledge such as prior distributionor sparse quality about the original image we can incorporatesuch information into the restoration process via some sortof regularization [2] This makes the solution of IPLIPpossible A large class of regularization approaches leads tothe following minimization problem

minu

Φ (u) + 120582

2

Hu minus f22 (2)

where u is the estimate of uclean and 120582 is the so-calledregularization parameter The first term of (2) represents theregularization term whereas the second represents the data-fidelity term The regularization has the quality of numericalstabilizing and encourages the result to have some desirable

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 617026 9 pageshttpdxdoiorg1011552014617026

2 Mathematical Problems in Engineering

properties The positive regularization parameter 120582 plays therole of balancing the relative weight of the two terms

Among the various regularization methods the total-variation (TV) regularization is famed for its attractive edgepreservation ability It was introduced into image restorationby Rudin et al [3] in 1992 From then on the TV regular-ization has been arousing significant attention [4ndash7] and sofar it has resulted in several variants [8ndash10] The objectivefunctional of the TV restoration problem is given by

minu

sum

119894

1003817100381710038171003817D119894u10038171003817100381710038172+

120582

2

Hu minus f22 (3)

where the first term is the so-called TV seminorm of u andD119894u (its detailed definition is in Section 2) is the discrete

gradient of u at pixel 119894 In minimization functional (3) theTV is either isotropic if sdot is 2-norm or anisotropic if it is1-norm We emphasize here that our method is applicable toboth isotropic and anisotropic cases However we will onlytreat the isotropic one for simplicity since the treatment forthe other one is completely analogous Despite the advantageof edge preservation the minimization of functional (3) istroublesome and it has no closed form solution at all Variousmethods have been proposed to minimize (3) includingtime-marching schemes [3] primal-dual based methods[11ndash13] fixed point iteration approaches [14] and variablesplitting algorithms [15ndash17] In particular the split Bregmanmethod adopted in this paper is an instance of the variablesplitting based algorithms

Another critical issue in TV regularization is the selectionof the regularization parameter 120582 since it plays a veryimportant role If 120582 is too large the regularized solution willbe undersmoothed and on the contrary if 120582 is too smallthe regularized solution will not fit the observation properlyMost works in the literature only consider a fixed 120582 andwhen applying these methods to image restoration problemsone should adjust 120582 manually to get a satisfying solution Sofar a few strategies are proposed for the adaptive estimationof parameter 120582 for example the L-curve method [18] thevariational Bayesian approach [19] the generalized cross-validation (GCV) method [20] and Morozovrsquos discrepancyprinciple [21]

If the noise level is available or can be estimated firstMorozovrsquos discrepancy principle is a good choice for theselection of 120582 According to this rule the TV image restora-tion problem can be described as

minusum

119894

1003817100381710038171003817D119894u10038171003817100381710038172

st u isin S (4)

where S = u Hu minus f22le 119888 with 119888 = 120591119898119899120590

2 is the feasibleset in accordance with the discrepancy principle Althoughit is much easier to solve the unconstrained problem (3)than the constrained problem (4) formulation (4) has a clearphysics meaning (119888 is proportional to the noise variance)and this makes the estimation of 120582 easier In fact referringto the theory of Lagrangian methods if u is a solution ofconstrained problem (4) it will also be a solution of (3) for aparticular choice of 120582 ge 0 which is the Lagrangianmultiplier

corresponding to the constraint in (4) To minimize (4) wehave either u isin S for 120582 = 0 or

Hu minus f22= 119888 (5)

for 120582 gt 0 In fact if 120582 = 0 minimizing (3) is equivalentto minimizing sum

119894D119894u2 which means that the solution

is a constant image Obviously this will not happen to anature image Therefore only 120582 gt 0 will happen in practicalapplications

There exists no closed form solution of functional (3)or (4) and up to now several papers pay attention tothe numerical solving of problem (4) In [22] the authorsprovided a modular solver to update 120582 for making use ofexisting methods for the unconstrained problems Afonsoet al [17] proposed an alternating direction method ofmultipliers (ADMM) based approach and suggested usingChambollersquos dual method [23] to adaptively restore thedegraded image In [13] Wen and Chan proposed a primal-dual based method to solve the constrained problem (4)The minimization problem was transformed into a saddlepoint problem of the primal-dual model of (4) and then theproximal point method [24] was applied to find the saddlepoint When dealing with the updating of 120582 they resortedto a Newtonrsquos inner iteration All these methods mentionedabove have the same limitation in order to adaptively update120582 an inner iteration is introduced and this results in extracomputing cost

In this paper based on the split Bregman schemewe propose a fast algorithm to solve the constrained TVrestoration problem (4) When referring to the variablesplitting technique we introduce two auxiliary variables torepresent Du and the TV norm respectively and thereforethe constrained problem (4) can be solved efficiently witha separable structure without any inner iteration Differingfrom the previous works focusing on the adaptive regulariza-tion parameter estimation in TV restoration problems ourmethod involves no inner iteration and adjusts the regular-ization parameter in a closed form in each iterationThus fastcomputation speed is achieved The simulation results in TVrestoration problems indicate that our method outperformssome famous methods in accuracy and especially in speedAccording to the equivalence of split Bregman methodADMM andDouglas-Rachford splitting algorithmunder theassumption of linear constraints [25ndash27] our algorithm canalso be seen as an instance of ADMM or Douglas-Rachfordsplitting algorithm

In the rest of this paper the basic notation is presentedin Section 2 Section 3 gives the derivation leading to theproposed algorithm and some practical parameter settingstrategies In Section 4 several experiments are reportedto demonstrate the effectiveness of our algorithm FinallySection 5 draws a short conclusion of this paper

2 Basic Notation

Let us describe the notation that we will be using throughoutthis paper Euclidean space 119877

119898119899 is denoted as P whereasEuclidean space 119877119898119899times119898119899 is denoted as T = P times P The 119894th

Mathematical Problems in Engineering 3

components of x isin P and y isin T are denoted as 119909119894isin 119877 and

y119894= (119910(1)

119894 119910(2)

119894)

119879

isin 1198772 respectively Define inner products

⟨x x⟩P = sum119894119909119894119909119894 ⟨y y⟩T = sum

119894sum2

119896=1119910(119896)

119894119910(119896)

119894 and norm

x2= radic⟨x x⟩P y2 = radic⟨y y⟩T For each u isin P we define

D119894u = [(D(1)u)

119894 (D(2)u)

119894]

119879

with

(D(1)u)119894=

119906119894+119899

minus 119906119894 if 1 le 119894 le 119899 (119898 minus 1)

119906 mod (119894119899) minus 119906119894 otherwise

(6)

(D(2)u)119894=

119906119894+1

minus 119906119894 if mod (119894 119899) = 0

119906119894minus119899+1

minus 119906119894 otherwise

(7)

where D(1)D(2) isin 119877119898119899times119898119899 are 119898119899 times 119898119899 matrices in the

vertical and horizontal directions and obviously it holds thatD(1)u isin P and D(2)u isin P D

119894isin 1198772times119898119899 is a tow-row matrix

formed by stacking the 119894th rows of D(1) and D(2) togetherDefine the global first-order finite difference operator asD =

[(D(1))119879

(D(2))119879

]

119879

isin 1198772119898119899times119898119899 and we considerDu isin T From

(6) and (7) we see that the periodic boundary condition isassumed here

Given a convex functional 119869(z) the subdifferential 120597119869(z1)

of 119869(z) at z1is defined as

120597119869 (z1) = q isin P ⟨q z minus z

1⟩ le 119869 (z) minus 119869 (z

1) forallz isin P

(8)

And the Bregman distance between z and z1is defined as

119863(z1)

119869= 119869 (z) minus 119869 (z

1) minus ⟨q z minus z

1⟩ (9)

From the definition of Bregman distance we learn that it ispositive all the time

3 Methodology

31 Deduction of the Proposed Algorithm We refer to thevariable splitting technique [28] for liberating the discreteoperator D

119894u out from nondifferentiability and simplifying

the regularization parameterrsquos updating An auxiliary variablex isin P is introduced for Hu and another auxiliary variabley isin T is introduced to represent Du (or y

119894isin 1198772 for D

119894u

resp) Therefore functional (3) is equivalent to

minuxy

sum

119894

1003817100381710038171003817y119894

10038171003817100381710038172+

120582

2

x minus f22

subject to Hu = x y119894= D119894u 119894 = 1 2 119898119899

(10)

Define Bregman functional

119869 (u x y) = sum

119894

1003817100381710038171003817y119894

10038171003817100381710038172+

120582

2

x minus f22 (11)

Then the Bregman distance of 119869(u x y) is

119863

(p119896u p119896

x p119896

y)

119869(u x y u119896 x119896 y119896) = 119869 (u x y) minus 119869 (u119896 x119896 y119896)

minus ⟨p119896u u minus u119896⟩minus⟨p119896x x minus x119896⟩

minus ⟨p119896y y minus y119896⟩ (12)

According to the split Bregman method [16 29] we obtainthe following iterative scheme

(u119896+1 x119896+1 y119896+1)

= argminuxy

119863

(p119896u p119896

x p119896

y)

119869(u x y u119896 x119896 y119896)

+

1205731

2

x minusHu22+

1205732

2

1003817100381710038171003817y minusDu100381710038171003817

1003817

2

2

(13)

p119896+1u = p119896u + 1205731H119879 (x119896+1 minusHu119896+1) + 120573

2D119879 (y119896+1 minusDu119896+1)

(14)

p119896+1x = p119896x + 1205731(Hu119896+1 minus x119896+1) (15)

p119896+1y = p119896y + 1205732(Du119896+1 minus y119896+1) (16)

if we define that

p0u = minus1205731H119879b0 minus 120573

2D119879d0

p0x = 1205731b0

p0y = 1205732d0

(17)

for any elements b0 isin P and d0 isin T and then according to(14)ndash(16) it holds that

p119896u = minus1205731H119879b119896 minus 120573

2D119879d119896 p119896x = 120573

1b119896 p119896y = 120573

2d119896

119896 = 0 1

(18)

and we obtain the following iterative scheme

(u119896+1 x119896+1 y119896+1)

= argminuxy

120582

2

x minus f22+

1205731

2

10038171003817100381710038171003817x minusHu minus b1198961003817100381710038171003817

1003817

2

2

+sum

119894

1003817100381710038171003817y119894

10038171003817100381710038172+

1205732

2

10038171003817100381710038171003817y minusDu minus d1198961003817100381710038171003817

1003817

2

2

b119896+1 = b119896 +Hu119896+1 minus x119896+1

d119896+1 = d119896 +Du119896+1 minus y119896+1

(19)

In iterative scheme (19) the problem yielding (u119896+1x119896+1 y119896+1) exactly is difficult since it needs an inner iterativescheme Here we adopt the alternating direction method(ADM) to approximately calculate u119896+1 x119896+1 and y119896+1in each iteration and this leads to the following iterativeframework

u119896+1 = argminu

1205731

2

10038171003817100381710038171003817x119896 minusHu minus b1198961003817100381710038171003817

1003817

2

2+

1205732

2

10038171003817100381710038171003817y119896 minusDu minus d1198961003817100381710038171003817

1003817

2

2

(20)

y119896+1 = argminy

sum

119894

1003817100381710038171003817y119894

10038171003817100381710038172+

1205732

2

10038171003817100381710038171003817y minusDu minus d1198961003817100381710038171003817

1003817

2

2 (21)


x119896+1 = argminx

120582119896+1

2

x minus f22+

1205731

2

10038171003817100381710038171003817x minusHu119896+1 minus b1198961003817100381710038171003817

1003817

2

2

(22)

b119896+1 = b119896 +Hu119896+1 minus x119896+1 (23)

d119896+1 = d119896 +Du119896+1 minus y119896+1 (24)

In the following we will discuss how to solve problems (20)ndash(22) efficiently

The minimization subproblem with respect to u is in theform of least square From functional (20) we obtain

(

1205731

1205732

H119879H +D119879D) u =

1205731

1205732

H119879 (x119896 minus b119896) +D119879 (y119896 minus d119896)

(25)

Under the periodic boundary conditionmatricesHD(1)and D(2) are block-circulant so they can be diagonalizedby a Discrete Fourier Transforms (DFTs) matrix Using theconvolution theorem of Fourier Transforms we obtain

u119896+1 = Fminus1(((

1205731

1205732

)Flowast(H) ∘F (x119896 minus b119896)

+Flowast(D(1))F((y119896)

(1)

minus (d119896)(1)

)


(2)

minus (d119896)(2)

))

∘ ((

1205731

1205732

)Flowast(H) ∘F (H) +F

lowast(D(1))

∘F (D(1)) +Flowast(D(2)) ∘F (D(2)) )

minus1

)

(26)

where F denotes the DFT ldquolowastrdquo denotes complex conju-gate and ldquo∘rdquo represents componentwise multiplication Thereciprocal notation is also componentwise here Thereforeproblem (20) can be solved by two Fast Fourier Transforms(FFTs) and one inverse FFT in 119874(119898119899 log(119898119899)) operations

Functional (21) is a proximal minimization problemand it can be solved componentwise by a two-dimensionshrinkage as follows

y119896+1119894

= max10038171003817100381710038171003817D119894u119896+1 + d119896

119894

100381710038171003817100381710038172minus

1

1205732

0

D119894u119896+1 + d119896

119894

1003817100381710038171003817D119894u119896+1 + d119896

119894

10038171003817100381710038172

(27)

During the calculation we employ the convention 0 times (00) =0 to avoid getting results of no meaning

When dealing with problem (22) we assume that w119896+1 =Hu119896+1+b119896 first It is obvious that x is 120582 related and it plays therole of Hu Therefore in each iteration we should examinewhether x minus f2

2le 119888 holds true that is whether x meets the

discrepancy principleThe solutions of 120582 and x fall into two cases according to

the range of w119896+1

(1) If10038171003817100381710038171003817w119896+1 minus f1003817100381710038171003817

1003817

2

2le 119888 (28)

holds true we set 120582119896+1 = 0 and x119896+1 = w119896+1 Obvi-ously this x119896+1 satisfies the discrepancy principle

(2) If w119896+1 minus f2

2gt 119888 according to the discrepancy

principle we should solve equation

10038171003817100381710038171003817w119896+1 minus f1003817100381710038171003817

1003817

2

2= 119888 (29)

Since the minimization problem (22) with respect to x isquadratic it has a closed form solution

x119896+1 =(120582119896+1f + 120573

1w119896+1)

(120582119896+1

+ 1205731)

(30)

Substituting x119896+1 in (29) with (30) we obtain

120582119896+1

=

1205731

10038171003817100381710038171003817f minus w119896+11003817100381710038171003817

10038172

radic119888

minus 1205731 (31)

The above discussion can be summed up by Algorithm 1

In algorithm APE-SBA by introducing the auxiliaryvariable x Hu is liberated out from the constraint of thediscrepancy principle and consequently a closed form toupdate 120582 is obtained without any inner iteration This is themajor difference between APE-SBA and the methods in [13]and [17] Since the procedure of solving (26) correspondingto the u subproblem consumes the most the calculation costof our algorithm is about 119874(119898119899 log(119898119899)) FFT operations Infact our algorithm is an instance of the classical split Bregmanmethod so the convergence of it is guaranteed by the theoremproposed by Eckstein and Bertsekas [30] We summarize theconvergence of our algorithm as follows

Theorem 1 For 1205731 1205732gt 0 the sequence u119896 x119896 y119896 b119896 d119896 120582119896

generated by Algorithm APE-SBA from any initial point(u0 x0 b0 d0) converges to (ulowast xlowast ylowast blowast dlowast 120582lowast) where(ulowast xlowast ylowast) is a solution of the functional (10) In particularulowast is the minimizer of functional (4) and 120582lowast is the Lagrangemultiplier corresponding to constraint u isin S according to theunconstrained problem (3)

32 Parameter Setting In this paper the noise level is denotedby the following defined blurred signal-to-noise ratio (BSNR)

BSNR = 10 log10(

10038171003817100381710038171003817f minus f1003817100381710038171003817

1003817

2

2

1198981198991205902) (32)

where f denotes the mean of f In minimization problem (4) the noise dependent upper

bound 119888 is very important since a good choice of it canconstrain the error between the restored image and theoriginal image to a reasonable level To our best knowledgethe choice of this parameter is an open problem which hasnot been solved theoretically One approach to choose 119888 isreferring to the equivalent degrees of freedom (DF) but thecalculation of DF is a difficult problem and we can only get


Input fH 119888(1) Initialize u0 x0 b0 d0 Set 119896 = 0 and 120573

1gt 0 and 120573

2gt 0

(2) while stopping criterion is not satisfied do(3) Compute u119896+1 according to (26)(4) Compute y119896+1 according to (27)(5) if (28) holds then(6) 120582

119896+1= 0 and x119896+1 = w119896+1

(7) else(8) Update 120582119896+1 and x119896+1 according to (31) and (30)(9) end if(10) Update b119896+1 and d119896+1 according to (23) and (24)(11) 119896 = 119896 + 1(12) end while(13) return 120582

119896+1 and u119896+1

Algorithm 1 APE-SBA Adaptive Parameter Estimation Split Bregman Algorithm

Figure 1 Test images Cameraman Lena Shepp-Logan phantom and Abdomen of size 256 times 256

an estimate of it A simple strategy of choosing 119888 is to employa curve approximating the relation between the noise leveland 120591 By fitting experimental data with a straight line in thispaper we suggest setting

120591 = minus 0006 times BSNR + 109 (33)

Besides the setting of 120591 the choice of 1205731and 120573

2is essential

to our algorithm We suggest setting 1205731= 10(BSNR10minus1)

times 1205732

where 1205732= 1 This parameter setting is obtained by large

numbers of experiments Actually 1205731 1205732gt 0 is sufficient for

the convergence of the proposed algorithm but why 1205731and

1205732play different important role when the BSNR varies The

reason is that when the BSNR becomes higher the distancebetweenHu and f is nearer Fromminimization problem (10)we learn that auxiliary variable x plays the role of Hu and ahigher BSNR means a larger 120573

1

4 Numerical Results

In this section two experiments are presented to demon-strate the effectiveness of the proposed method They wereperformed under MATLAB v780 and Windows 7 on a PCwith Intel Core (TM) i5 CUP (320GHz) and 8GB of RAMThe improved signal-to-noise ratio (ISNR) is used tomeasurethe quality of the restoration results It is defined as

ISNR = 10 log10(

1003817100381710038171003817f minus uclean

1003817100381710038171003817

2

2

1003817100381710038171003817u minus uclean

1003817100381710038171003817

2

2

) (34)

During the experiments the four images shown in Figure 1were used they are named Cameraman Lena Shepp-Loganphantom and Abdomen all of size 256 times 256

41 Experiment 1 In this experiment we examine whetherthe regularization parameter is well estimated by the prosedalgorithm We compare APE-SBA with some famous TV-based methods in the literature and they are denoted by BFO[5] BMK [19] and LLN [20] We make use of MATLABcommands ldquofspecial (ldquoaveragerdquo 9)rdquo and ldquofspecial (ldquoGaussianrdquo[9 9] 3)rdquo to blur the Lena Cameraman and Shepp-Loganphantom images first and then the images are contami-nated with Gaussian noises such that the BSNRs of theobserved images are 20 dB 30 dB and 40 dB We adoptu119896+1 minus u119896

2

2u1198962

2le 10minus6 as the stopping criteria for our

algorithm where u119896 is the restored image in the 119896th iterationTable 1 presents the ISNRs of the restoration results of

different methods Symbol ldquomdashrdquomeans that the results are notpresented in the original reference and bold type numbersrepresent the best results among the four methods FromTable 1 we see that our algorithm is more competitive thanthe other three and only in one case our result is worse thanbut close to the bestThis also indicates that the regularizationparameter obtained by our method is good

42 Experiment 2 In this subsection we compare our algo-rithm with the other two state-of-the-art algorithms theprimal-dual based method in [13] named AutoRegSel andthe ADMM based method in [17] named C-SALSA The


Table 1 ISNRs obtained by different methods

BSNR Method Lena Cameraman Shepp-LoganISNR (dB) ISNR (dB) ISNR (dB)9 times 9 uniform blur

20

BFO [5] 405 327 625BMK [19] 372 242 301LLN [20] 315 288 mdashAPE-SBA 409 388 760

30


40


9 times 9 Gaussian blur

20


30


40


Table 2 Comparison between different methods in terms of ISNR iterations and runtime

Problem Method Abdomen 256 times 256 Lena 256 times 256ISNR (dB) Iterations Runtime (s) ISNR (dB) Iterations Runtime (s)

Prob 1APE-SBA 963 261 231 744 201 178

AutoRegSel [13] 941 435 653 739 392 594C-SALSA [17] 914 773 2044 698 658 1742

Prob 2APE-SBA 554 551 490 408 421 375


Prob 3APE-SBA 887 263 234 678 197 175


stopping criterion of all methods is u119896+1 minus u1198962

2u1198962

2le

10minus6 or the number of iterations is larger than 1000 We

consider the three image restoration problems adopted in[17] In the first problem the PSF is a 9 times 9 uniform blur withnoise variance 0562 (Prob 1) in the second problem the PSFis a 9 times 9 Gaussian blur with noise variance 2 (Prob 2) in thethird problem the PSF is given by ℎ

119894119895= 1(1 + 119894

2+ 1198952) with

noise variance 2 (Prob 3) where 119894 119895 = minus7 7The plots of ISNR (in dB) versus runtime (in second)

are shown in Figure 2 Table 2 presents the ISNR valuesthe number of iterations and the total runtime to reach

convergence We again use the bold type numbers to repre-sent the best results From the results we see that APE-SBAproduces the best ISNRs compared with the other methodswithin the least runtime Besides in most cases APE-SBAobtains the best ISNR within the least iterations Only whendealing with the Abdomen image under Prob 2 APE-SBAtakes more iterations but less runtime to reach convergencethan C-SALSA and the total iteration number for thesetwo is close to each other For achieving the adaptive imagerestoration both C-SALSA and AutoRegSel introduce inan inner iterative scheme whereas APE-SBA contains no


0 5 10 15 200

2

4

6

8

10

Runtime (s)

ISN

R (d

B)

(a)

0 5 10 15 200

2

4

6

8

Runtime (s)

ISN

R (d

B)

(b)

0 5 10 150

2

4

6

Runtime (s)

ISN

R (d

B)

(c)

0 5 10 150

1

2

3

4

Runtime (s)

ISN

R (d

B)

(d)

0 2 4 6 8 10 120

2

4

6

8

10

Runtime (s)

ISN

R (d

B)

AutoRegSelC-SALSAAPE-SBA

(e)

0 2 4 6 8 10 12 140

2

4

6

Runtime (s)

ISN

R (d

B)


(f)

Figure 2 ISNR versus runtime for the (left) Abdomen image and (right) Lena image which are blurred by a 9 times 9 uniform blur with noisevariance 0562 (first row) by a 9times 9Gaussian blur with noise variance 2 (second row) and by PSF given by ℎ

119894119895= 1(1+119894

2+1198952) (119894 119895 = minus7 7)

with noise variance 2 (third row)


Observed image BSNR 30871dB

(a)

Restored image by APE-SBA ISNR 554dB

(b)

Restored image by AutoRegSel ISNR 524dB

(c)

Restored image by C-SALSA ISNR 500dB

(d)

Figure 3The observed image (a) which is degraded by a 9 times 9 Gaussian blur with noise variance 2 and the restored images by APE-SBA (b)by AutoRegSel (c) and by C-SALSA (d) of the Abdomen image under Prob 2

inner iteration Obviously the superiority in speed of ourmethod will be enlarged when the image size becomes largerFigure 3 shows the blurred image and the restored resultsby different methods in Prob 2 of the Abdomen image Ouralgorithm results in the best ISNR and for other problems inExperiment 2 we obtain the similar results

5 Conclusions

We developed a split Bregman based algorithm to solve theTV image restorationdeconvolution problem Unlike someother methods in the literature without any inner iterationour method achieves the updating of the regularizationparameter and the restoration of the blurred image simul-taneously by referring to the operator splitting techniqueand introducing two auxiliary variables for both the data-fidelity term and the TV regularization term Therefore thealgorithm can run without any manual interference Thenumerical results have indicated that the proposed algorithm

outperforms some state-of-the-art methods in both speedand accuracy

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61203189 and 61304001and the National Science Fund for Distinguished YoungScholars of China under Grant 61025014

References

[1] H Andrew and B Hunt Digital Image Restoration Prentice-Hall Englewood Cliffs NJ USA 1977


[2] C R Vogel Computational Methods for Inverse Problems vol23 Society for Industrial and Applied Mathematics Philadel-phia Pa USA 2002

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

[4] T Chan S Esedoglu F Park and A Yip ldquoRecent developmentsin total variation image restorationrdquo inMathematical Models ofComputer Vision Springer New York NY USA 2005

[5] J M Bioucas-Dias M A T Figueiredo and J P Oliveira ldquoAda-ptive total variation image deblurring a majorization-mini-mization approachrdquo in Proceedings of the European SignalProcessing Conference (EUSIPCO rsquo06) Florence Italy 2006

[6] W K Allard ldquoTotal variation regularization for image denois-ing III Examplesrdquo SIAM Journal on Imaging Sciences vol 2 no2 pp 532ndash568 2009

[7] W Stefan R A Renaut and A Gelb ldquoImproved total variation-type regularization using higher order edge detectorsrdquo SIAMJournal on Imaging Sciences vol 3 no 2 pp 232ndash251 2010

[8] 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

[9] K Bredies K Kunisch and T Pock ldquoTotal generalized varia-tionrdquo SIAM Journal on Imaging Sciences vol 3 no 3 pp 492ndash526 2010

[10] K Bredies Y Dong andM Hintermuller ldquoSpatially dependentregularization parameter selection in total generalized variationmodels for image restorationrdquo International Journal of Com-puter Mathematics vol 90 no 1 pp 109ndash123 2013

[11] T F Chan G H Golub and P Mulet ldquoA nonlinear primal-dualmethod for total variation-based image restorationrdquo SIAM Jou-rnal on Scientific Computing vol 20 no 6 pp 1964ndash1977 1999

[12] B He and X Yuan ldquoConvergence analysis of primal-dual algor-ithms for a saddle-point problem from contraction perspec-tiverdquo SIAM Journal on Imaging Sciences vol 5 no 1 pp 119ndash1492012

[13] Y-W Wen and R H Chan ldquoParameter selection for total-var-iation-based image restoration using discrepancy principlerdquoIEEE Transactions on Image Processing vol 21 no 4 pp 1770ndash1781 2012

[14] C R Vogel and M E Oman ldquoIterative methods for total vari-ation denoisingrdquo SIAM Journal on Scientific Computing vol 17no 1 pp 227ndash238 1996

[15] YWang J YangW Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008

[16] T Goldstein and S Osher ldquoThe split Bregman method for L1-

regularized problemsrdquo SIAM Journal on Imaging Sciences vol2 no 2 pp 323ndash343 2009

[17] M V Afonso J M Bioucas-Dias and M A T Figueiredo ldquoAnaugmented Lagrangian approach to the constrained optimiza-tion formulation of imaging inverse problemsrdquo IEEE Transac-tions on Image Processing vol 20 no 3 pp 681ndash695 2011

[18] H W Engl and W Grever ldquoUsing the 119871-curve for determiningoptimal regularization parametersrdquo Numerische Mathematikvol 69 no 1 pp 25ndash31 1994

[19] S D Babacan R Molina and A K Katsaggelos ldquoVariationalBayesian blind deconvolution using a total variation priorrdquoIEEE Transactions on Image Processing vol 18 no 1 pp 12ndash262009

[20] H Liao F Li and M K Ng ldquoSelection of regularization par-ameter in total variation image restorationrdquo Journal of the Opt-ical Society of America A vol 26 no 11 pp 2311ndash2320 2009

[21] V A Morozov Methods for Solving Incorrectly Posed ProblemsSpringer NewYork NYUSA 1984 translated from the Russianby A B Aries translation edited by Z Nashed

[22] P Blomgren and T F Chan ldquoModular solvers for image restora-tion problems using the discrepancy principlerdquo NumericalLinear Algebra with Applications vol 9 no 5 pp 347ndash358 2002

[23] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004 Special issue on mathematicsand image analysis

[24] H H Bauschke and P L CombettesConvex Analysis andMon-otone Operator Theory in Hilbert Spaces Springer New YorkNY USA 2011

[25] W Yin S Osher D Goldfarb and J Darbon ldquoBregman itera-tive algorithms for L

1-minimization with applications to com-

pressed sensingrdquo SIAM Journal on Imaging Sciences vol 1 no1 pp 143ndash168 2008

[26] CWu andX-C Tai ldquoAugmented Lagrangianmethod dualme-thods and split Bregman iteration for ROF vectorial TV andhigh order modelsrdquo SIAM Journal on Imaging Sciences vol 3no 3 pp 300ndash339 2010

[27] S Setzer ldquoOperator splittings Bregmanmethods and frame shr-inkage in image processingrdquo International Journal of ComputerVision vol 92 no 3 pp 265ndash280 2011

[28] R Glowinski and P Le Tallec Augmented Lagrangian and Ope-rator-Splitting Methods in Nonlinear Mechanics vol 9 of SIAMStudies in Applied Mathematics Society for Industrial and App-lied Mathematics Philadelphia Pa USA 1989

[29] S Osher M Burger D Goldfarb J Xu and W Yin ldquoAn iter-ative regularizationmethod for total variation-based image res-torationrdquo Multiscale Modeling amp Simulation vol 4 no 2 pp460ndash489 2005

[30] J Eckstein andD Bertsekas ldquoOn theDouglas Rachford splittingmethod and the proximal point algorithm for maximal mono-tone operatorsrdquo Mathematical Programming vol 55 no 1 pp293ndash318 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


properties The positive regularization parameter 120582 plays therole of balancing the relative weight of the two terms

Among the various regularization methods the total-variation (TV) regularization is famed for its attractive edgepreservation ability It was introduced into image restorationby Rudin et al [3] in 1992 From then on the TV regular-ization has been arousing significant attention [4ndash7] and sofar it has resulted in several variants [8ndash10] The objectivefunctional of the TV restoration problem is given by

minu

sum

119894

1003817100381710038171003817D119894u10038171003817100381710038172+

120582

2

Hu minus f22 (3)

where the first term is the so-called TV seminorm of u andD119894u (its detailed definition is in Section 2) is the discrete

gradient of u at pixel 119894 In minimization functional (3) theTV is either isotropic if sdot is 2-norm or anisotropic if it is1-norm We emphasize here that our method is applicable toboth isotropic and anisotropic cases However we will onlytreat the isotropic one for simplicity since the treatment forthe other one is completely analogous Despite the advantageof edge preservation the minimization of functional (3) istroublesome and it has no closed form solution at all Variousmethods have been proposed to minimize (3) includingtime-marching schemes [3] primal-dual based methods[11ndash13] fixed point iteration approaches [14] and variablesplitting algorithms [15ndash17] In particular the split Bregmanmethod adopted in this paper is an instance of the variablesplitting based algorithms

Another critical issue in TV regularization is the selectionof the regularization parameter 120582 since it plays a veryimportant role If 120582 is too large the regularized solution willbe undersmoothed and on the contrary if 120582 is too smallthe regularized solution will not fit the observation properlyMost works in the literature only consider a fixed 120582 andwhen applying these methods to image restoration problemsone should adjust 120582 manually to get a satisfying solution Sofar a few strategies are proposed for the adaptive estimationof parameter 120582 for example the L-curve method [18] thevariational Bayesian approach [19] the generalized cross-validation (GCV) method [20] and Morozovrsquos discrepancyprinciple [21]

If the noise level is available or can be estimated firstMorozovrsquos discrepancy principle is a good choice for theselection of 120582 According to this rule the TV image restora-tion problem can be described as

minusum

119894

1003817100381710038171003817D119894u10038171003817100381710038172

st u isin S (4)

where S = u Hu minus f22le 119888 with 119888 = 120591119898119899120590

2 is the feasibleset in accordance with the discrepancy principle Althoughit is much easier to solve the unconstrained problem (3)than the constrained problem (4) formulation (4) has a clearphysics meaning (119888 is proportional to the noise variance)and this makes the estimation of 120582 easier In fact referringto the theory of Lagrangian methods if u is a solution ofconstrained problem (4) it will also be a solution of (3) for aparticular choice of 120582 ge 0 which is the Lagrangianmultiplier

corresponding to the constraint in (4) To minimize (4) wehave either u isin S for 120582 = 0 or

Hu minus f22= 119888 (5)

for 120582 gt 0 In fact if 120582 = 0 minimizing (3) is equivalentto minimizing sum

119894D119894u2 which means that the solution

is a constant image Obviously this will not happen to anature image Therefore only 120582 gt 0 will happen in practicalapplications

There exists no closed form solution of functional (3)or (4) and up to now several papers pay attention tothe numerical solving of problem (4) In [22] the authorsprovided a modular solver to update 120582 for making use ofexisting methods for the unconstrained problems Afonsoet al [17] proposed an alternating direction method ofmultipliers (ADMM) based approach and suggested usingChambollersquos dual method [23] to adaptively restore thedegraded image In [13] Wen and Chan proposed a primal-dual based method to solve the constrained problem (4)The minimization problem was transformed into a saddlepoint problem of the primal-dual model of (4) and then theproximal point method [24] was applied to find the saddlepoint When dealing with the updating of 120582 they resortedto a Newtonrsquos inner iteration All these methods mentionedabove have the same limitation in order to adaptively update120582 an inner iteration is introduced and this results in extracomputing cost

In this paper based on the split Bregman schemewe propose a fast algorithm to solve the constrained TVrestoration problem (4) When referring to the variablesplitting technique we introduce two auxiliary variables torepresent Du and the TV norm respectively and thereforethe constrained problem (4) can be solved efficiently witha separable structure without any inner iteration Differingfrom the previous works focusing on the adaptive regulariza-tion parameter estimation in TV restoration problems ourmethod involves no inner iteration and adjusts the regular-ization parameter in a closed form in each iterationThus fastcomputation speed is achieved The simulation results in TVrestoration problems indicate that our method outperformssome famous methods in accuracy and especially in speedAccording to the equivalence of split Bregman methodADMM andDouglas-Rachford splitting algorithmunder theassumption of linear constraints [25ndash27] our algorithm canalso be seen as an instance of ADMM or Douglas-Rachfordsplitting algorithm

In the rest of this paper the basic notation is presentedin Section 2 Section 3 gives the derivation leading to theproposed algorithm and some practical parameter settingstrategies In Section 4 several experiments are reportedto demonstrate the effectiveness of our algorithm FinallySection 5 draws a short conclusion of this paper

2 Basic Notation

Let us describe the notation that we will be using throughoutthis paper Euclidean space 119877

119898119899 is denoted as P whereasEuclidean space 119877119898119899times119898119899 is denoted as T = P times P The 119894th



y119894= (119910(1)

119894 119910(2)

119894)

119879


⟨x x⟩P = sum119894119909119894119909119894 ⟨y y⟩T = sum

119894sum2

119896=1119910(119896)

119894119910(119896)

119894 and norm


D119894u = [(D(1)u)

119894 (D(2)u)

119894]

119879

with

(D(1)u)119894=

119906119894+119899



(6)

(D(2)u)119894=

119906119894+1

minus 119906119894 if mod (119894 119899) = 0

119906119894minus119899+1


(7)





[(D(1))119879

(D(2))119879

]

119879






1⟩ le 119869 (z) minus 119869 (z

1) forallz isin P

(8)


119863(z1)

119869= 119869 (z) minus 119869 (z


1⟩ (9)


3 Methodology




119894isin 1198772 for D

119894u


minuxy

sum

119894

1003817100381710038171003817y119894

10038171003817100381710038172+

120582

2

x minus f22


(10)


119869 (u x y) = sum

119894

1003817100381710038171003817y119894

10038171003817100381710038172+

120582

2

x minus f22 (11)


119863

(p119896u p119896

x p119896

y)



minus ⟨p119896y y minus y119896⟩ (12)


(u119896+1 x119896+1 y119896+1)

= argminuxy

119863

(p119896u p119896

x p119896

y)

119869(u x y u119896 x119896 y119896)

+

1205731

2

x minusHu22+

1205732

2

1003817100381710038171003817y minusDu100381710038171003817

1003817

2

2

(13)

p119896+1u = p119896u + 1205731H119879 (x119896+1 minusHu119896+1) + 120573

2D119879 (y119896+1 minusDu119896+1)

(14)

p119896+1x = p119896x + 1205731(Hu119896+1 minus x119896+1) (15)

p119896+1y = p119896y + 1205732(Du119896+1 minus y119896+1) (16)

if we define that

p0u = minus1205731H119879b0 minus 120573

2D119879d0

p0x = 1205731b0

p0y = 1205732d0

(17)


p119896u = minus1205731H119879b119896 minus 120573

2D119879d119896 p119896x = 120573

1b119896 p119896y = 120573

2d119896

119896 = 0 1

(18)


(u119896+1 x119896+1 y119896+1)

= argminuxy

120582

2

x minus f22+

1205731

2

10038171003817100381710038171003817x minusHu minus b1198961003817100381710038171003817

1003817

2

2

+sum

119894

1003817100381710038171003817y119894

10038171003817100381710038172+

1205732

2

10038171003817100381710038171003817y minusDu minus d1198961003817100381710038171003817

1003817

2

2

b119896+1 = b119896 +Hu119896+1 minus x119896+1

d119896+1 = d119896 +Du119896+1 minus y119896+1

(19)


u119896+1 = argminu

1205731

2

10038171003817100381710038171003817x119896 minusHu minus b1198961003817100381710038171003817

1003817

2

2+

1205732

2

10038171003817100381710038171003817y119896 minusDu minus d1198961003817100381710038171003817

1003817

2

2

(20)

y119896+1 = argminy

sum

119894

1003817100381710038171003817y119894

10038171003817100381710038172+

1205732

2

10038171003817100381710038171003817y minusDu minus d1198961003817100381710038171003817

1003817

2

2 (21)


x119896+1 = argminx

120582119896+1

2

x minus f22+

1205731

2

10038171003817100381710038171003817x minusHu119896+1 minus b1198961003817100381710038171003817

1003817

2

2

(22)

b119896+1 = b119896 +Hu119896+1 minus x119896+1 (23)

d119896+1 = d119896 +Du119896+1 minus y119896+1 (24)



(

1205731

1205732

H119879H +D119879D) u =

1205731

1205732


(25)


u119896+1 = Fminus1(((

1205731

1205732



(1)

minus (d119896)(1)

)


(2)

minus (d119896)(2)

))

∘ ((

1205731

1205732


lowast(D(1))

∘F (D(1)) +Flowast(D(2)) ∘F (D(2)) )

minus1

)

(26)



y119896+1119894

= max10038171003817100381710038171003817D119894u119896+1 + d119896

119894

100381710038171003817100381710038172minus

1

1205732

0

D119894u119896+1 + d119896

119894

1003817100381710038171003817D119894u119896+1 + d119896

119894

10038171003817100381710038172

(27)






(1) If10038171003817100381710038171003817w119896+1 minus f1003817100381710038171003817

1003817

2

2le 119888 (28)


(2) If w119896+1 minus f2



10038171003817100381710038171003817w119896+1 minus f1003817100381710038171003817

1003817

2

2= 119888 (29)


x119896+1 =(120582119896+1f + 120573

1w119896+1)

(120582119896+1

+ 1205731)

(30)


120582119896+1

=

1205731

10038171003817100381710038171003817f minus w119896+11003817100381710038171003817

10038172

radic119888

minus 1205731 (31)






BSNR = 10 log10(

10038171003817100381710038171003817f minus f1003817100381710038171003817

1003817

2

2

1198981198991205902) (32)





1gt 0 and 120573

2gt 0


119896+1= 0 and x119896+1 = w119896+1


119896+1 and u119896+1




120591 = minus 0006 times BSNR + 109 (33)


2is essential


times 1205732






1

4 Numerical Results


ISNR = 10 log10(

1003817100381710038171003817f minus uclean

1003817100381710038171003817

2

2

1003817100381710038171003817u minus uclean

1003817100381710038171003817

2

2

) (34)



2

2u1198962








20


30


40



20


30


40




Prob 1APE-SBA 963 261 231 744 201 178


Prob 2APE-SBA 554 551 490 408 421 375


Prob 3APE-SBA 887 263 234 678 197 175



2u1198962

2le



119894119895= 1(1 + 119894

2+ 1198952) with





0 5 10 15 200

2

4

6

8

10

Runtime (s)

ISN

R (d

B)

(a)

0 5 10 15 200

2

4

6

8

Runtime (s)

ISN

R (d

B)

(b)

0 5 10 150

2

4

6

Runtime (s)

ISN

R (d

B)

(c)

0 5 10 150

1

2

3

4

Runtime (s)

ISN

R (d

B)

(d)

0 2 4 6 8 10 120

2

4

6

8

10

Runtime (s)

ISN

R (d

B)


(e)

0 2 4 6 8 10 12 140

2

4

6

Runtime (s)

ISN

R (d

B)


(f)


119894119895= 1(1+119894

2+1198952) (119894 119895 = minus7 7)




(a)


(b)


(c)


(d)



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











y119894= (119910(1)

119894 119910(2)

119894)

119879


⟨x x⟩P = sum119894119909119894119909119894 ⟨y y⟩T = sum

119894sum2

119896=1119910(119896)

119894119910(119896)

119894 and norm


D119894u = [(D(1)u)

119894 (D(2)u)

119894]

119879

with

(D(1)u)119894=

119906119894+119899



(6)

(D(2)u)119894=

119906119894+1

minus 119906119894 if mod (119894 119899) = 0

119906119894minus119899+1


(7)





[(D(1))119879

(D(2))119879

]

119879






1⟩ le 119869 (z) minus 119869 (z

1) forallz isin P

(8)


119863(z1)

119869= 119869 (z) minus 119869 (z


1⟩ (9)


3 Methodology




119894isin 1198772 for D

119894u


minuxy

sum

119894

1003817100381710038171003817y119894

10038171003817100381710038172+

120582

2

x minus f22


(10)


119869 (u x y) = sum

119894

1003817100381710038171003817y119894

10038171003817100381710038172+

120582

2

x minus f22 (11)


119863

(p119896u p119896

x p119896

y)



minus ⟨p119896y y minus y119896⟩ (12)


(u119896+1 x119896+1 y119896+1)

= argminuxy

119863

(p119896u p119896

x p119896

y)

119869(u x y u119896 x119896 y119896)

+

1205731

2

x minusHu22+

1205732

2

1003817100381710038171003817y minusDu100381710038171003817

1003817

2

2

(13)

p119896+1u = p119896u + 1205731H119879 (x119896+1 minusHu119896+1) + 120573

2D119879 (y119896+1 minusDu119896+1)

(14)

p119896+1x = p119896x + 1205731(Hu119896+1 minus x119896+1) (15)

p119896+1y = p119896y + 1205732(Du119896+1 minus y119896+1) (16)

if we define that

p0u = minus1205731H119879b0 minus 120573

2D119879d0

p0x = 1205731b0

p0y = 1205732d0

(17)


p119896u = minus1205731H119879b119896 minus 120573

2D119879d119896 p119896x = 120573

1b119896 p119896y = 120573

2d119896

119896 = 0 1

(18)


(u119896+1 x119896+1 y119896+1)

= argminuxy

120582

2

x minus f22+

1205731

2

10038171003817100381710038171003817x minusHu minus b1198961003817100381710038171003817

1003817

2

2

+sum

119894

1003817100381710038171003817y119894

10038171003817100381710038172+

1205732

2

10038171003817100381710038171003817y minusDu minus d1198961003817100381710038171003817

1003817

2

2

b119896+1 = b119896 +Hu119896+1 minus x119896+1

d119896+1 = d119896 +Du119896+1 minus y119896+1

(19)


u119896+1 = argminu

1205731

2

10038171003817100381710038171003817x119896 minusHu minus b1198961003817100381710038171003817

1003817

2

2+

1205732

2

10038171003817100381710038171003817y119896 minusDu minus d1198961003817100381710038171003817

1003817

2

2

(20)

y119896+1 = argminy

sum

119894

1003817100381710038171003817y119894

10038171003817100381710038172+

1205732

2

10038171003817100381710038171003817y minusDu minus d1198961003817100381710038171003817

1003817

2

2 (21)


x119896+1 = argminx

120582119896+1

2

x minus f22+

1205731

2

10038171003817100381710038171003817x minusHu119896+1 minus b1198961003817100381710038171003817

1003817

2

2

(22)

b119896+1 = b119896 +Hu119896+1 minus x119896+1 (23)

d119896+1 = d119896 +Du119896+1 minus y119896+1 (24)



(

1205731

1205732

H119879H +D119879D) u =

1205731

1205732


(25)


u119896+1 = Fminus1(((

1205731

1205732



(1)

minus (d119896)(1)

)


(2)

minus (d119896)(2)

))

∘ ((

1205731

1205732


lowast(D(1))

∘F (D(1)) +Flowast(D(2)) ∘F (D(2)) )

minus1

)

(26)



y119896+1119894

= max10038171003817100381710038171003817D119894u119896+1 + d119896

119894

100381710038171003817100381710038172minus

1

1205732

0

D119894u119896+1 + d119896

119894

1003817100381710038171003817D119894u119896+1 + d119896

119894

10038171003817100381710038172

(27)






(1) If10038171003817100381710038171003817w119896+1 minus f1003817100381710038171003817

1003817

2

2le 119888 (28)


(2) If w119896+1 minus f2



10038171003817100381710038171003817w119896+1 minus f1003817100381710038171003817

1003817

2

2= 119888 (29)


x119896+1 =(120582119896+1f + 120573

1w119896+1)

(120582119896+1

+ 1205731)

(30)


120582119896+1

=

1205731

10038171003817100381710038171003817f minus w119896+11003817100381710038171003817

10038172

radic119888

minus 1205731 (31)






BSNR = 10 log10(

10038171003817100381710038171003817f minus f1003817100381710038171003817

1003817

2

2

1198981198991205902) (32)





1gt 0 and 120573

2gt 0


119896+1= 0 and x119896+1 = w119896+1


119896+1 and u119896+1




120591 = minus 0006 times BSNR + 109 (33)


2is essential


times 1205732






1

4 Numerical Results


ISNR = 10 log10(

1003817100381710038171003817f minus uclean

1003817100381710038171003817

2

2

1003817100381710038171003817u minus uclean

1003817100381710038171003817

2

2

) (34)



2

2u1198962








20


30


40



20


30


40




Prob 1APE-SBA 963 261 231 744 201 178


Prob 2APE-SBA 554 551 490 408 421 375


Prob 3APE-SBA 887 263 234 678 197 175



2u1198962

2le



119894119895= 1(1 + 119894

2+ 1198952) with





0 5 10 15 200

2

4

6

8

10

Runtime (s)

ISN

R (d

B)

(a)

0 5 10 15 200

2

4

6

8

Runtime (s)

ISN

R (d

B)

(b)

0 5 10 150

2

4

6

Runtime (s)

ISN

R (d

B)

(c)

0 5 10 150

1

2

3

4

Runtime (s)

ISN

R (d

B)

(d)

0 2 4 6 8 10 120

2

4

6

8

10

Runtime (s)

ISN

R (d

B)


(e)

0 2 4 6 8 10 12 140

2

4

6

Runtime (s)

ISN

R (d

B)


(f)


119894119895= 1(1+119894

2+1198952) (119894 119895 = minus7 7)




(a)


(b)


(c)


(d)



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x119896+1 = argminx

120582119896+1

2

x minus f22+

1205731

2

10038171003817100381710038171003817x minusHu119896+1 minus b1198961003817100381710038171003817

1003817

2

2

(22)

b119896+1 = b119896 +Hu119896+1 minus x119896+1 (23)

d119896+1 = d119896 +Du119896+1 minus y119896+1 (24)



(

1205731

1205732

H119879H +D119879D) u =

1205731

1205732


(25)


u119896+1 = Fminus1(((

1205731

1205732



(1)

minus (d119896)(1)

)


(2)

minus (d119896)(2)

))

∘ ((

1205731

1205732


lowast(D(1))

∘F (D(1)) +Flowast(D(2)) ∘F (D(2)) )

minus1

)

(26)



y119896+1119894

= max10038171003817100381710038171003817D119894u119896+1 + d119896

119894

100381710038171003817100381710038172minus

1

1205732

0

D119894u119896+1 + d119896

119894

1003817100381710038171003817D119894u119896+1 + d119896

119894

10038171003817100381710038172

(27)






(1) If10038171003817100381710038171003817w119896+1 minus f1003817100381710038171003817

1003817

2

2le 119888 (28)


(2) If w119896+1 minus f2



10038171003817100381710038171003817w119896+1 minus f1003817100381710038171003817

1003817

2

2= 119888 (29)


x119896+1 =(120582119896+1f + 120573

1w119896+1)

(120582119896+1

+ 1205731)

(30)


120582119896+1

=

1205731

10038171003817100381710038171003817f minus w119896+11003817100381710038171003817

10038172

radic119888

minus 1205731 (31)






BSNR = 10 log10(

10038171003817100381710038171003817f minus f1003817100381710038171003817

1003817

2

2

1198981198991205902) (32)





1gt 0 and 120573

2gt 0


119896+1= 0 and x119896+1 = w119896+1


119896+1 and u119896+1




120591 = minus 0006 times BSNR + 109 (33)


2is essential


times 1205732






1

4 Numerical Results


ISNR = 10 log10(

1003817100381710038171003817f minus uclean

1003817100381710038171003817

2

2

1003817100381710038171003817u minus uclean

1003817100381710038171003817

2

2

) (34)



2

2u1198962








20


30


40



20


30


40




Prob 1APE-SBA 963 261 231 744 201 178


Prob 2APE-SBA 554 551 490 408 421 375


Prob 3APE-SBA 887 263 234 678 197 175



2u1198962

2le



119894119895= 1(1 + 119894

2+ 1198952) with





0 5 10 15 200

2

4

6

8

10

Runtime (s)

ISN

R (d

B)

(a)

0 5 10 15 200

2

4

6

8

Runtime (s)

ISN

R (d

B)

(b)

0 5 10 150

2

4

6

Runtime (s)

ISN

R (d

B)

(c)

0 5 10 150

1

2

3

4

Runtime (s)

ISN

R (d

B)

(d)

0 2 4 6 8 10 120

2

4

6

8

10

Runtime (s)

ISN

R (d

B)


(e)

0 2 4 6 8 10 12 140

2

4

6

Runtime (s)

ISN

R (d

B)


(f)


119894119895= 1(1+119894

2+1198952) (119894 119895 = minus7 7)




(a)


(b)


(c)


(d)



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1gt 0 and 120573

2gt 0


119896+1= 0 and x119896+1 = w119896+1


119896+1 and u119896+1




120591 = minus 0006 times BSNR + 109 (33)


2is essential


times 1205732






1

4 Numerical Results


ISNR = 10 log10(

1003817100381710038171003817f minus uclean

1003817100381710038171003817

2

2

1003817100381710038171003817u minus uclean

1003817100381710038171003817

2

2

) (34)



2

2u1198962








20


30


40



20


30


40




Prob 1APE-SBA 963 261 231 744 201 178


Prob 2APE-SBA 554 551 490 408 421 375


Prob 3APE-SBA 887 263 234 678 197 175



2u1198962

2le



119894119895= 1(1 + 119894

2+ 1198952) with





0 5 10 15 200

2

4

6

8

10

Runtime (s)

ISN

R (d

B)

(a)

0 5 10 15 200

2

4

6

8

Runtime (s)

ISN

R (d

B)

(b)

0 5 10 150

2

4

6

Runtime (s)

ISN

R (d

B)

(c)

0 5 10 150

1

2

3

4

Runtime (s)

ISN

R (d

B)

(d)

0 2 4 6 8 10 120

2

4

6

8

10

Runtime (s)

ISN

R (d

B)


(e)

0 2 4 6 8 10 12 140

2

4

6

Runtime (s)

ISN

R (d

B)


(f)


119894119895= 1(1+119894

2+1198952) (119894 119895 = minus7 7)




(a)


(b)


(c)


(d)



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












20


30


40



20


30


40




Prob 1APE-SBA 963 261 231 744 201 178


Prob 2APE-SBA 554 551 490 408 421 375


Prob 3APE-SBA 887 263 234 678 197 175



2u1198962

2le



119894119895= 1(1 + 119894

2+ 1198952) with





0 5 10 15 200

2

4

6

8

10

Runtime (s)

ISN

R (d

B)

(a)

0 5 10 15 200

2

4

6

8

Runtime (s)

ISN

R (d

B)

(b)

0 5 10 150

2

4

6

Runtime (s)

ISN

R (d

B)

(c)

0 5 10 150

1

2

3

4

Runtime (s)

ISN

R (d

B)

(d)

0 2 4 6 8 10 120

2

4

6

8

10

Runtime (s)

ISN

R (d

B)


(e)

0 2 4 6 8 10 12 140

2

4

6

Runtime (s)

ISN

R (d

B)


(f)


119894119895= 1(1+119894

2+1198952) (119894 119895 = minus7 7)




(a)


(b)


(c)


(d)



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 200

2

4

6

8

10

Runtime (s)

ISN

R (d

B)

(a)

0 5 10 15 200

2

4

6

8

Runtime (s)

ISN

R (d

B)

(b)

0 5 10 150

2

4

6

Runtime (s)

ISN

R (d

B)

(c)

0 5 10 150

1

2

3

4

Runtime (s)

ISN

R (d

B)

(d)

0 2 4 6 8 10 120

2

4

6

8

10

Runtime (s)

ISN

R (d

B)


(e)

0 2 4 6 8 10 12 140

2

4

6

Runtime (s)

ISN

R (d

B)


(f)


119894119895= 1(1+119894

2+1198952) (119894 119895 = minus7 7)




(a)


(b)


(c)


(d)



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(a)


(b)


(c)


(d)



5 Conclusions





Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Fast Total-Variation Image Deconvolution with … · 2019. 7. 31. · Research Article Fast Total-Variation Image Deconvolution with Adaptive Parameter Estimation