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Research ArticleA Total Variation Model Based on the Strictly ConvexModification for Image Denoising
Boying Wu,1 Elisha Achieng Ogada,1,2 Jiebao Sun,1 and Zhichang Guo1
1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China2Department of Mathematics, Egerton University, P.O. Box 536-20115, Egerton, Kenya
Correspondence should be addressed to Jiebao Sun; [email protected]
Received 18 April 2014; Revised 16 May 2014; Accepted 26 May 2014; Published 18 June 2014
Academic Editor: Adrian Petrusel
Copyright ยฉ 2014 Boying Wu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We propose a strictly convex functional in which the regular term consists of the total variation term and an adaptive logarithmbased convex modification term. We prove the existence and uniqueness of the minimizer for the proposed variational problem.The existence, uniqueness, and long-time behavior of the solution of the associated evolution system is also established. Finally, wepresent experimental results to illustrate the effectiveness of the model in noise reduction, and a comparison is made in relation tothe more classical methods of the traditional total variation (TV), the Perona-Malik (PM), and the more recent D-๐ผ-PM method.Additional distinction from the other methods is that the parameters, for manual manipulation, in the proposed algorithm arereduced to basically only one.
1. Introduction
Noise removal, edge detection, contrast enhancement, in-painting, and segmentation have been the subject of intensemathematical image analysis and processing research fornearly three decades. Several methods have been pursuedover the passage of time. These include wavelet transform[1, 2], curvelet shrinkage methods [3โ6], and variationalpartial differential equation (PDE) based methods [7, 8].These methods generate processes that can easily be dividedinto either linear and nonlinear processes or isotropic andnonisotropic processes [9].
Due its ability to preserve crucial image features, suchas edges, nonlinear anisotropic diffusion is favored overisotropic diffusion [10]. Much interest, therefore, has focusedon understanding operations andmathematical properties ofthe nonlinear anisotropic diffusion and associated variationalformulations [11, 12], formulation of well-posed and stableequations [8, 11], extending and modifying anisotropic dif-fusion [11, 13, 14], and studying the relationships that existbetween the various image processing techniques [11, 15, 16].
The objective of any image denoising process should notfocus only on the removal of noise, but it should also ensurethat no spurious details are created on the restored image and
that the edges are preserved or sharpened [7, 17, 18]. It is,therefore, necessary to develop formulations which are sen-sitive to the local image structure, especially edges/contours[19]. Consequently, a number of edge indicators have beenproposed and logically grafted into the partial differentialequation (PDE) based evolution equations [7, 11, 20].
Some of these PDEs originate from variational problems.For instance, Rudin et al. [8] proposed a minimizationfunctional, widely referred to as the total variation (TV)functional, of the form
min๐ข
{๐น (๐ข) = โซ
ฮฉ
|โ๐ข| ๐๐ฅ + ๐โซ
ฮฉ
(๐ข โ ๐)2
๐๐ฅ} , (1)
where ๐ is the fidelity parameter, ๐ = ๐(๐ฅ) denotes thenoise image, and ฮฉ is an open bounded subset of R2. TVfunctionals are defined in the space of functions of boundedvariation (BV) and, therefore, do not necessarily requireimage functions to be continuous and smooth. This factmakes them allow jumps or discontinuities, and hence theyare able to preserve edges.
The original TV formulation has certain weaknesses.Firstly, the formulation is susceptible to backward diffusionsince it is not strictly convex. Secondly, the numerical imple-mentation cannot be accomplished without additional small
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 948392, 16 pageshttp://dx.doi.org/10.1155/2014/948392
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2 Abstract and Applied Analysis
perturbation quantity, say ๐, at the denominator [21โ23].Otherwise a spike/singularity is suddenly generated when|โ๐ข| = 0 in the homogeneous regions.This perturbation phe-nomenon is believed to contribute to some loss of accuracy inthe results of the restoration process.Moreover, the additionalparameter unnecessarily increases the number of parameters,therebymaking it difficult to determinewhich permutation ofparameter values will give optimal result.
Additionally, given that the method is more efficient inpreserving edges of uniform and small curvature, it mayexcessively smoothen and possibly destroy small scale fea-tures having more pronounced curvature edges [24]. TVregularization approach may also result in a loss of contrastand geometry of the final output images, even in noise freeobserved images [9, 25]. Furthermore, TV regularization hasdifficulties recovering texture, and there is also evidence ofenhanced noise when the fidelity parameter is chosen so thattexture is not removed [26]. Lastly, the formulation favorspiecewise constant solutions. This has the material effectof causing staircases (false edges) on the resultant image,especially from images severely degraded by noise [25].
However, given the strength of TV based techniques,especially in edge preservation, various modifications havebeen proposed. For instance, Vogel in [22, 27] proposed atotal variation penalty method of the form
min๐ข
โซ
ฮฉ
(๐ด๐ข โ ๐ง)2
๐๐ฅ + ๐ผโซ
ฮฉ
(โ|โ๐ข|2
+ ๐ฝ)๐๐ฅ, (2)
where ๐ฝ โฅ 0, ๐ด is a linear operator, and ๐ผ > 0 is thepenalty parameter. The formulation becomes total variationformulation of theRudin et al. form [8]when๐ฝ = 0 and there-fore largely suffers the same shortcoming of the ordinary TVformulation. Hence, it has no significant practical advantageover TV.
Strong and Chan [28] proposed an adaptive total varia-tion based regularization model of the form
min๐ขโBV(ฮฉ)
โซ
ฮฉ
๐ผ (๐ฅ) |โ๐ข| , (3)
where 0 โค ๐ผ(๐ฅ) โค 1 is a control factor which controlsthe speed of diffusion depending on whether the region ishomogeneous or an edge. This model demonstrated fairlygood results. However, since it is also not strictly convex,it is still susceptible to backward diffusion, which has thepotential of introducing blurs in the restored image.
Chambolle and Lions [29] proposed to minimize acombination of total variation and the integral of the squarednorm of the gradient and thus have
min๐ขโBV(ฮฉ)
1
2๐
โซ
|โ๐ข|โค๐
|โ๐ข|2
+ โซ
|โ๐ข|โฅ๐
|โ๐ข| โ
๐
2
, (4)
where ๐ is a parameter. In this formulation |โ๐ข| โฅ ๐ signalsedges, while |โ๐ข| โค ๐ signals homogeneous regions. Thismodel is successful in restoring images where homogeneousregions are separated by distinct edges but may becomesensitive to the thresholding parameter ๐ in the event ofnonuniform image intensities or heavy degradation [24].
A variable exponent adaptive model was proposed byChen et al. in [24]. It has the form
min๐ขโBV(ฮฉ)
โซ
ฮฉ
|๐ท๐ข|๐(๐ฅ)
+
๐
2
(๐ข โ ๐ผ)2
, ๐ > 0, (5)
where 1 < ๐ผ โค ๐(๐ฅ) โค 2, with ๐(๐ฅ) proposed as ๐(๐ฅ) =1 + 1/(1 + ๐|โ๐บ
๐โ ๐ผ(๐ฅ)|
2
), ๐บ๐is the Gaussian filter, and
๐ > 0, ๐ > 0 are fixed parameters. It is observed that themethod exploits the benefits of Gaussian smoothing when๐(๐ฅ) = 2 and the strength of TV regularization when ๐(๐ฅ) =1. This method also demonstrates good results and is indeeda great improvement of the earlier models. However, theconvolution of the image with a Gaussian kernel before theevolution process diminishes the accuracy of these models.This is because of the introduction of the scale variance of theGaussian; the scale variance is itself an additional parameterthat is subject to manual manipulation [12]. Furthermore, thediffusion process becomes ill-posed if scale variance is toosmall, while image features become smeared if the Gaussianvariance is too large [30]. Therefore, optimal selection of thescale variance remains a challenge. Parameter permutationgiving optimal result is a challenge due to the number of suchparameters involved.
Further literature surveys attest to the fact that research ineffective regularization functionals which have the ability togenerate diffusion processes that restore images, while simul-taneously preserving critical images features, the analysis, andpractical implementation of suchmodels, is still an extremelyactive concern.
Consequently, in this paper, we propose a new adaptivetotal variation (TV) formulation for image denoising, whichis strictly convex. The only parameter ๐พ, which is a thresh-olding parameter to be tweaked, is such that it depends on theevolution parameter ๐ก and is therefore not as grossly limitedas the choice of numerical values. The other parameter ๐ isdynamically obtained and is therefore not manually tuned.With all this the number of parameters is reduced to basicallyonly one. Our method approximates TV model, for highervalues of the thresholding parameter๐พ. Experimental results,presented here, demonstrate the effectiveness of our modelover the classical models of TV, PM, and D-๐ผ-PM.
The structure of this paper is as follows. In Section 2, wepresent the proposed model (6), its properties, justificationsof the model, and the associated evolution equation. InSection 3, we give certain preliminary definitions we relyon from time to time in this paper; we also prove theexistence and uniqueness of the solution to the minimizationproblem (6). In Section 4, we discuss the associated evolutionequation to theminimization problem (6). Also, we define theweak solution to the evolution problem (12)โ(14), derive theestimates for the solution of an approximating problem (32),prove the existence and uniqueness of the weak solution ofthe evolution problem (12)โ(14), and discuss the asymptoticbehaviour of the weak solution as ๐ก โ โ. In Section 5,we give the numerical schemes and experimental results todemonstrate the strength and effectiveness of our method.We have also presented a brief comparative discussion ofour results with the PM, original TV methods, and D-๐ผ-PMmethods. A brief summary concludes the paper in Section 6.
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Abstract and Applied Analysis 3
2. Proposed Model
In this section, motivated by the methods of [11, 21, 24],we propose a modified version of total variation denoisingmodel. The model is based on minimization of a totalvariation functional with a logarithm-based strictly convexmodification. First, we present the model and certain impor-tant properties of it.
2.1. The New Energy Functional. The new strictly convexenergy functional is given as follows:
min๐ขโBV(ฮฉ)โฉ๐ฟ2(ฮฉ)
{๐ผ (๐ข) = โซ
ฮฉ
ฮฆ (|โ๐ข|) ๐๐ฅ +
๐
2
โซ
ฮฉ
(๐ข โ ๐)2
๐๐ฅ} ,
(6)
where
ฮฆ (๐ ) = ๐ โ
1
๐พ
ln (1 + ๐พ๐ ) , (7)
where๐พ is a positive parameter, and ๐ is the noise image.Calculating directly, we can obtain the following propo-
sition.
Proposition 1. The function ฮฆ(๐ ) satisfies the following prop-erties:
(i) ฮฆ(0) = 0, ฮฆ(๐ ) = ๐พ๐ /(1 + ๐พ๐ ), and therefore ฮฆ(๐ ) isa strictly nondecreasing function for ๐ > 0;
(ii) ฮฆ(๐ ) = ๐พ/(1 + ๐พ๐ )2 > 0, and thereforeฮฆ(๐ ) is strictlyconvex;
(iii) ฮฆ(๐ ) = (๐พ/2)๐ 2 + ๐(๐ 2) as ๐ โ 0;(iv) limsโ+โ(ฮฆ(๐ )/๐ ) = 1, ๐ผ|๐ | โค ฮฆ(|๐ |) < |๐ |, 0 < ๐ผ < 1,
and therefore ฮฆ(๐ ) is the linear growth.
Compared with the work in [31], the new model satisfiesthe linear growth condition, which is much more natural.And the existence result for the flow associated with theminimization problem in [31] is only in one and two dimen-sions because themethods employed there use general resultson maximal monotone operators and evolution operators inHilbert spaces.
Remark 2. In this method,ฮฆ satisfies someminimal hypoth-eses; namely:
(H1) ฮฆ is a strictly convex, nondecreasing function fromR+ to R+, with ฮฆ(0) = 0;
(H2) because in the homogeneous regions weak gradientsshould be smoothed, there should be weak penaliza-tion in these region. Thus as |โ๐ข| = ๐ โ 0, ฮฆ(๐ ) โ๐พ๐
2; this signals isotropic diffusion;(H3) because edges, signaled by regions of strong gradients,
should be protected, regularization near the edgeswillbe penalized strongly. Thus, as |โ๐ข| = ๐ โ +โ,๐ผ|๐ | < ฮฆ(|๐ |) < ฮ|๐ |, 0 < ๐ผ โค ฮ; this not onlydemonstrates the linear growth nature of the modelbut also signals the TV edge preservation behaviourof the model.
In [27], the authors obtained
lim๐ฝโ0
๐ฝ๐ฝ(๐ข) = ๐ฝ
0(๐ข) , (8)
where ๐ฝ๐ฝ(๐ข) = โซ
ฮฉ
โ๐ฝ + |โ๐ข|2
๐๐ฅ. Actually, the perturba-tion ๐ฝ is always used to the eliminate singularity of theterm div(๐ท๐ข/|๐ท๐ข|) in the numerical experiments. With theinclusion or addition of ๐ฝ, what is then implemented isnot actually the TV formulation, but an approximation ofit. And, so, because of the approximation, the time step inthe discrete scheme becomes limited only to smaller values,as ๐ฝ diminishes [32]. The proposed model beats all thesechallenges and therefore has an easier design for numericalimplementation without the need for lifting parameters.Moreover, it is noticed that, for ๐ โ R,
lim๐พโ+โ
ฮฆ (|๐ |) = |๐ | , (9)
which is the form of the TV model. This means that anymerits accruing from the TV model could be still obtainedas ๐พ becomes larger and larger.
Remark 3. (1) Since TV potential (i.e., ฮฆ(๐ ) = ๐ ) is onlyconvex (i.e., ฮฆ(๐ ) = 0), it gives a local minimum, but itcannot guarantee uniqueness of the minimum energy. Theproposed energy potential (i.e.,ฮฆ(๐ ) = ๐ โ (1/๐พ) ln(1 + ๐พ๐ )),however, is strictly convex (i.e., ฮฆ(๐ ) = ๐พ/(1 + ๐พ๐ )2 >0). And, with additional strict convexity in the fidelity part,๐ป(๐ง) = (๐ง โ ๐)
2, the resultant energy functional is strictlyconvex in both |โ๐ข| and ๐ข, thereby giving a global minimumenergy and hence guaranteeing uniqueness of the results.
(2) The proposed method seeks to reduce the number ofparameters subject to manual manipulation. In the imple-mentation,๐พ ismade to depend on time, thereby leaving time(evolution parameter) as the only parameter to be tweaked.
Other modifications of the TV model are only convex in|โ๐ข| and therefore do not guarantee uniqueness of solutions;hence uniqueness of results is not easy to assure. Besides, theevolution system of the models contains 1/|โ๐ข| component,which runs into a spike when |โ๐ข| = 0, in smoothregions. This then makes only the implementation of theapproximations of the corresponding formulations possible(see [13, 20, 24, 27, 29]).
2.2. The Associated Evolution Equation. The Euler-Lagrangeequation for the energy functional (6) is
0 = โ div( ๐พโ๐ข1 + ๐พ |โ๐ข|
) + ๐ (๐ข โ ๐) , ๐ฅ โ ฮฉ, (10)
with
๐๐ข
๐ โ๐
= 0, ๐ฅ โ ๐ฮฉ. (11)
To compute a solution of (10) numerically, it is embeddedinto a dynamical scheme, where ๐ก is used as the evolution
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4 Abstract and Applied Analysis
parameter. Hence, corresponding to the proposed minimiza-tion model (6), we have the following evolution system:
๐ข๐ก= div( ๐พโ๐ข
1 + ๐พ |โ๐ข|
) โ ๐ (๐ข โ ๐) , (๐ฅ, ๐ก) โ ฮฉ ร (0, ๐) ,
(12)
๐ข (๐ฅ, 0) = ๐ (๐ฅ) , ๐ฅ โ ฮฉ, (13)
๐๐ข
๐ โ๐
= 0, (๐ฅ, ๐ก) โ ๐ฮฉ ร [0, ๐] . (14)
Note 1. The modified formulation also does not have thedisadvantage of running into a singularity, since the denom-inator does not become zero. This particular observationmakes it more convenient to design a numerical scheme forthe model.
Furthermore, to see more clearly the action of the diffu-sion operator (kernel), we decompose the divergence term onthe premise of the local image structures. That is, we breakit into the tangential (๐) and normal (๐) directions to theisophote lines. Consequently we have
div( โ๐ข1 + ๐พ |โ๐ข|
) =
1
(1 + ๐พ |โ๐ข|)2๐ข๐๐
+
1
1 + ๐พ |โ๐ข|
๐ข๐๐,
(15)
where
๐ข๐๐
=
1
|โ๐ข|2(๐ข
2
๐ฅ1
๐ข๐ฅ1๐ฅ1
+ ๐ข2
๐ฅ2
๐ข๐ฅ2๐ฅ2
+ 2๐ข๐ฅ1
๐ข๐ฅ2
๐ข๐ฅ1๐ฅ2
) ,
๐ข๐๐
=
1
|โ๐ข|2(๐ข
2
๐ฅ1
๐ข๐ฅ2๐ฅ2
+ ๐ข2
๐ฅ2
๐ข๐ฅ1๐ฅ1
โ 2๐ข๐ฅ1
๐ข๐ฅ2
๐ข๐ฅ1๐ฅ2
) .
(16)
The divergence term is visibly a weighted sum of the direc-tional derivatives, along the normal and tangential directions.Since we are also seeking to preserve the edges and otherimportant features of the image, smoothing in the tangentialdirection should be encouragedmore than that in the normaldirection, in the regions near the boundaries (edges).Observein the above formulation that, as |โ๐ข| increases, the coefficientof ๐ข
๐๐diminishes faster, reducing diffusion in the normal
direction, thus preserving edges. The edges are signalled byhigher values of the magnitude of gradient of ๐ข. However, as|โ๐ข| decreases, there is relatively uniform diffusion in both๐ข๐๐
and ๐ข๐๐
directions, thereby achieving isotropic diffusionin homogeneous regions. The homogeneous regions aresignaled by diminishing values of the magnitude of gradientof image ๐ข. The proposed model is, therefore, sensitive tothe local image structure. However, TV based denoisingmodel and most other modifications do not have that ๐ข
๐๐
component.This leads to a situation where the homogeneousparts are processed into piecewise constant regions, whoseboundaries reflect staircases or false edges in the image. Thissituation, for instance, affects the results of a modification byChen et al. [24] when ๐ = 1, and the model becomes thetraditional TV model.
3. The Minimization Problem
In this section, we prove the existence and uniqueness of theminimization problem (6). But, first, we give the followingpreliminary presentations which will guide our reasoning inthe subsequent sections of this paper.
3.1. Preliminaries
Definition 4. Let ฮฉ be an open subset of R๐. A function ๐ข โ๐ฟ1 has a bounded variation inฮฉ if
sup {โซฮฉ
๐ข div ๐๐๐ฅ : ๐ โ ๐ถ10(ฮฉ;R
๐
) ,๐โค 1} < โ, (17)
where BV(ฮฉ) denotes the space of such functions. Then BV-norm is given by
โโ๐ขโBV = โซฮฉ
|โ๐ข| + |๐ข|๐ฟ1(ฮฉ). (18)
Definition 5 (see [33]). If ๐ข โ BV(ฮฉ), then
๐ท๐ข = โ๐ข โ L๐
+ ๐ท๐
๐ข (19)
and ๐ท๐ข is a radon measure, where โ๐ข is the density of theabsolutely continuous part of๐ท๐ขwith respect to the Lebesguemeasure,L๐, and๐ท๐ ๐ข is the singular part.
Lemma 6 (see [34]). Let ๐ : R โ R+ be convex, even,nondecreasing onR+ with linear growth at infinity. Also letฮฆโbe the recession function of ฮฆ defined by
ฮฆโ
(๐) = lim๐ โโ
ฮฆ (๐ ๐)
๐
. (20)
Then for ๐ข โ ๐ต๐(ฮฉ) and setting ฮฆ(๐) = ฮฆ(|๐|) we have
โซ
ฮฉ
ฮฆ (๐ท๐ข) = โซ
ฮฉ
ฮฆ (|โ๐ข|) ๐๐ฅ + ฮฆโ
(1) โซ
ฮฉ
๐ท๐
๐ข. (21)
This implies that ๐ข โ โซฮฉ
ฮฆ(๐ท๐ข) is lower semicontinuous forthe ๐ต๐(ฮฉ) weakโ topology.
3.2. Existence and Uniqueness of Solution to the MinimizationProblem. The linear growth condition makes it natural toconsider a solution in the space
๐ = {๐ข โ ๐ฟ2
(ฮฉ) ; โ๐ข โ ๐ฟ1
(ฮฉ)2
} . (22)
However, this space is not reflexive. But sequences boundedin๐ are also bounded in BV(ฮฉ) and are therefore compact forBV weakโ topology. Moreover, due to the fact that ๐ฟ1-spaceis separable, weakโ topology allows us to obtain compactnessresults even if the space is not reflexive [34โ37]. Hence,by denoting BV(ฮฉ) weakโ topology simply as BV(ฮฉ), wetherefore seek the solution to the minimization problem (6)in the space BV(ฮฉ) โฉ ๐ฟ2(ฮฉ).
Theorem 7. The minimization problem ๐ผ(๐ข) (refer to (6)) hasa unique solution ๐ข โ ๐ต๐(ฮฉ) โฉ ๐ฟ2(ฮฉ).
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Abstract and Applied Analysis 5
Proof . From the linear growth property ofฮฆ(๐ ) we have
๐ผ |๐ | โค ฮฆ (|๐ |) < |๐ | , for 0 < ๐ผ < 1, (23)
which implies that
lim๐ โ+โ
ฮฆ (๐ ) = +โ. (24)
Thus ๐ผ(๐ข) is coercive. Therefore, let ๐ข๐be a minimizing
sequence in BV(ฮฉ) โฉ ๐ฟ2(ฮฉ). Then
๐ผ (๐ข๐) โค ๐, (25)
where๐ denotes a generic constant that may differ from lineto line. It clearly follows from above that
โซ
ฮฉ
ฮฆ(โ๐ข
๐
) ๐๐ฅ โค ๐ผ (๐ข
๐) โค ๐, โซ
ฮฉ
๐ข๐
2
๐๐ฅ โค ๐. (26)
This implies that {๐ข๐} is bounded in BV(ฮฉ) โฉ ๐ฟ2(ฮฉ). Hence
there exists a subsequence {๐ข๐๐
} of {๐ข๐} and a function ๐ข โ
BV(ฮฉ) โฉ ๐ฟ2(ฮฉ) such that
๐ข๐๐
โ ๐ข, strongly in ๐ฟ1 (ฮฉ) ,
๐ข๐๐
โ ๐ข, weakly in ๐ฟ2 (ฮฉ) .(27)
And, by Lemma 6 and the weak lower semicontinuity of the๐ฟ2-norm, we have that
๐ผ (๐ข) โค lim inf๐โโ
๐ผ (๐ข๐๐
) = minBV(ฮฉ)โฉ๐ฟ2(ฮฉ)
๐ผ (V) . (28)
Hence, there exists a solution of the minimization problem.Uniqueness of the solution follows from the strict convexityof the functional.
4. Existence and Uniqueness forthe Evolution Equation
In this section, we define a weak solution for the evolutionequations (12)โ(14). That is, if (6) has a minimum point ๐ข,then it should formally satisfy the Euler-Lagrange equation(12), subject to the boundary conditions (13)-(14). We definea weak solution that we are seeking for the equation system(12)โ(14). We then propose an approximating problem andderive certain a priori estimates to the approximating prob-lem. These will aid the process of proving the existence anduniqueness of the weak solution. Finally, we show the largetime behavior (asymptotic stability) of the weak solution.
4.1. Definition of Weak Solution. Let V โ ๐ฟ2(0, ๐;๐ป1(ฮฉ)), ๐ โBV(ฮฉ), V > 0, and ๐V/๐ โ๐ = 0, where ฮฉ
๐= ฮฉ ร [0, ๐] and
๐ข is a solution to (12)โ(14). Multiplying (12) by (V โ ๐ข) andintegrating overฮฉ, we have
โซ
ฮฉ
๐๐ก๐ข (V โ ๐ข) ๐๐ฅ + โซ
ฮฉ
๐พโ๐ข
1 + ๐พ |โ๐ข|
(โV โ โ๐ข) ๐๐ฅ
= โ๐โซ
ฮฉ
(๐ข โ ๐) (V โ ๐ข) ๐๐ฅ.(29)
Then applying the convexity condition of ฮฆ, that is, ฮฆ(๐ฅ) โฮฆ(๐ฆ) โฅ ฮฆ
(๐ฆ)(๐ฅ โ ๐ฆ), on both the second term on the leftside and on the right side of the above equation gives
โซ
ฮฉ
๐๐ก๐ข (V โ ๐ข) ๐๐ฅ + โซ
ฮฉ
ฮฆ (|โ๐ข|) ๐๐ฅ +
๐
2
โซ
ฮฉ
(V โ ๐)2๐๐ฅ
โฅ โซ
ฮฉ
ฮฆ (|โ๐ข|) +
๐
2
โซ
ฮฉ
(๐ข โ ๐)2
๐๐ฅ.
(30)
Integrating (30) with respect to ๐ก we get
โซ
๐ก
0
โซ
ฮฉ
๐๐ก๐ข (V โ ๐ข) ๐๐ฅ ๐๐ก + โซ
๐ก
0
๐ผ (V) ๐๐ก โฅ โซ๐ก
0
๐ผ (๐ข) ๐๐ก. (31)
Now, since V โ ๐ฟ2(0, ๐;๐ป1(ฮฉ)), we may choose V = ๐ข + ๐๐,where ๐ โ ๐ถโ
0(ฮฉ). We observe that the left-hand side of (31)
has aminimum at ๐ = 0.This shows that ๐ข is a solution of (12)in the distributional sense.The facts raised abovemotivate thefollowing definition for a weak solution to (12)โ(14).
Definition 8. A function ๐ข โ ๐ฟ2(0, ๐;BV(ฮฉ) โฉ ๐ฟ2) is called aweak solution of (12)โ(14) if ๐
๐ก๐ข โ ๐ฟ
2
(ฮฉ๐), ๐ข(๐ฅ, 0) = ๐(๐ฅ), on
ฮฉ and ๐ข satisfies (31) for every V โ ๐ฟ2((0, ๐);BV(ฮฉ)โฉ๐ฟ2) and๐ก โ [0, ๐].
4.2. Approximating Energy Functional. Let ฮฉ be an openbounded subset of R๐ and let ๐ โ BV(ฮฉ) โฉ ๐ฟโ. Then let usconsider the following approximating energy functional for1 < ๐ โค 2:
๐ผ๐(๐ข) = โซ
ฮฉ
ฮฆ๐(โ๐ข) ๐๐ฅ +
๐
2
โซ
ฮฉ
(๐ข โ ๐)2
๐๐ฅ, (32)
where
ฮฆ๐(๐ ) =
1
๐
(|๐ |๐
โ
1
๐พ
ln (1 + ๐พ|๐ |๐)) , (33)
with the following properties:
(1) ฮฆ๐(๐ ) = ๐พ|๐ |
2๐โ2
๐ /(1+๐พ|๐ |๐
) andฮฆ๐(๐ ) โฅ 0 โ๐ โ R๐;
(2) ฮฆ๐(๐ ) = ๐พ|๐ |
2๐โ2
[(2๐ โ 1) + ๐พ(๐ โ
1)|๐ |๐
]/[1 + ๐พ|๐ |๐
]
2
> 0. Hence ฮฆ(๐ ) is strictlyconvex in ๐ .
Remark 9. Since ๐ โ BV(ฮฉ) we can have a sequence ๐๐ โ
๐1,๐
(ฮฉ) with
๐๐ โ ๐ as ๐ โ 0 in ๐ฟ2 (ฮฉ) ,
โซ
ฮฉ
โ๐
๐
โ โซ
ฮฉ
โ๐.
(34)
And as a consequence of Theorem 2.9 in [20] we have
โซ
ฮฉ
โ๐
๐
โค ๐ถ. (35)
From property (iii) ofฮฆ(๐ ) it is easy to see that
ฮฆ(โ๐) โคโ๐. (36)
-
6 Abstract and Applied Analysis
This implies that
ฮฆ(โ๐๐ ) โค
โ๐
๐
โค ๐ถ. (37)
Hence
โซ
ฮฉ
ฮฆ(โ๐๐ ) โ โซ
ฮฉ
ฮฆ(โ๐) , as ๐ โ 0. (38)
On the strength of (32) and Remark 9 let us consider theapproximate evolution problem
๐๐ก๐ข๐ ,๐
= div (ฮฆ๐(โ๐ข
๐ ,๐)) โ ๐ (๐ข
๐ ,๐โ ๐
๐ ) , in ฮฉ ร [0, ๐] ,
(39)
๐๐ข๐ ,๐
๐ โ๐
= 0; on ๐ฮฉ ร [0, ๐] , (40)
๐ข๐ ,๐
(๐ฅ, 0) = ๐๐ , ๐ฅ โ ฮฉ. (41)
Let the following๐ฟโ bound also hold for the solution of (39)โ(41).Then, the following lemma indicates the boundedness ofthe solution๐ข
๐ ,๐of the above approximate evolution problem.
Lemma 10. Suppose ๐๐ โ ๐ฟ
โ
(ฮฉ) โฉ ๐ต๐(ฮฉ) and that ๐ข๐ ,๐
is asolution to the problem (39)โ(41); then
inf ๐๐ โค ๐ข
๐ ,๐โค sup๐
๐ . (42)
Proof. By a method similar to that of Zhou in [33], let ๐ =sup๐
๐ and define (๐ข
๐ ,๐โ๐)
+such that
(๐ข๐ ,๐
โ๐)+
:= {
(๐ข๐ ,๐
โ๐) , if ๐ข๐ ,๐
โ๐ โฅ 0,
0, otherwise;(43)
then multiplying (39) and integrating overฮฉ yields
โซ
ฮฉ
๐๐ก๐ข๐ ,๐(๐ข
๐ ,๐โ๐)
+
๐๐ฅ + โซ
ฮฉ
ฮฆ
๐(โ๐ข
๐ ,๐) โ โ๐ข
๐ ,๐๐๐ฅ
+ ๐โซ
ฮฉ
(๐ข๐ ,๐
โ ๐๐ ) (๐ข
๐ ,๐โ๐)
+
๐๐ฅ = 0.
(44)
Observe that the last two integrals in the above equation arenonnegative, based on the definition of (๐ข
๐ ,๐โ๐)
+
and thefact thatฮฆ
๐(๐ ) โ ๐ โฅ 0. Therefore, we have
1
2
โซ
ฮฉ
๐
๐๐ก
(๐ข๐ ,๐
โ๐)
2
+
๐๐ฅ โค 0, (45)
which indicates that ๐ฝ(๐ก) = (1/2) โซฮฉ
(๐ข๐ ,๐
โ๐)2
+
๐๐ฅ is adecreasing function in ๐ก. Since ๐ฝ(0) = 0 we have that
๐ฝ (๐ก) = 0, โ๐ก โ [0, ๐] . (46)
Hence ๐ข๐ ,๐
โค ๐ = sup๐๐ . Conversely, multiplying (39)
by (๐ข๐ ,๐
โ ๐)โ
, where ๐ = inf ๐๐ , and employing a similar
argument, we obtain that ๐ข๐ ,๐
โฅ โ๐, โ๐ก โ [0, ๐]. Therefore,inf ๐
๐ โค ๐ข
๐ ,๐โค sup๐
๐ , โ๐ก โ [0, ๐].
Another estimate (bound) is obtained via the followinglemma.
Lemma 11. The approximate evolution problem (39)โ(41) hasa unique solution ๐ข
๐ ,๐โ ๐ฟ
โ
(0, ๐;BV(ฮฉ)) with ๐๐ก๐ข๐ ,๐
โ
๐ฟ2
(0, ๐; ๐ฟ2
(ฮฉ)) such that
โซ
๐ก
0
โซ
ฮฉ
๐๐ก๐ข๐ ,๐
(V โ ๐ข๐ ,๐) ๐๐ฅ ๐๐ก + โซ
๐ก
0
๐ผ๐(V) ๐๐ก โฅ โซ
๐ก
0
๐ผ๐(๐ข
๐ ,๐) ๐๐ก,
(47)
for any ๐ก โ [0, ๐] and V โ ๐ฟ2(0, ๐;๐1,๐(ฮฉ)) with ๐V/๐ โ๐ = 0.Moreover
โซ
โ
0
โซ
ฮฉ
๐๐ก๐ข๐ ,๐
2
๐๐ฅ ๐๐ก
+ sup๐กโ[0,๐]
{โซ
ฮฉ
ฮฆ๐(โ๐ข
๐ ,๐) ๐๐ฅ +
๐
2
โซ
ฮฉ
(๐ข๐ ,๐
โ ๐๐ )
2
๐๐ฅ}
โค ๐ผ๐(โ๐
๐ ) .
(48)
Proof. Since ฮฆ(โ๐ข) is a lower semicontinuous and strictlyconvex function, then, by definitions provided in [34, 38, 39],โ div(ฮฆ(โ๐ข)), which is a subdifferential of โซ
ฮฉ
ฮฆ(โ๐ข)๐๐ฅ, ismaximal monotone. The approximate problem (39)โ(41) hasa unique solution such that ๐ข
๐ ,๐โ ๐ฟ
โ
(0, ๐;๐1,๐
(ฮฉ)). Now,to show that ๐ข
๐ ,๐is the weak solution to the approximating
problem (39)โ(41), as governed by (47), we multiply (39) byVโ๐ข
๐ ,๐for V โ ๐ฟ2(0, ๐;๐1,๐(ฮฉ))with ๐V/๐ โ๐ = 0 and integrate
overฮฉ and ๐ก โ [0, ๐]. Thus
โซ
๐ก
0
โซ
ฮฉ
๐๐ก๐ข๐ ,๐
(V โ ๐ข๐ ,๐) ๐๐ฅ ๐๐ก
= โซ
๐ก
0
โซ
ฮฉ
(div (ฮฆ๐(โ๐ข
๐ ,๐)) (V โ ๐ข
๐ ,๐)) ๐๐ฅ ๐๐ก
โ ๐โซ
๐ก
0
โซ
ฮฉ
((๐ข๐ ,๐
โ ๐๐ ) (V โ ๐ข
๐ ,๐)) ๐๐ฅ ๐๐ก,
โซ
๐ก
0
โซ
ฮฉ
๐๐ก๐ข๐ ,๐
(V โ ๐ข๐ ,๐) ๐๐ฅ ๐๐ก
+ โซ
๐ก
0
โซ
ฮฉ
ฮฆ
๐(โ๐ข
๐ ,๐) (โV โ โ๐ข
๐ ,๐) ๐๐ฅ ๐๐ก
= โ๐โซ
๐ก
0
โซ
ฮฉ
(๐ข๐ ,๐
โ ๐๐ ) (V โ ๐ข
๐ ,๐) ๐๐ฅ ๐๐ก.
(49)
Applying convexity condition on both sides of the aboveequation we obtain
โซ
๐ก
0
โซ
ฮฉ
๐๐ก๐ข๐ ,๐
(V โ ๐ข๐ ,๐) ๐๐ฅ ๐๐ก + โซ
๐ก
0
โซ
ฮฉ
ฮฆ๐(โV) ๐๐ฅ ๐๐ก
โ
๐
2
โซ
๐ก
0
โซ
ฮฉ
(V โ ๐๐ )2
๐๐ฅ ๐๐ก
โฅ โซ
๐ก
0
โซ
ฮฉ
ฮฆ๐(โ๐ข
๐ ,๐) ๐๐ฅ ๐๐ก +
๐
2
โซ
๐ก
0
โซ
ฮฉ
(๐ข๐ ,๐
โ ๐๐)
2
๐๐ฅ ๐๐ก,
(50)
which implies (47).
-
Abstract and Applied Analysis 7
Then, multiplying (39) by ๏ฟฝฬ๏ฟฝ๐ ,๐
and integrating overฮฉ and๐ก โ [0, ๐] we obtain
โซ
๐ก
0
โซ
ฮฉ
(๐๐ก๐ข๐ ,๐)
2
๐๐ฅ ๐๐ก = โซ
๐ก
0
โซ
ฮฉ
div (ฮฆ๐(โ๐ข
๐ ,๐)) ๏ฟฝฬ๏ฟฝ
๐ ,๐๐๐ฅ ๐๐ก
โ ๐โซ
๐ก
0
โซ
ฮฉ
(๐ข๐ ,๐
โ ๐๐ ) ๏ฟฝฬ๏ฟฝ
๐ ,๐๐๐ฅ ๐๐ก,
(51)
which yields
โซ
๐ก
0
โซ
ฮฉ
(๐๐ก๐ข๐ ,๐)
2
๐๐ฅ ๐๐ก + โซ
๐ก
0
๐๐ก(โซ
ฮฉ
ฮฆ๐(โ๐ข
๐ ,๐) ๐๐ฅ) ๐๐ก
+
๐
2
โซ
๐ก
0
๐๐ก(โซ
ฮฉ
(๐ข๐ ,๐
โ ๐๐ ) ๐๐ฅ)
2
๐๐ก = 0.
(52)
From the above equation together with (41) we obtain
โซ
๐ก
0
โซ
ฮฉ
(๐๐ก๐ข๐ ,๐)
2
๐๐ฅ ๐๐ก + โซ
ฮฉ
ฮฆ๐(โ๐ข
๐ ,๐)
๐ก
0
๐๐ฅ
+
๐
2
โซ
ฮฉ
(๐ข๐ ,๐
โ ๐๐ )
2
๐ก
0
๐๐ฅ = 0,
(53)
which by (40) and (41) produces (48).
4.3. Existence and Uniqueness of the Solution of the EvolutionProblem. Next, using the a priori estimates obtained above,through the approximate problem, we proceed to prove theexistence and uniqueness of the solution of the evolutionproblem (12)โ(14).
Theorem 12. Let ๐ โ ๐ต๐(ฮฉ) โฉ ๐ฟโ(ฮฉ). Then there exists aunique weak solution ๐ข โ ๐ฟโ(0,โ; ๐ต๐(ฮฉ) โฉ ๐ฟโ(ฮฉ)), ๐
๐ก๐ข โ
๐ฟ2
(๐โ), and ๐ข(๐ฅ, 0) = ๐ such that
โซ
โ
0
โซ
ฮฉ
๐๐ก๐ข
2
๐๐ฅ ๐๐ก + sup๐กโ[0,โ)
{๐ผ (๐ข)} โค ๐ผ (๐) , (54)
where ๐ก โ [0,โ), ๐โ:= ฮฉ ร [0,โ).
Proof. From Lemmas 10 and 11 we see that
๐๐ก๐ข๐ ,๐
๐ฟ2(๐โ)
+
๐ข๐ ,๐
๐ฟโ(๐โ)
+
๐ข๐ ,๐
๐ฟโ(0,โ;๐
1,๐(ฮฉ)โฉ๐ฟ
โ(ฮฉ))
โค ๐,
(55)
where๐ is some constant that may vary from line to line. Let{๐ข
๐ ,๐} by Lemma 10 be a bounded sequence of solutions of the
problem (39)โ(41). Then there exists a subsequence denoted
by {๐ข๐ ,๐๐
} of {๐ข๐ ,๐} and a function ๐ข
๐ โ ๐ฟ
โ
(ฮฉ), with ๏ฟฝฬ๏ฟฝ๐ โ
๐ฟ2
(ฮฉ; [0, ๐]), such that, as ๐๐โ 1,
๐ข๐ ,๐๐
โ ๐ข๐
strongly in ๐ฟ1 (ฮฉ) for each ๐ก โ [0,โ) , (56)
๐๐ก๐ข๐ ,๐๐
โ ๐๐ก๐ข๐
weakly in ๐ฟ2 (๐โ) , (57)
๐ข๐ ,๐๐
โ
โ ๐ข๐
weakโ in ๐ฟโ (๐โ) , (58)
๐ข๐ ,๐๐
โ ๐ข๐
in ๐ฟ2 (ฮฉ) uniformly in ๐ก, (59)
lim๐กโ0+
โซ
ฮฉ
๐ข๐ ,๐๐
(๐ฅ, ๐ก) โ ๐๐ (๐ฅ)
2
๐๐ฅ = 0. (60)
Indeed, from (55), there is a sequence {๐ข๐ ,๐๐
} and a func-tion ๐ข
๐ โ ๐ฟ
โ
(๐โ) with ๏ฟฝฬ๏ฟฝ
๐ โ ๐ฟ
2
(๐โ) such that (57) and (58)
hold. Observe also that for any ๐ โ ๐ฟ2(ฮฉ), as ๐ โ โ,
โซ
ฮฉ
(๐ข๐ ,๐๐(๐ฅ, ๐ก) โ ๐
๐ (๐ฅ)) ๐ (๐ฅ) ๐๐ฅ
= โซ
๐ก
0
๐๐ (โซ
ฮฉ
๐ข๐ ,๐๐(๐ฅ, ๐ ) ๐ (๐ฅ) ๐๐ฅ) ๐๐
โ โซ
๐ก
0
๐๐ (โซ
ฮฉ
๐ข๐ (๐ฅ, ๐ ) ๐ (๐ฅ) ๐๐ฅ) ๐๐
= โซ
ฮฉ
(๐ข๐ (๐ฅ, ๐ก) โ ๐
๐ (๐ฅ)) ๐ (๐ฅ) ๐๐ฅ,
(61)
which indicates that for each ๐ก๐ข๐ ,๐๐
โ ๐ข๐
in ๐ฟ2 (ฮฉ) . (62)By Lemmas 10 and 11, for each ๐ก โ [0,โ), {๐ข
๐ ,๐๐
(๐ฅ, ๐ก)} is abounded sequence in๐1,1(ฮฉ). Combining that fact with (62)we obtain that for each ๐ก as ๐
๐โ 1
๐ข๐ ,๐๐
โ ๐ข๐
in ๐ฟ1 (ฮฉ) . (63)Moreover, (60) follows from the fact that๐ข๐ ,๐๐
(โ , ๐ก) โ ๐ข๐ ,๐๐
(โ , ๐ก
)
2
๐ฟ2(ฮฉ)
โค
๐ก โ ๐ก
โซ
๐ก
0
โซ
ฮฉ
(๐๐ก๐ข๐ ,๐๐
)
2
๐๐ฅ ๐๐ก.
(64)
From (64) ๐ก โ ๐ข๐ ,๐๐
(โ , ๐ก) โ ๐ฟ2
(ฮฉ) is equicontinuous, and from(55) and (58) we have that
๐ข๐ ,๐๐
โ ๐ข๐
in ๐ฟ2 (ฮฉ) . (65)It then follows by standard argument thatwe canhave๐ข
๐ ,๐๐
โ
๐ข๐ in ๐ฟ2(ฮฉ) uniformly in ๐ก, giving us (59). From Lemma 10
and (58) we obtain that ๐ข๐ โ ๐ฟ
โ
(0,โ,BV(ฮฉ)โฉ๐ฟโ(ฮฉ))with๏ฟฝฬ๏ฟฝ๐ โ ๐ฟ
2
(๐โ).
Next we show that, for all V โ ๐ฟ2(0, ๐,๐1,๐(ฮฉ) โฉ ๐ฟ2(ฮฉ)),V > 0 and ๐V/๐ โ๐ = 0 and for each ๐ก โ [0,โ)
โซ
๐ก
0
โซ
ฮฉ
๐๐ก๐ข๐ (V โ ๐ข
๐ ) ๐๐ฅ ๐๐ก + โซ
๐ก
0
โซ
ฮฉ
ฮฆ (โV) ๐๐ฅ ๐๐ก
โ
๐
2
โซ
๐ก
0
โซ
ฮฉ
(V โ ๐๐ )2
๐๐ฅ ๐๐ก
โฅ โซ
๐ก
0
โซ
ฮฉ
ฮฆ(โ๐ข๐ ) ๐๐ฅ ๐๐ก โ
๐
2
โซ
๐ก
0
โซ
ฮฉ
(๐ข๐ โ ๐
๐)2
๐๐ฅ ๐๐ก.
(66)
-
8 Abstract and Applied Analysis
To show this end, from Lemma 11, we obtain
โซ
ฮฉ
๐๐ก๐ข๐ ,๐๐
(V โ ๐ข๐ ,๐๐
) ๐๐ฅ + โซ
ฮฉ
ฮฆ๐๐(โV) ๐๐ฅ
+
๐
2
โซ
ฮฉ
(V โ ๐๐ )2
๐๐ฅ
โฅ โซ
ฮฉ
ฮฆ๐๐
(โ๐ข๐ ,๐๐
) ๐๐ฅ +
๐
2
โซ
ฮฉ
(๐ข๐ ,๐๐
โ ๐๐ )
2
๐๐ฅ.
(67)
From (56) and (58) we deduce that there exists a subsequence{๐ข
๐ ,๐๐
} such that
๐ข๐ ,๐๐
โ ๐ข๐
strongly in ๐ฟ2 (ฮฉ; [0, ๐]) . (68)
Using (57), (59), and (68) andwe let๐๐โ 1; in (67)we obtain
โซ
ฮฉ
๐๐ก๐ข๐ (V โ ๐ข
๐ ) ๐๐ฅ + โซ
ฮฉ
ฮฆ๐๐(โV) ๐๐ฅ +
๐
2
โซ
ฮฉ
(V โ ๐)2๐๐ฅ
โฅ lim inf๐โโ
โซ
ฮฉ
ฮฆ๐๐
(โ๐ข๐ ,๐๐
) ๐๐ฅ +
๐
2
โซ
ฮฉ
(๐ข๐ โ ๐
๐ )2
๐๐ฅ.
(69)
But we define from the lower semicontinuity theorem that
โซ
ฮฉ
ฮฆ(โ๐ข๐ ) ๐๐ฅ โค lim inf
๐โโ
โซ
ฮฉ
ฮฆ๐๐
(โ๐ข๐ ,๐๐
) ๐๐ฅ. (70)
Using (70) in (69) and integrating over ๐ โ [0, ๐] we obtain
โซ
๐
0
โซ
ฮฉ
๐๐ก๐ข๐ (V โ ๐ข
๐ ) ๐๐ฅ ๐๐ก + โซ
๐
0
โซ
ฮฉ
ฮฆ (โV) ๐๐ฅ ๐๐ก
+
๐
2
โซ
๐
0
โซ
ฮฉ
(V โ ๐๐ )2
๐๐ฅ ๐๐ก
โฅ โซ
๐
0
โซ
ฮฉ
ฮฆ๐ (โ๐ข
๐ ) ๐๐ฅ ๐๐ก +
๐
2
โซ
๐
0
โซ
ฮฉ
(๐ข๐ โ ๐
๐ )2
๐๐ฅ ๐๐ก,
(71)
which confirms (66).Now, to complete the proof of the existence of solution to
(12)โ(14), it remains to pass to the limit as ๐ โ 0 in (71).Replacing ๐ by ๐
๐in (48), letting ๐ โ โ (๐
๐โ 1), and
using (57)โ(59) and (70) we obtain
โซ
โ
0
โซ
ฮฉ
๐๐ก๐ข๐
2
๐๐ฅ ๐๐ก
+ sup๐กโ[0,โ)
{โซ
ฮฉ
ฮฆ(โ๐ข๐ ) ๐๐ฅ +
๐
2
โซ
ฮฉ
(๐ข๐ โ ๐
๐ )2
๐๐ฅ}
โค โซ
ฮฉ
ฮฆ(โ๐๐ ) ๐๐ฅ.
(72)
From Lemma 10 and also from the above inequality we seethat ๐ข
๐ is uniformly bounded in ๐ฟโ(0,โ,BV(ฮฉ) โฉ ๐ฟโ(ฮฉ))
and ๐๐ก๐ข๐ is uniformly bounded in ๐ฟ2(๐
โ).Thismeanswe can
extract a subsequence {๐ข๐ ๐
} of {๐ข๐ } such that as ๐ โ โ (๐
๐โ
0) we have
๐๐ก๐ข๐ ๐
โ ๐๐ก๐ข weakly in ๐ฟ2 (๐
โ) ,
๐ข๐ ๐
โ
โ ๐ข weakโ in ๐ฟโ (ฮฉโ) ,
๐ข๐ ๐
โ ๐ข strongly in ๐ฟ1 (ฮฉ, [0, ๐]) , for each ๐ก โ [0,โ) ,
๐ข๐ ,๐๐
โ ๐ข๐
in ๐ฟ2 (ฮฉ) uniformly in ๐ก,
lim๐กโ0+
โซ
ฮฉ
๐ข๐ ,๐๐
(๐ฅ, ๐ก) โ ๐๐ (๐ฅ)
2
๐๐ฅ = 0.
(73)
Now, replacing ๐ with ๐ ๐in (71), letting ๐ โ โ (๐
๐โ 0),
and applying the lower semicontinuity in (70) we obtain
โซ
๐
0
โซ
ฮฉ
๐๐ก๐ข (V โ ๐ข) ๐๐ฅ ๐๐ก + โซ
ฮฉ
๐ผ (V) ๐๐ก โฅ โซฮฉ
๐ผ (๐ข) ๐๐ก, (74)
for all V โ ๐ฟ2(0, ๐,๐1,๐(ฮฉ)โฉ๐ฟ2(ฮฉ)), V > 0 and ๐V/๐ โ๐ = 0 andfor each ๐ก โ [0,โ). Hence ๐ข is a weak solution to (12)โ(14).Replacing ๐ by ๐
๐in (72), letting ๐ โ โ(๐
๐โ 0), and using
(57)โ(59) and (70) we obtain (54).
Uniqueness of the Weak Solution. With reference to the def-inition of solution inequality as given in (31), let ๐ข
1and ๐ข
2
be two weak solutions to the problem (12) to (14) such that๐ข1(๐ฅ, 0) = ๐ข
2(๐ฅ, 0) = ๐. Then we have, for two solutions, two
inequalities:
โซ
๐
0
โซ
ฮฉ
๐๐ก๐ข1(๐ข
2โ ๐ข
1) ๐๐ฅ ๐๐ก+ โซ
๐
0
๐ผ (๐ข2) ๐๐ก โฅ โซ
๐
0
๐ผ (๐ข1) ๐๐ก,
โซ
๐
0
โซ
ฮฉ
๐๐ก๐ข2(๐ข
1โ ๐ข
2) ๐๐ฅ ๐๐ก+ โซ
๐
0
๐ผ (๐ข1) ๐๐ก โฅ โซ
๐
0
๐ผ (๐ข2) ๐๐ก.
(75)
Now, adding the two inequalities (75) we obtain a more com-pact inequality given by
โซ
๐
0
โซ
ฮฉ
(๐๐ก๐ข2โ ๐
๐ก๐ข1) (๐ข
1โ ๐ข
2) ๐๐ฅ ๐๐ก โฅ 0. (76)
This implies that
โซ
๐
0
๐
๐๐ก
โซ
ฮฉ
(๐ข1โ ๐ข
2)2
๐๐ฅ ๐๐ก โค 0, (77)
which implies โ๐ข1โ๐ข
2โ = 0 for a.e. (๐ฅ, ๐ก) โ ๐
โ.This confirms
the uniqueness of the weak solution.
4.4. Large Time Behavior. Finally, we will assess the asymp-totic limit of the weak solution ๐ข(โ , ๐ก) as ๐ก โ โ. The essenceof this part is to demonstrate that over time the solution ofthe evolution problem (12)โ(14) ultimately converges to theunique minimizer of the functional (6).
Theorem 13. As ๐ก โ โ the weak solution ๐ข of the evolutionequation (12)โ(14) converges weakly in ๐ฟ2(ฮฉ) to a minimizer ๐ขof the functional ๐ผ(๐ข) in (6).
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Abstract and Applied Analysis 9
Proof. Let V โ BV(ฮฉ) โฉ ๐ฟโ(ฮฉ) into (31) to obtain
โ
1
2
โซ
ฮฉ
(V (๐ฅ) โ ๐ข)2
๐
0
๐๐ฅ + โซ
๐
0
๐ผ (V (๐ฅ)) ๐๐ก โฅ โซ๐
0
๐ผ (๐ข) ๐๐ก. (78)
The equation above simplifies to
โซ
ฮฉ
(๐ข (๐ฅ, ๐ ) โ ๐) V (๐ฅ) ๐๐ฅ
โ
1
2
โซ
ฮฉ
(๐ข2
(๐ฅ, ๐ ) โ ๐2
) ๐๐ฅ + ๐ ๐ผ (V (๐ฅ))
โฅ โซ
๐
0
๐ผ (๐ข) ๐๐ก.
(79)
By taking ๐ข(๐ฅ, ๐ ) = (1/๐ ) โซ๐ 0
๐ข(๐ฅ, ๐ก)๐๐ก, and have for each ๐ ,๐ข(๐ฅ, ๐ ) โ BV(ฮฉ)โฉ๐ฟโ(ฮฉ). Given that ๐ข is uniformly boundedin BV(ฮฉ), we conclude that there exists sequence ๐ข(๐ฅ, ๐
๐)
and its subsequence is still denoted by {๐ข(๐ฅ, ๐ ๐)} such that
๐ข(๐ฅ, ๐ ๐) โ ๐ข(๐ฅ, ๐ ) in ๐ฟ1(ฮฉ) and ๐ข(๐ฅ, ๐
๐) โ ๐ข(๐ฅ, ๐ ) weakโ in
BV(ฮฉ), as ๐ ๐โ โ, respectively. Hence dividing inequality
(79) by ๐ and taking the limit as ๐ ๐โ โ we obtain
๐ผ (V) โฅ ๐ผ (๐ข) . (80)
This demonstrates that indeed ๐ข is a weak solution to (12)โ(14) and also the unique minimizer of problem (6).
5. Numerical Experiments
In this section, we present the performance of our methodin denoising images involving a Gaussian white noise. Wehave then compared our results with the ones obtained by theclassical methods of PMmethod [7], TVmethod [8], and themore recent D-๐ผ-PM method [11].
5.1. Numerical Scheme. In the following two subsections, twonumerical discrete schemes, the PM scheme (PMS) and theadditive operator splitting (AOS) scheme, have been pro-posed.
5.1.1. PM Scheme. Here, we have proposed a numericalscheme similar to the original PMmethod, whereby (12)โ(14)are discretized as follows:
๐ถ๐
๐๐,๐=
๐พ
1 + ๐พ
โ๐๐ข๐,๐
, ๐ถ๐
๐๐,๐=
๐พ
1 + ๐พ
โ๐๐ข๐,๐
,
๐ถ๐
๐ธ๐,๐=
๐พ
1 + ๐พ
โ๐ธ๐ข๐,๐
, ๐ถ๐
๐๐,๐=
๐พ
1 + ๐พ
โ๐๐ข๐,๐
,
div๐๐,๐= [๐ถ
๐
๐๐,๐โ๐๐ข๐,๐+ ๐ถ
๐
๐๐,๐โ๐๐ข๐,๐+ ๐ถ
๐
๐ธ๐,๐โ๐ธ๐ข๐,๐
+ ๐ถ๐
๐๐,๐โ๐๐ข๐,๐] ,
(81)
and ๐ is dynamically determined according to the followingdiscretization scheme:
๐๐
=
1
๐2|ฮฉ|
โ
๐,๐
div๐๐,๐(๐ข
๐,๐โ ๐
๐,๐) ,
where |ฮฉ| = ๐๐ is the size of image.
(82)
Hence from (81) and (82) we have
๐ข๐+1
๐,๐= ๐ข
๐
๐,๐+ ๐div๐
๐,๐โ ๐
๐
๐ (๐ข๐,๐โ ๐
๐,๐) ,
๐ข0
๐,๐= ๐
๐,๐, ๐ข
๐
๐,0= ๐ข
๐
๐,1, ๐ข
๐
0,๐= ๐ข
๐
1,๐,
๐ข๐
๐,๐= ๐ข
๐
๐โ1,๐, ๐ข
๐
๐,๐= ๐ข
๐
๐,๐โ1,
(83)
where
โ๐๐ข๐,๐= ๐ข
๐โ1,๐โ ๐ข
๐,๐, โ
๐๐ข๐,๐= ๐ข
๐+1,๐โ ๐ข
๐,๐,
โ๐ธ๐ข๐,๐= ๐ข
๐,๐+1โ ๐ข
๐,๐, โ
๐๐ข๐,๐= ๐ข
๐,๐โ1โ ๐ข
๐,๐,
(84)
for ๐ = 0, 1, 2, . . . , ๐ and ๐ = 0, 1, 2, . . . ,๐.
5.1.2. AOS Scheme. In this part, using a AOS scheme, theproblem (12)โ(14) has been discretized as follows:
๐0
= 0,
๐ข๐+1
=
1
๐
๐
โ
๐=1
[๐ผ โ ๐๐๐ด๐(๐ข
๐
)]
โ1
[๐ข๐
+ ๐๐ (๐ โ ๐ข๐
)] ,
div๐ =(๐ข
๐+1
โ ๐ข๐
)
๐
,
๐๐
=
1
๐2๐๐
(๐ข โ ๐) div๐,
๐ข0
๐,๐= ๐
๐,๐= ๐ (๐โ, ๐โ) , ๐ข
๐
๐,0= ๐ข
๐
๐,1,
๐ข๐
0,๐= ๐ข
๐
1,๐, ๐ข
๐
๐ผ,๐= ๐ข
๐
๐ผโ1,๐, ๐ข
๐
๐,๐ฝ= ๐ข
๐
๐,๐ฝโ1,
(85)
where ๐ด๐(๐ข
๐
) = [๐๐,๐(๐ข
๐
)],
๐๐,๐(๐ข
๐
) :=
{{{{{{
{{{{{{
{
๐ถ๐
๐+ ๐ถ
๐
๐
2โ2
, [๐ โ N (๐)] ,
โ โ
๐โN(๐)
๐ถ๐
๐+ ๐ถ
๐
๐
2โ2
, (๐ = ๐) ,
0, (else) ,
๐ถ๐
๐:=
๐พ
1 + ๐พ
โ๐ข
๐
๐,๐
,
(86)
where
โ๐ข
๐
๐,๐
=
1
2
โ
๐,๐โN(๐)
๐ข๐
๐โ ๐ข
๐
๐
2โ
, (87)
whereN(๐) is the set of the two neighbors of pixel ๐ (boundarypixels have only one neighbor).
It is observed that AOS schemes with large time stepsstill reveal average grey value invariance, stability based onextremum principle, Lyapunov functionals, and convergenceto a constant steady state [10]. The AOS scheme is lessthan twice the typical effort needed for the PM scheme, arather low price for gaining absolute stability [10]. It is worthnoting that if we use div(โ๐ข/โ๐ + |โ๐ข|2) to approximate
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10 Abstract and Applied Analysis
Table 1: Numerical results for synthetic image (300 ร 300).
Algorithm Parameters Number of steps CPU time (sec) PSNR MAE PSNRE SSIM๐ ๐พ ๐
PM 30 12 0.25 232 12.73 34.23 2.84 25.26 0.9829TV 30 n/a 0.2 203 17.86 32.79 3.76 15.76 0.9748D-๐ผ-PM 30 1 0.25 42 2.40 37.30 3.38 24.76 0.9838PMS 30 n/a 0.20 262 12.01 35.75 2.15 25.60 0.9870AOS 30 n/a 3.00 17 2.51 33.30 2.92 25.40 0.9839
Table 2: Numerical results for Lena image (300 ร 300).
Algorithm Parameters Number of steps CPU time (sec) PSNR MAE PSNRE SSIM๐ ๐พ ๐
PM 30 12 0.25 60 2.43 27.10 7.75 21.70 0.7234TV 30 n/a 0.2 149 13.87 27.46 7.49 22.63 0.7960D-๐ผ-PM 30 4 2 13 0.90 28.17 7.35 24.78 0.7880PMS 30 n/a 0.2 122 5.80 27.73 7.45 26.60 0.8060AOS 30 n/a 2.0 7 1.24 28.06 7.27 27.80 0.8509
div(โ๐ข/|โ๐ข|) (TV kernel), in numerical scheme, the AOSscheme may be unstable because of the small number ๐.Our approximation, however, effectively avoids instabilitiesarising from such a scenario, as there is no need for liftingthe denominator, since it cannot not be equal to zero. Thishas been an additional motivation for considering AOS forour numerical experiments.The possibility of occurrence of azero denominator in the evolution problem is a phenomenonthatmakes the numerical implementation of TVproblematic.
5.2. Comparison with Other Methods. The experiments inthis work were performed on a Compaq610 computer, havingIntel Core 2 Duo CPU T5870 each 2.00GHz, physical RAMof 4.00GB, and Professional Windows 8 64-bit OperatingSystem, on MATLAB R2013b. The image restoration per-formance was measured in terms of the peak-signal-to-noise ratio (PSNR), mean absolute deviation/error (MAE),structural similarity index measure (SSIM), the measure ofsimilarity of edges (PSNRE), and visual effects. The iterationstopping mechanism was based on the maximal PSNR or thelowestMAE.The PSNR andMAE values were obtained usingthe formula by Durand et al. [40] and are given by
PSNR = 10log10
๐๐max ๐ข
0โmin ๐ข
0
2
๐ข โ ๐ข
0
2
๐ฟ2
๐๐ต,
MAE =๐ข โ ๐ข
0
๐ฟ1
๐๐
.
(88)
Also, taking the edge map EM(๐ข) from the evolution equa-tion, we have
EM (๐ข) = ๐พ(1 + ๐พ |โ๐ข|)
. (89)
And corresponding edge similarity measure is given, accord-ing to the formulation by Guo et al. [11], by
PSNRE = PSNR (EM (๐ข) ,EM (๐ข0)) ๐๐ต. (90)
The SSIM measures have been obtained according to theformula by Wang et al. [41] given by
SSIM (๐ข, ๐ข0) = ๐ฟ (๐ข, ๐ข
0) โ ๐ถ (๐ข, ๐ข
0) โ ๐ (๐ข, ๐ข
0) , (91)
where ๐ข0denotes the noise-free image, ๐ข is the denoised
image,๐ร๐ is the dimension of image, and |max ๐ข0โmin ๐ข
0|
yields the gray scale range of the original image. In addition,๐ฟ(๐ข, ๐ข
0) = (2๐
๐ข๐๐ข0
+ ๐1)/(๐
2
๐ข+ ๐
2
๐ข0
+ ๐1) compares the two
imagesโ mean luminances ๐๐ขand ๐
๐ข0
. The maximal value of๐ฟ(๐ข, ๐ข
0) = 1, if ๐
๐ข= ๐
๐ข0
and ๐ถ(๐ข, ๐ข0) = (2๐
๐ข๐๐ข0
+ ๐2)/(๐
2
๐ข+
๐2
๐ข0
+ ๐2), measures the closeness of contrast of the two
images ๐ข and ๐ข0. Contrast is determined in terms of standard
deviation, ๐. Contrast comparison measure ๐ถ(๐ข, ๐ข0) = 1
maximally if and only if ๐๐ข= ๐
๐ข0
, that is, when the imageshave equal contrast.
๐ (๐ข, ๐ข0) = (๐
๐ข๐ข0
+ ๐3)/(๐
๐ข๐๐ข0
+ ๐3), where ๐
๐ข๐ข0
is covari-ance between ๐ข and ๐ข
0, is a structure comparison measure
which determines the correlation between the images ๐ข and๐ข0. It attains maximal value of 1 if structurally the two images
coincide, but its value is equal to zero when there is absolutelyno structural coincidence. The quantities ๐
1, ๐
2, and ๐
3are
small positive constants that avert the possibility of havingzero denominators.
The results of our method were compared to PMmethod[7], TV method [8], and the D-๐ผ-PM method [11]. Tables 1and 2, respectively, give a summary of the results from theexperiments, using synthetic image in Figure 1 and Lenaimage in Figure 3.The parameters considered here are thresh-olding parameter ๐พ, variance ๐ฟ, the time step parameter ๐,and the convolution parameter ๐
1applied in the D-๐ผ-PM
method. The fidelity parameter ๐ was dynamically obtainedaccording to (82) or under the AOS scheme in Section 5.1.2,while the rest of the parameters were chosen to guaranteestability and attainment of optimal results.
For nontexture images as displayed in Figure 1 and theresults of our method using PMS scheme, as shown in
-
Abstract and Applied Analysis 11
(a) Original image (b) Noisy image (c) Proposed model I (PMS)
(d) Proposed model I (AOS) (e) PM model (f) TV model
(g) D-๐ผ-PM model
Figure 1: Synthetic image (300 ร 300). (a) Original image. (b) Noisy image corrupted by Gaussian noise for ๐ = 30. (c) Our algorithm byPMS; ๐ = 0.2 (262 steps). (d) Our algorithm by AOS scheme; ๐ = 3 (17 steps). (e) PMmethod;๐พ = 5 and ๐ = 0.25 (232 steps). (f) TVmethod;๐ = 0.2 (203 steps). (g) D-๐ผ-PM method; ๐
1= 0.5, ๐ = 30, ๐ = 0.25, and ๐พ = 1 (42 steps).
the fourth row of Table 1, demonstrate better performancethan those PM and TV as indicated by the higher PSNR,PSNRE, and SSIM values and lower MAE. And, in spiteof higher iterative steps, the corresponding CPU time islower compared to TV and the traditional Perona-Malik(PM) model. And although D-๐ผ-PM model shows betterresults than PSNR and MAE results, it can be observed thatin terms of edge feature recovery measure (PSNRE) andgeneral structural coincidence measure (SSIM) our methodperforms better. Moreover, implementing our model by theOAS scheme revealed faster execution in terms of both the
CPU time and iteration steps (see Table 1). The MAE haslower and even better PSNR results than TV (see Table 1).Looking at visual results of our method using PM scheme(PMS) and AOS, respectively, in Figures 1(c) and 1(d), thereis manifestly better visual appeal for our method comparedto the PM method (see Figure 3(e)) which shows somespeckles, TVmethod (see Figure 1(f)) which exhibits staircaseeffects and slight loss in contrast, and the D-๐ผ-method (seeFigure 3(g)) which shows slightly deformed edges.
However, for real image such as the given Lena image inFigure 3, the results as shown in Table 2 indicate that our
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12 Abstract and Applied Analysis
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2
3
4
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ree o
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ilarit
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Original imageNew AOS
ร104
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ร104
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ร104
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6
7
Pixel count (horizontal dimension)
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ree o
f sim
ilarit
y
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ร104
(e) D-๐ผ-PM model
Figure 2: Synthetic image (300ร300). Similarity graphs between the original image and results of our method (AOS and PMS), PMmethod,TV method, and D-๐ผ-PM method, respectively.
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Abstract and Applied Analysis 13
(a) Original image (b) Noisy image (c) Proposed model I (PMS)
(d) Proposed model I (AOS) (e) PM model (f) TV model
(g) D-๐ผ-PM model
Figure 3: Lena image (300 ร 300). (a) Original image. (b) Noisy image corrupted by Gaussian noise for ๐ = 30. (c) Our algorithm by PMS;๐ = 0.2 (122 steps). (d) Our algorithm by AOS scheme; ๐ = 2 (7 steps). (e) PM method; ๐พ = 12 and ๐ = 0.25 (60 steps). (f) TV method;๐ = 0.2 (149 steps). (g) D-๐ผ-PM method; ๐
1= 0.5, ๐ = 30, ๐ = 2, and ๐พ = 4 (13 steps).
method by AOS scheme gives superior results as shown bythe extremely lower iteration steps (7 steps), very short CPUprocessing time (1.24 sec), better PSNR (28.06), and evenlower MAE (7.27) compared to those of the TV method andPM method. Note that our model implemented using AOSperforms better than the samemodel implemented using PMscheme (PMS) for real images. And, in spite of the slightlysuperior PSNR and MAE values by the D-๐ผ-PM method,our method not only gives better edge preservation as evi-denced by the higher PSNRE, but also gives closer structuralcoincidence than the other three models. Further, the visual
appeal of our method, whether using PMS (Figure 3(c)) orAOS scheme (Figure 3(d)), excels those of the traditionalPerona-Malik (PM) (see Figure 3(e)) method which showsspeckles and a bit of blur, TVmethod (see Figure 3(f)) whichintroduces staircasing effects on the denoised image, andeven the D-๐ผ-PMmethod, which shows blockiness and someslight speckle effects.
In addition, from the similarity curves given in Figures 2and 4, it can be observed in the light of the marked areas,for instance, that this model performs better than its com-parisons, both for the synthetic image and real image. Note
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14 Abstract and Applied Analysis
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Figure 4: Lena image (300ร 300). Similarity graphs between the original image and results of our method (AOS and PMS), PMmethod, TVmethod, and D-๐ผ-PM method, respectively.
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Abstract and Applied Analysis 15
also that, even though in performance metrics, especially interms of PSNR and MAE, the D-๐ผ-PM method seems toperform better, the similarity curves attest that our methodgenerates restored images that more closely match the origi-nal image than the results obtained by the D-๐ผ-PM method(see Figures 2(e) and 4(e)).
6. Conclusion
In this paper, we have proposed a modified total variationmodel based on the strictly convex modification for imagedenoising. The main idea was to offer a better image restora-tionmodel that is strictly convex and is, therefore, not subjectto backward diffusion which has the potential of introducingblurs and to limit the number of parameters available formanualmanipulation.This, probably, explains why, in spite ofthe fact that ourmodel tends toTVas๐พ tends to+โ, its prac-tical implementation enjoys better image visual sharpness.This visual sharpness is comparably visible in the results byPM method, but it is still marred behind the oversmoothingand speckle effects in PM images. In fact, the reason why theD-๐ผ-PM method tends to give images that have some blurand deformed edges is attributable to the backward diffusionthat it introduces in the course of diffusion. Indeed, it is diffi-cult to assure convergence to a solution of minimum energygiven the backward-forward motion of the diffusion processinduced by the D-๐ผ-PM algorithm. At the practical level, theincreased number of parameters in the D-๐ผ-PM algorithmtends to make it difficult to arrive at a definite permutation ofparameter values that would give an optimal result.
For the proposed method, we have demonstrated theexistence and uniqueness of the solution of the model.Moreover, numerical implementation of our model also doesnot suffer potential inaccuracies typical of the TV method,since we do not need to add any small perturbation constantin TV method. Our thresholding parameter ๐พ dependson the evolution parameter ๐ก and therefore does not haveto be constrained to smaller values as in the case of PMmodel. The resultant evolution equation has been discretizedand implemented using PM scheme and AOS scheme todemonstrate the performance of our algorithm. From thegiven experimental results, the PSNR values, MAE values,Iterative steps, the CPU processing time, PSNRE, SSIM, andvisual appeal of our denoised images all testify that ourmethod is actually a good balance of the PM, TV, and D-๐ผ-PM methods and hence a better image restoration model.
However, in real life application, we observe that the suc-cess of any denoising algorithm depends on the type of imagebeing considered, the type of noise, the application intendedfor the results of the restoration, the extent of degradation,and indeed the implementation platform. And although wehave only considered additive Gaussian white noise, differentkinds of noise (whether Poisson, speckle, salt, or paper,among others) will require different formulations for thefidelity part and may even demand the application of morethan just one formulation for effectiveness. The choice ofplatform and scheme of implementation must also be appro-priately made for efficient performance of the formulation.
With respect to the use of the restoration results, thereare situations where the noise removal, generally, may becounterproductive. This usually occurs when the oscillationsdue to the noise are of comparable scale to those of thefeatures being targeted for preservation. A case like this mayrequire a combination of formulations [42, 43].
For images that are heavily degraded, it might be nec-essary to do a preconvolution, to blur the noise effects, andthen to use an effective formulation such as this one torecover semantically important features such as the edges andcontours of the image.
Conflict of Interests
The authors declare that they have no conflict of interestswhatsoever and do approve the publication of this paper.
Acknowledgments
This work is partially supported by the National ScienceFoundation of China (11271100 and 11301113), the Ph.D.Programs Foundation of Ministry of Education of China(no. 20132302120057), the class General Financial Grantfrom the China Postdoctoral Science Foundation (Grantno. 2012M510933), the Fundamental Research Funds for theCentral Universities (Grant nos. HIT. NSRIF. 2011003 andHIT. A. 201412), the Program for Innovation Research ofScience in Harbin Institute of Technology (Grant no. PIRSOF HIT A201403), and Harbin Science and TechnologyInnovative Talents Project of Special Fund (2013RFXYJ044).
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