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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 939131, 16 pages
http://dx.doi.org/10.1155/2014/939131
Research Article

An Alternative Variational Framework for Image Denoising

1Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China
2Department of Mathematics, Egerton University, Egerton, Kenya

Received 13 March 2014; Accepted 6 April 2014; Published 5 May 2014

Academic Editor: Carlos Lizama

Copyright © 2014 Elisha Achieng Ogada 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.

Abstract

We propose an alternative framework for total variation based image denoising models. The model is based on the minimization of the total variation with a functional coefficient, where, in this case, the functional coefficient is a function of the magnitude of image gradient. We determine the considerations to bear on the choice of the functional coefficient. With the use of an example functional, we demonstrate the effectiveness of a model chosen based on the proposed consideration. In addition, for the illustrative model, we prove the existence and uniqueness of the minimizer of the variational problem. The existence and uniqueness of the solution associated evolution equation are also established. Experimental results are included to demonstrate the effectiveness of the selected model in image restoration over the traditional methods of Perona-Malik (PM), total variation (TV), and the D-α-PM method.

1. Introduction

The objective of any image restoration process should not focus only on the removal of noise, but it should also observe, as Perona and Malik [1], Koenderink [2], and Witkin [3] determined, that no new spurious details are created in the restored image; at each scale-space representation, the boundaries/edges are sharp or preserved, and, at all scales, intraregion smoothing is preferred to interregion smoothing.

In the light of the above considerations, researchers have observed that it is then logical to obtain or develop edge indicators that would be adapted to the local image structure [4]. Consequently, a number of edge indicators have been proposed and logically grafted into the partial differential equation (PDE) based evolution equations [1, 5]. In addition, energy minimization problems are continually being formulated which focus on producing adaptive partial differential equations [6, 7].

Total variation (TV) method, widely considered a powerful technique for smoothing, edge preservation, and general image restoration, was first proposed by Rudin et al. [8]. The method is based on the strength of the argument that TV norms principally are -norms of derivatives and that -norms provide the proper basis for image restoration [8, 9]. TV functionals are defined in the space of functions of bounded variation (BV) and therefore do not necessarily require image functions to be continuous and smooth. This fact makes them allow for “jumps” or discontinuities and thus be able to protect edges.

The -norm of the gradient of image allows removing noise; however, it has the adverse effect of penalizing too much the gradients corresponding to edges [10]. A functional based on the -norm does not permit discontinuities in the solution, and thus edges cannot be recovered properly [11].

The total variation (TV) norm in [8] is a regularization functional of the form Although TV regularization above allows for edge recovery, it has some demerits. Firstly, the formulation favours solutions which are piecewise constant. This has the effect of causing staircase effects and may even generate false edges on the image [12]. Secondly, the method has the effect of reducing contrast even in regions of the same pixel intensity or in noise-free observed images [13, 14].

Various modifications have, therefore, been proposed, in an attempt to address the drawbacks of the TV model, to make it as adaptive as possible to the local image structure. For instance, Strong and Chan in [15] proposed the spatially adaptive regularization functional of the form where the control factor is designed to slow diffusion in the neighbourhood of edges. Then, we have the efforts, according to Blomgren et al. in [13], which produced a denoising functional of the form where is a nondecreasing function of the magnitude of gradient, as and as . This model is designed to automatically tap into the benefits of both isotropic diffusion and TV regularization. For other modifications, we refer the reader to, among others, the works in [12, 16, 17].

In this paper, however, we propose an alternative framework of variational model for image denoising, where the regularization potential is a product of a gradient based functional coefficient and the norm of gradient of image (potential function for TV); namely, . That is, the coefficient function is a function of the magnitude of the gradient of the image. We propose the criteria for the choice of the coefficient. Finally, we have selected a model based on the proposed criteria, as an example for further analysis and demonstration of experimental results.

The structure of this paper is as follows. In Section 2, we present the proposed model (4) and discuss the general criteria for choosing the functional coefficient to the traditional TV potential. Section 2 is concluded by considering, based on the proposed criteria, an example functional for further analysis. In Section 3, we give certain preliminary definitions and lemmas we rely on, variously, in this paper. In addition, we prove the existence and uniqueness of the solution to the minimization problem (11) and the associated evolution equation (27)–(29). In Section 4, we define the weak solution to the evolution problem (27)–(11). Furthermore, we present the formulation of the approximate evolution equation (32)–(36), prove the existence of solutions to the approximate evolution problem, and conclude Section 4 with the existence and uniqueness of the solution to the evolution problem (27)–(29). In Section 5, we give the numerical schemes and experimental results to demonstrate the strength and effectiveness of our method. Additionally, we have presented a brief discussional comparison of our results with those of other methods like PM, original TV, and D--PM method by Guo et al. [5]. A brief summary concludes the paper in Section 6.

2. Proposed Model

In this section, inspired by the works of Chan and Shen [18], Vese [19], and Chen et al. [12], among others, we propose an alternative framework for total variation based denoising model. The model is based on minimization of a functional, where the regularization potential is a product of the total variation potential and a gradient based functional coefficient. The coefficient, which acts to penalize the norm of gradient and detect any edges, is also taken as function of the norm of the gradient of image. So, we present the general form of the model and determine certain properties of the functional coefficient, especially in terms of linearity, sublinearity, and superlinearity growth at infinity. Ultimately, we have selected a specific example of model, based on the proposed guidelines, for further analysis.

2.1. The New Framework for Energy Functional

The proposed energy functional is given in the following general form: where is a function that detects edges and penalizes the norm of gradient and is the noise image. Observe that if we set , then the kernel of becomes a variational problem of the -norm that is known not to allow for discontinuities, since it leads to the traditional edge obliterating isotropic diffusion [20], while if the problem becomes the usual TV functional, which allows a diffusion mechanism that is strictly normal to the image gradient [12].

The functional in (4) should verify the Euler-Lagrange equation Next, we establish certain properties of from (5) that should influence its choice. For the purposes of denoising, to be able to smooth the image in homogeneous regions, that is, regions where is expected such that . On the other hand, to preserve discontinuities (edges) in the image, that is, in areas where , takes a form such that .

Now, for a little more precision on the expected behaviour of , we decompose divergence part in terms of the tangent and normal directions to the isophote lines. Equation (5) then becomes Then, to be able to achieve isotropic diffusion within homogeneous regions, from (6), we may impose the condition that where is a constant. The above condition yields from (6) a diffusion model that diffuses isotropically.

In the regions neighboring the edges, the model should dissipate diffusion effects across the edges and favor diffusion along the edges. This implies that as ,  , where is an arbitrary positive constant, while . These conditions might be difficult to meet simultaneously for . Therefore, motivated by the works in [10, 19], a weaker compromise is imposed by demanding that both terms approach zero, but at different rates, with diffusion along the direction normal to the isophote lines approaching zero faster than the diffusion along the tangent to isophote lines. This leads to the condition that

Hence, we observe that, for to become an effective coefficient in the functional in (4), it should be such that conditions (7) and (8) are satisfied.

For simplicity of notation in subsequent stages, we will denote, from (5),

Furthermore, we are interested in the properties of with regard to its growth at infinity and its overall impact on the functional : Note that is a constant and growth.

2.1.1. Summary on the Characterization of Growth of   

The aim of this section is to discuss the property of the function coefficient that will lead to a functional of linear growth at infinity, but, of course, subject to conditions (7)-(8). The characterization is based upon the results of process (10).

Observe that setting the coefficient in the regularization kernel, as a function of the norm of gradient of the image to be linear growth, leads to a functional in (4), which is of superlinear growth. Such a functional yields associated diffusion equation which either diffuses images uniformly, without being sensitive to discontinuities or edges, or generally does not yield good denoising results [10, 21].

Choosing that is of superlinear growth leads to which is invariably of superlinear growth. For instance, if we set which is of superlinear growth at infinity, then we obtain the function . This functional is of superlinear growth and does not give good results in image denoising. This is because, with such a choice of , the derivative of would yield a nondegenerate elliptic differential operator of the second order, which would have an oversmoothing effect under the optimality condition [22].

Bildhauer and Fuchs [21] reckon that a regularization function need not necessarily be of power growth. Therefore, another example for of superlinear growth is , where is viewed as some kind of compromise between the cases and of coefficient of power growth [21]. This formulation too does not promise any better results.

Consequently, we observe that to achieve maximal results, which entails reconstructing a noise image in such a way that the edges are protected while diffusion within homogeneous regions proceeds rather uniformly, it is required that be of sublinear growth at infinity. Also, the chosen should be such that the functional generated is of linear growth. Moreover, should be such that the conditions in (7)-(8) are satisfied. Functionals which arise from such potential functions tend to give better image denoising results. Some examples of the functions of sublinear growth are , and . The function is a good candidate for further consideration.

In this paper, therefore, we choose the control function , which is function of sublinear growth at infinity. This leads to a nondecreasing potential function for , which is strictly convex and of linear growth at infinity given in the form .

With this choice, we observe the following further properties of ;(i);(ii);(iii) is nondecreasing for . This, as observed from (6), ensures only forward diffusion along the tangent to the isophote lines.

3. The Minimization Problem

Our model based on the presentation above is then given by where Hence, in this section, we study the existence and uniqueness of the minimization problem (11). The following preliminary information prefaces our reasoning here and in the subsequent sections of this paper.

3.1. Preliminaries

Definition 1. Let be an open subset of . A function has a bounded variation in , denoted by , if where denotes the space of such functions. Then, BV-norm is given by

Definition 2 (see [23]). If , then And writing and its total variation on , one has (see [7]) is a Radon measure, where is the density of the absolutely continuous part of with respect to the Lebesgue measure, , and is the singular part.

Lemma 3 (see [24]). Let . The function is convex if and only if and if , then is convex if and only if .

Lemma 4 (see [10]). Let be convex, even, and nondecreasing on with linear growth at infinity. Also, let be the recession function of defined by Then, for and setting , one has This implies that is lower semicontinuous for the topology.

3.2. Existence and Uniqueness of Solution to the Minimization Problem

Since the objective here is to preserve edges, which are viewed as discontinuities, the natural function space to seek the solution is the space. In image analysis, the space allows for discontinuities across the edges of images. Furthermore, due to nonreflexivity of the space, which would have been a more natural space within which to seek the solution, it is noted that the solution to the minimization problem might not exist there. Therefore, the -, which denotes - topology, provides the most reasonable alternative space for the existence of the solution. In what follows, we will, for convenience of notation, refer to - simply as . This space allows obtaining compactness because of the separability of space even though it is not reflexive [10]. Hence, we state and prove the following existence and uniqueness theorem for our problem.

Theorem 5. Given as assigned above, there exists which is a unique solution to the minimization problem (11).

Proof. Since as specifically defined above for (11) is of linear growth and given the fact that, it implies that it is also coercive. Consequently, there exists a sequence such that . Hence, we have , and, from the inequality of the second component of , namely, , if we define , , we observe that . We then deduce from [10] that and .
This indicates that is bounded in and . And from it is also clear that is abounded in . Thus, there exists a subsequence of such that And on the strength of the lemma on convex functions of measures (see Lemma 4) and the weak lower semicontinuity of the -norm for the second component of , we deduce that is lower semicontinuous. In fact, the recession function in the definition of convex function of measures is finite for our functional. That is, . Hence, we obtain that or We, thus, have Thus, is a minimizer of the problem (11). Uniqueness of is drawn from the strict convexity of and convexity of the second component of , which implies the overall strict convexity of the functional . Additionally, a strictly convex functional admits at most one minimum. This, then, implies that is a unique minimizer of (11).

3.3. The Associated Evolution Equation

From the energy minimization problem as assigned above, namely, we have the associated Euler-Lagrange equation given by with the Neumann boundary condition We compute for in the Euler-Lagrange equation (25) by putting it into a dynamical system, where the time is used as an evolution parameter. Hence, the evolution equation associated with the minimization problem (4) is given by where is the noise image, , and .

In order to see whether the potential , as defined above, respects the general principle of image reconstruction, where it is required that reconstructed image be formed by homogeneous regions separated by sharp edges, we decompose the divergence term of (25) using the local image structures like tangent and normal directions to the isophote lines. Writing it in its nonconservative form, analogous to (6), yields Notice from (30) that as , signaling homogeneous regions, the potential behaves like a linear isotropic diffusion encouraging uniform smoothing in both the and directions. However, as , corresponding to the neighbourhood of the edges, diffusion rate along the normal (or ) direction is diminished, while the diffusion along the tangent (or ) direction is preferred, thereby preserving the edges. The model therefore is well-behaved, since it reasonably satisfies the principle cited above.

4. Weak Solution to the Flow Associated with the Minimization Problem

In this section, we present the definition of weak solution to the evolution problem (27)–(29), propose an approximating evolution equation, establish existence result for the solution of the approximate evolution equation, and, then, by logical mathematical manipulation and passing to the limits, present a proof of the existence and uniqueness of the solution of the evolution problem (27)–(29).

We refer to the works in [2528] as the motivation for our definition of the weak solution to the evolution problem (27)–(29).

Definition 6. A measurable function , is called an entropy solution of (27)–(29) if ,  , in in the trace sense, and if there exists ,   , in such that for every , a.e. for .

4.1. Existence of the Solution of the Approximate Evolution Problem (32)–(34)

Before we study the existence and uniqueness of the evolution problem (27)–(29) above, let us consider the following approximate evolution problem: for and , we construct the approximation where which implies that

Theorem 7. The approximate evolution equation (32)–(34) with (35) and (36) has a weak solution , such that where , for any , for every ,  , a.e. on , and that .

Proof. We apply Rothe’s method in [29] to construct an approximating solution sequence for the approximate evolution problem (32)–(34). We divide the interval into equal parts, where . For any , for any integer and a function , we have for .
We then consider the difference approximating equation of (32) as follows: Denoting , then the above equation becomes The idea here is to prove that if the value of is known and , then (41) admits a weak solution .
From (41), we may back-project to the general functional defined in , given by It can be shown that above is convex and lower semicontinuous in . Hence, there exists a minimizing sequence for such that . For simplicity and without loss of generality, we will let . Then, for any test function with (40) and integration by parts, we have To obtain the approximate solution to the whole domain , we denote where is the indicator function of , Equation (43) implies that for .
In the steps that follow, we obtain some estimates for and . For this purpose, let us choose the test function in (43) such that . We then obtain Applying convexity leads to
Summing (48) from to for yields which implies that In addition, let us consider that for some appropriately defined constant . Observe from the above inequality that Application of Hölder inequality yields the following inequality: It is then clear from (53) and (54) that Hence, we can conclude from (50)–(54) that
Now, summing (48) from to leads to Equation (57) implies that Moreover, by definition of , we see that which by (58) implies that It can be deduced from (50) that Now, observe from (57) that Using Minkowski’s inequality on the strength of (62), we deduce that which leads to From the above estimate, it can be deduced that Denoting , then from (56), (60), (61), (64), and (65), it can be seen that the sequences ,  , , and are bounded. Hence, there exist subsequences of , , , and , respectively, denoted by the same sequences such that, as , for some , and .
Next, we show that . Observe from (44)-(45) that which by (58) leads to This implies that as . Since from the subsequences above it can be seen that converges to and converges to , both in , it follows from (68) that .
Furthermore, from (56), (60), (64), and (65) and letting , it can be deduced that Thus, the above convergence results imply that, as and for any , (46) yields
Now, it remains to show that . Here, we follow the works in [25, 30, 31].
Recalling that , then for any and for from to , by the monotonicity condition, we obtain the inequality
Now, letting in (43), we obtain Applying Young’s inequality on the first term of (72) together with the inequality (71) and integrating over, we obtain Summing up (73) for from to , we obtain Recalling the convergence sets in (66) and letting , (74) yields Equation (75) may be rewritten in the form Now, setting in (70), we obtain Then, substituting equation (77) into (76) leads to Since is arbitrary, we may set where and . We then have Sending , we obtain Observe that, in fact, equality holds if we set in the inequality above. We thus deduce that . Hence, a.e. in . Therefore, (70) together with (80) leads to the identity (37). Thus, is a weak solution to (32)–(34), taken together with (35) and (36).
To prove the second part, that is, the relation (38), we let in (70) and then , for. Consider Observe that from the above equation we have We deduce from the above equation that

4.2. Existence and Uniqueness of the Solution to the Evolution Problem (27)–(29)

Theorem 8. Let . Then, there exists a unique weak solution , , and in the trace sense.

Proof. By Theorem 7, there exists , which is a weak solution to the approximate problem (32)–(34) and a constant such that From (84), we deduce that there exists a subsequence of denoted by itself and a function   with such that, as ,
From (84), we may also deduce that Since , continuously [7, 27, 31, 32] and also considering (84), we deduce that Applying the method in [25], we next show that is weakly compact in . Using Jensen’s inequality and Hölder’s inequality, we may have that Hence, is bounded and equi-integrable in and is therefore weakly compact in . Thus, we may deduce that Hence, we obtain for every and in .
Now, it remains to show that .
For any and setting , we have Then, as above, there exist a function and the indicator function of such that And, for any , it can be shown that Letting , we obtain Now, taking into consideration equations (91), (93) and (94), we deduce that letting , we get that converges weakly to some function with . And since for any we may write , with satisfying condition (94), it follows that .
Next, we verify the solution definition inequality (31). By setting , (37) leads to Taking the limit as gives the inequality Taking the limit as for every and recognizing the arbitrary nature of , we obtain for every , a.e. in . That concludes the proof of the existence of the entropy solution of problem (27)–(29).
Uniqueness of Weak Solution. Let and be two entropy solutions of the problem (27)–(29), such that, respectively, their initial data are and . Then, for every , a.e. on , there exists such that In addition, let and be approximations, respectively, for and , such that a.e. on . By setting in (99) and in (100), adding the two equations, applying Lemma 3, and rearranging the result, we obtain Then, integrating the inequality above from to and taking the limit as , yield This establishes the uniqueness of the entropy solution.

5. Numerical Experiments

In this section, we show the performance of the proposed formulation in denoising images involving a Gaussian white noise. The results are then compared with those obtained by the classical methods of Perona and Malik (PM) [1], Rudin et al. (TV) [8], and Guo et al. [5].

The following numerical scheme has been proposed for the implementation of the model.

5.1. Additive Operator Splitting (AOS) Scheme

Here, we have implemented the evolution problem (27)–(29) using AOS scheme proposed by Weickert et al. in [33]. Thus, the equations are discretized as follows: where , where where is the set of the two neighbors of pixel (boundary pixels have only one neighbor) and is introduced, for convenience, as a tuning parameter in the implementation of the proposed formulation.

5.2. Discussion of the Results

The experiments in this work have been performed on a Compaq610 computer, having Intel(R) Core (TM)2 Duo CPU T5870 each 2.00 GHz, physical RAM of 4.00 GB, and Professional Windows 8 64-bit Operating System, on MATLAB R2013b. The image restoration performance has been measured in terms of the peak signal-to-noise ratio (PSNR), mean absolute deviation/error (MAE), structural similarity index measure (SSIM), the measure of similarity of edges (PSNRE), and visual effects. The iteration stopping mechanism is based on the maximal PSNR.

At the end of iteration process, the PSNR, MAE, SSIM, and PSNRE values are recorded. PSNR and MAE values as discussed in [34] are, respectively, given by the following formulas: where denotes the noise-free image, is the denoised image, is the dimension of image, and yields the gray scale range of the original image.

SSIM, designed by Wang et al. [35], is a quality metric used to measure the similarity between any two images. It is widely considered to work in a manner analogous to the human visual system. As opposed to the PSNR, SNR, MAE, and MSE, which are error based measures, SSIM models image distortion as a combination of loss of correlation, luminance degradation, and contrast distortion. Given any two images and , SSIM measure is given by the formula