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

Elisha Achieng Ogada,1,2 Zhichang Guo,¹ and Boying Wu¹

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 [25–28] 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

compares the two images’ mean luminance and . The maximal value of , if , , measures the closeness of contrast of the two images and . Contrast is determined in terms of standard deviation, . Contrast comparison measure maximally if and only if , that is, when the images have equal contrast.

, where is covariance between and , is a structure comparison measure which determines the correlation between the images and . It attains maximal value of if structurally the two images coincide, but its value is equal to zero when there is absolutely no structural coincidence. The quantities , , and are small positive constants that avert the possibility of having zero denominators.

Taking the edge map from (27) with the tuning parameter, we have

Hence, the corresponding edge similarity measure is given by

All these measures have been deployed not only to judge the extent of signal recovery, but also to see the extent of structural coincidence between the original image and the reconstruction image. It should be noted, however, that there is no hard and fast rule for selecting between the error based measures and the structural similarity index measure [36]. Therefore, any or both can be used at the same time to measure the quality of image recovery.

We have compared results of the construction by our method to those of Perona-Malik method [1], total variation method of Rudin et al. [8], and the D--PM algorithm proposed by [5]. Adjustable parameters in the experiments were time step , thresholding\tuning parameter , and others depending on the respective model. However, fidelity parameter was dynamically obtained in the course and process of iteration.

In Tables 1 and 2, we have, respectively, presented the results of the reconstruction for the synthetic image, Figure 1, and those of the textured image, Lena, Figure 2. The comparisons are based upon PSNR, MAE, SSIM, and PSNRE values and visual effects.

Table 1: Numerical results for synthetic image experiments.

Algorithm

Parameters
Number of steps
CPU time (sec)
PSNR
MAE
SSIM
PSNRE

PM
30
4
0.25
356
1.56
36.57
3.78
0.9817
25.06

TV
30

0.25
205
1.66
34.56
4.73
0.9725
14.75

D--PM
30
1
0.25
40
2.33
37.2
3.18
0.9825
24.7

Our method
30
0.01
5
65
3.98
38.52
3.16
0.9905
24.90

Table 2: Numerical results for Lena image experiments.

Algorithm

Parameters
Number of steps
CPU time (sec)
PSNR
MAE
SSIM
PSNRE

PM
30
5
0.25
113
0.45
25.94
12.88
0.6939
20.69

TV
30

0.25
129
1.15
27.75
10.45
0.7698
20.70

D--PM
30
4
2
10
0.50
28.15
7.21
0.7780
24.57

Our method
30
0.1
5
15
1.07
27.84
10.20
0.7805
26.34

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Synthetic image (). (a) Original image. (b) Noisy image corrupted by Gaussian noise for . (c) PM algorithm, , (356 steps). (d) TV algorithm, (205 steps). (e) Our method, ,   (65 steps). (f) D--PM algorithm, ,  , (40 steps).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Lena image (). (a) Original image. (b) Noisy image corrupted by Gaussian noise for . (c) PM algorithm, , (113 steps). (d) TV method, (129 steps). (e) Our method, ,   (15 steps). (f) D--PM algorithm, ,   (10 steps).

For the nontextured synthetic image, Figure 1, it is observed that, in spite of higher CPU time, our method enjoys superior signal and structural recovery, at , , and (see Table 1). In addition, as evidenced by Figure 1(e), reconstructions due the proposed model do not show the relative blurriness speckles evident in Figure 1(c) (by PM method); it does not manifest the relative blockiness visible on Figure 1(d) (by TV method) neither does it show heavily jagged edges apparent in Figure 1(f) due to the D--PM algorithm. Comparing Figure 1(e) produced by our method and Figure 1(a)of the original image, it can be observed that the differences are extremely marginal. The edges also seem better preserved and sharper.

Furthermore, although in terms of numerical metrics our method compares fairly well with those obtained by the D--PM, it should be noted that the increased number of parameters in the D- that have to be manually tweaked or manipulated to obtain optimal results aggregately hinders the effectiveness of the method.

For the textured image of Lena in Figure 2, the metrics evidence that our method still gives better signal recovery and structural coincidence compared to its comparisons (see Table 2). By visual observation, Figure 2(c) produced by PM method shows speckles and blur, while Figure 2(d) shows heavy presence of staircase effect, and Figure 2(f) shows heavier ramp-like features. However, Figure 2(e) produced by our method not only shows no speckles and blur but also manifests significant reduction of staircase effect.

In respect of the edge similarity measure PSNRE, for the nontexture image (see Figure 1), we observe a phenomenon where the PM method only marginally beats the proposed method. However, the method far outperforms the TV method. For the texture image (see Figure 2), experiments show that the proposed method generally outperforms the PM, TV, and D--PM methods in reconstruction of image from noise (see Tables 1 and 2).

In addition, experiments demonstrate that the tuning parameter introduced in the implementation of the proposed model is texture sensitive. In the nontextured images, tends to be very small, while as texture increases the value of also tends to increase. Generally observed is that the AOS implementation algorithm shows gray value invariance, stability based on extremum principle, Lyapunov functionals, and convergence even for larger values of the scale parameter [33]. This does not apply in the case of TV and PM, since they become unstable for . The D--PM survives this eventuality only when implemented by the AOS algorithm.

Generally, therefore, the proposed formulation performs better than TV method and PM method and indeed fairly better than the D--PM algorithm not only in the quantitative metric measures but also in terms of visual effect. The D-α-PM method competes fairly well with the proposed method. However, the increased number of parameters in the D-α-PM may disadvantage its practical ability to produce optimal results. Moreover, our formulation seems to handle texture images better than its comparison. Finally, given the nonmonotone nature of the flux function of the D--PM model proposed by Guo et al. [5], it is an extremely ill-posed problem, generating both backward and forward diffusion at various times in the evolution process. Therefore, analytically speaking, its ability to arrive at steady state and produce data with minimum energy is highly in mathematical doubt. Notwithstanding the foregoing, the discrete or numerical implementation of the D--PM method still does give an approximated solution.

6. Conclusion

In this paper, we have proposed an alternative framework for total variation based image denoising models. The model is based on minimization of total variation with a functional coefficient. We set out to determine factors to consider when choosing the functional coefficient for the minimization potential, and we have established that, for a well-behaved minimization functional, it is desirable that the functional coefficient be of sublinear growth at infinity. The chosen functional coefficient, apart from its sublinear growth at infinity, must be one that results in a potential which is nondecreasing, of linear growth at infinity, and convex. Convexity can then be used to obtain lower semicontinuity, as it is also instrumental in determination of existence of a minimizer to the functional.

We picked on an example to illustrate the effectiveness of the consideration. We have shown for the example that existence and uniqueness of minimizer can be established and that the evolution equation has a unique entropy solution. In addition, numerical implementation of the evolution equation on images demonstrates better denoising results than those offered by traditional competition due to PM method, TV method, and even the D--PM in [5].

The successful application or otherwise of any regularization formulation in practical denoising always depends on the type of image under consideration, the type of noise involved, the application intended for the restored image, and indeed the platform upon which the formulation is implemented. Consequently, some of the constraints that might affect the practical performance of a regularization formulation modeled along the lines proposed here, including the sample formulation used, include the nature of the noise and even the hardware in which the implementation has been carried out. For more efficient performance, we suggest that platform with higher processing capabilities be deployed. The extent of degradation and the level of preservation required are also other factors that determine the performance of the formulation. For instance, in case of heavy degradation, it is widely recommended that a preconvolution of the noise image with Gaussian be done before the implementation of the suggested formulation to recover edges and other semantically important features. In addition, there may also be cases where noise removal may be counterproductive. This may occur when the oscillation due to noise is of the same scale as that of the features or patterns for which preservation is intended [37]. To surmount such a challenge, one may require a combination of formulations.

However, generally, we have observed that models derived along the lines of the approach proposed here give formulations which not only generate better results but also submit rather progressively to mathematical analysis.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by the National Science Foundation of China (11271100 and 11301113), the Ph.D. Programs Foundation of Ministry of Education of China (no. 20132302120057), the class General Financial Grant from the China Postdoctoral Science Foundation (Grant no. 2012M510933), the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF. 2011003), the Program for Innovation Research of Science in Harbin Institute of Technology (Grant no. PIRS OF HIT A201403), and also the 985 Project of Harbin Institute of Technology.

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