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Mathematical Problems in Engineering
Volume 2015, Article ID 636295, 12 pages
http://dx.doi.org/10.1155/2015/636295
Research Article

Patch Similarity Modulus and Difference Curvature Based Fourth-Order Partial Differential Equation for Image Denoising

National Key Laboratory for Electronic Measurement Technology, North University of China, Taiyuan 030051, China

Received 15 October 2014; Revised 16 June 2015; Accepted 25 June 2015

Academic Editor: Ivanka Stamova

Copyright © 2015 Yunjiao Bai 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

The traditional fourth-order nonlinear diffusion denoising model suffers the isolated speckles and the loss of fine details in the processed image. For this reason, a new fourth-order partial differential equation based on the patch similarity modulus and the difference curvature is proposed for image denoising. First, based on the intensity similarity of neighbor pixels, this paper presents a new edge indicator called patch similarity modulus, which is strongly robust to noise. Furthermore, the difference curvature which can effectively distinguish between edges and noise is incorporated into the denoising algorithm to determine the diffusion process by adaptively adjusting the size of the diffusion coefficient. The experimental results show that the proposed algorithm can not only preserve edges and texture details, but also avoid isolated speckles and staircase effect while filtering out noise. And the proposed algorithm has a better performance for the images with abundant details. Additionally, the subjective visual quality and objective evaluation index of the denoised image obtained by the proposed algorithm are higher than the ones from the related methods.

1. Introduction

In the field of image processing, image denoising is the basis of image analysis, pattern recognition, and machine vision. It has been widely used in various applications, such as medical images, remote sensing images, and radiographic images. Therefore, image denoising has become a very intensive research topic.

The traditional denoising methods, such as the Gaussian filter, median filter, average filter, and wiener filter, can remove noise well. Unfortunately, these methods are not effective for preserving image edges and texture details. In recent years, novel image denoising methods based on the wavelet transform [1], the nonlocal mean [2], and the partial differential equation (PDE) [36] have been proposed to get clear and high-quality images. Thereinto, nonlinear anisotropic diffusion models based on the PDE are able to effectively relieve the contradiction between noise removal and edge preservation, which motivates the researchers’ considerable interest.

In 1990, Perona and Malik (PM) first proposed the classical anisotropic diffusion PDE [7] to improve the isotropic diffusion PDE, which is expressed aswhere is the divergence operator and is the absolute value of the gradient of the image . The diffusion coefficient is a nonnegative function that is given byorwhere is the gradient threshold. In 1992, Catté et al. [8] introduced the Gaussian smoothing kernel to regularize the PM model, which can effectively solve the ill-posed problem of the PM model. Recently, a series of research has been carried out on the PM model, which mainly focuses on how to preserve texture details and avoid staircase effect. In order to solve the problem that the PM model may obscure edges and fine details, the local gray-level variance [9], the local entropy [10], and the difference curvature [11] that can effectively distinguish between fine details and noise were introduced to dynamically change the size of the gradient threshold and then slow the diffusion process in the detail region, achieving the goal of removing noise and retaining more image details. On the other hand, You and Kaveh (YK) introduced a fourth-order PDE-based denoising model [12], in which the absolute value of Laplacian is used as the edge indicator. The YK model can suppress the staircase effect caused by the PM model. Even so, it has the weak capacity of edge preservation and tends to leave the processed image with isolated black and white speckles.

So far, much research regarding speckle removal and edge preservation has been devoted to improving the YK model [1317]. For instance, Liu et al. presented an adaptive fourth-order PDE filter [14] for image denoising, which can maintain the jump discontinuities while removing noise; Jidesh and George developed a gauss curvature driven fourth-order diffusion equation [16] adopted in image denoising, which can provide a natural look to the filtered image without causing staircase artifacts; a new PDE combining a second-order filter and a fourth-order filter [17] was proposed by Liu and Xiang, which can improve the quality of the denoised image. Nevertheless, it is difficult for this method to artificially select a series of parameters.

In this paper, we propose a new fourth-order PDE combining a new edge indicator called patch similarity modulus and the difference curvature for image denoising, which takes the superiority of the patch similarity modulus to remove noise and preserves edge features by the inherent characteristics of the difference curvature. Comparative studies with the most relevant image denoising methods in the literature demonstrate that the proposed algorithm can significantly improve the denoised image, and what is more, it can keep more weak edges and details.

The remainder of this paper is organized as follows. Section 2 reviews the related work about fourth-order PDE. Section 3 presents the new edge indicator, patch similarity modulus, and a new fourth-order PDE based on the patch similarity modulus and the difference curvature. Section 4 presents the numerical implementation of the proposed algorithm. The experimental results are demonstrated in Section 5, and the conclusion is drawn from this paper in Section 6.

2. Related Work about Fourth-Order PDE

It is well known that second-order PDEs, such as the PM model and total variation model, determine the diffusion degree of edge and nonedge regions with the gradients of different directions, which have achieved a good tradeoff between noise removal and edge preservation. However, these methods tend to exhibit staircase effect in the denoised image, which will mislead the postprocessing analysis and interpretation. In order to avoid staircase artifacts, You and Kaveh first proposed a fourth-order PDE for image denoising [12]. The energy functional introduced in [12] isthe corresponding Euler equation of which can be solved through the following gradient descent procedure:Here, is the image support, is the evolution time, is the absolute value of the Laplacian of the image , is an increasing function associated with the diffusion coefficient , that is, , and is a nonnegative and monotonically decreasing function of , defined aswhere is the contrast parameter. Although the YK model has been demonstrated to be capable of avoiding staircase effect, isolated speckles will be produced in the evolution process. Besides, the high-frequency components are also oversmoothed, which leads to losing the edge and detail characteristics of the image. Based on the YK model, Hajiaboli introduced the gradient modulus in place of the absolute value of Laplacian as the edge indicator in the diffusion coefficient, and the improved fourth-order PDE model [18] is given bywhereHere is the gradient modulus which is used to detect the characteristics of the image . It has been proved that (7) can remove noise without bringing about isolated speckles at the cost of weakening edges. In addition to this, taking into the edge orientation, Hajiaboli proposed the following nonlinear fourth-order diffusion equation [19]:where denotes the gradient direction and is perpendicular to , that is, the direction of level set. and are the second-order directional derivatives of the image in and , respectively. Due to the anisotropic diffusion that the diffusion coefficient is smaller in the gradient direction than in the direction of level set, this denoising method has a better edge-preserving capacity, compared with the method in the literature [18]. However, because of the diffusion inconformity in the directions of gradient and level set, the denoised image has staircase artifacts in the flat region when it comes to the distorted image corrupted by a high level of noise.

3. The Proposed Algorithm

3.1. New Edge Indicator

In the literature [8], Catté et al. introduced the Gaussian smoothing kernel to regularize the PM model, which can effectively smooth noise and then avoid staircase effect. Thereinto, the gradient information is based on the intensity similarity of each single pixel. Similarly, the existing edge indicators, such as, the gradient modulus and the absolute value of Laplacian, also depend on the intensity similarity of each single pixel, regardless of the change of the adjacent pixels around the processed pixel. Motivated by the idea in [20], we propose a new edge indicator employing the intensity similarity of neighbor pixels, that is, patch similarity modulus. The patch similarity modulus at the point of the image is defined aswhere

Image patches can represent structure information, such as edges and textures, while the single pixel cannot represent structure information [20]. Therefore, the patch similarity modulus based on image patches can well describe the characteristics of the image. In order to objectively analyze the performance of the patch similarity modulus, Figure 1(a) presents a geometry image, and Figure 1(b) is generated by adding Gaussian noise with the mean value of zero and standard deviation of to Figure 1(a). Figures 2(a) and 2(b) plot the intensity values of the gradient modulus, the absolute value of Laplacian, and the patch similarity modulus along Line 1 via the edge and along Line 2 within the flat region in Figure 1(b), respectively.

Figure 1: Test images: (a) original image () and (b) noisy image corrupted by Gaussian noise of .
Figure 2: The intensity values of the gradient modulus, the absolute value of Laplacian, and the patch similarity modulus: (a) along Line 1 in Figure 1(b) and (b) along Line 2 in Figure 1(b).

From Figure 2(a), we can observe that the patch similarity modulus can distinguish between noise and edges. As for the edge-preserving ability, the patch similarity modulus is relatively weaker than the gradient modulus and the absolute value of Laplacian on account of the smaller intensity values on the edges.

However, the patch similarity modulus has a better noise removal ability, compared with the gradient modulus and the absolute value of Laplacian, as seen in Figure 2(b). It can be seen that the intensity value of the absolute value of Laplacian changes dramatically with large peaks and troughs in the homogeneous region corrupted by noise. As a result, the YK model based on the absolute value of Laplacian is faced with the problem that a small threshold value will leave the noise in the processed image and a large value will oversmooth the details of the target object. Therefore, the YK model cannot achieve a good balance between noise removal and edge preservation. In contrast, the intensity value of the patch similarity modulus changes steadily without positive and negative pulses. Moreover, the fluctuation of the patch similarity modulus is the smallest among the three edge indicators. It reveals that the patch similarity modulus is robust to noise.

Through the above analysis, it can be seen that the patch similarity modulus can effectively distinguish the different characteristics of the image, such as edges and noise. And the denoising algorithm based on the patch similarity modulus can effectively remove noise.

3.2. New Fourth-Order PDE

In this paper, aiming at the shortages of the existing methods discussed in Section 2, we propose a new fourth-order PDE for image denoising.

Primarily, for removing noise effectively, we make full use of the advantage of the patch similarity modulus and propose a new diffusion coefficient expression given bywhere is the threshold value of . Therefore, the fourth-order PDE based on the patch similarity modulus is proposed asThen, for decreasing the diffusion coefficient of edges, the difference curvature [21] is introduced in the diffusion process to preferably preserve edge information. The difference curvature is formulated aswhere and represent the second derivatives in the direction normal and tangent to the level curves, respectively, and denotes the absolute function. Table 1 shows the characteristic analysis of the difference curvature in the three different regions of the distorted image. From Table 1, we can see that the difference curvature is only large on edge features and small in the homogeneous and noisy regions. Obviously, edges can be distinguished from noisy and flat regions by the value of . To have the same monotonicity as with respect to , the function concerning is defined asHere, is in inverse proportion to , which ensures that and are monotonically decreasing functions. is bounded in .

Table 1: The characteristic analysis of difference curvature.

All the facts discussed above motivated us to combine and to determine the diffusion process. Therefore, the new fourth-order PDE based on the patch similarity modulus and the difference curvature for image denoising is proposed as follows:

According to the analysis above, the performance of the proposed fourth-order PDE is described as follows. For noisy pixels, a large value is chosen for the small value of the patch similarity modulus and the value tends to be 1 for the small difference curvature . Then, the diffusion coefficient is approximately equal to , which will accelerate the diffusion process of noisy pixels. Therefore, the proposed algorithm can effectively filter out noise. Since the intensity value of at the noisy pixel is very small and changes smoothly in a small scope, a small threshold value can suppress noise, which will preserve more details at the same time. Aiming at edges, the value of is very large. Then, the value tends to be 0. Consequently, the diffusion coefficient is approximately equal to 0, which will slow down the diffusion process of edges to preserve the characteristics of the image. To sum up, the proposed algorithm can adaptively adjust the diffusion speed of edges, details, and noise in the distorted image and has a strong capability of removing noise and preserving edges and details.

To justify the contribution of the patch similarity modulus and the difference curvature, we conduct an experiment as shown in Figure 3. Figures 3(c) and 3(d) show the comparative results processed by (13) and (16), respectively. From Figure 3(c), we can see that the fourth-order PDE based on the patch similarity modulus can effectively remove noise. But this method obscures some edges. From Figure 3(d), we can see that the fourth-order PDE based on the patch similarity modulus and the difference curvature can not only remove noise, but also preserve edges. This means that the patch similarity modulus plays a crucial role in suppressing noise and the difference curvature is mainly used to preserve edges.

Figure 3: Comparison of the denoising results: (a) original image, (b) noisy image, (c) the denoising result with (13), and (d) the denoising result with (16).

For demonstrating the convergence of the proposed algorithm, the “normalized step difference energy” (NSDE) [22] is calculated at each iteration:where and denote the image at and iteration, respectively. Figure 4 shows the NSDE graph of the proposed algorithm for Figure 3(b). We can see that the NSDE graph decreases with the increase of the number of iterations, which shows that the proposed algorithm has a convergence.

Figure 4: The NSDE graph.

4. Numerical Implementation

We employ the explicit Euler numerical scheme for solving the PDE given in (16). Assuming that is the iterative time step and is the space grid size, we quantize the time and space coordinates as follows: is the size of the image support. In the paper, we use the grid size ; thus, the Laplacian at the point of the image after iterations is calculated byFor dealing with the boundary commodiously, the boundary conditions are given byLet ; then the Laplacian of can be calculated bywith symmetric boundary conditions:Then, (16) can be discretized asIn addition, the difference curvature at iteration is calculated bywhereHere, the central difference scheme is used for the spatial derivatives above; that is,

5. Experimental Results and Analysis

We have used two images with both textured and smooth regions, namely, Lena and Barbara, to test the performance of our algorithm. The original images are degraded by Gaussian noise with the mean value of zero and standard deviation of , , and , respectively. For the sake of demonstrating the accuracy and superiority of the proposed algorithm, related denoising methods, including the PM model in [7], the YK model in [12], the self-governing hybrid (SGH) model in [18], and the anisotropic fourth-order diffusion (AFOD) filter in [19], are carried out for comparative experiments. Moreover, the mean absolute error (MAE), peak signal to noise ratio (PSNR), and mean structural similarity (MSSIM) [23] are adopted to evaluate the denoising quality objectively.

The time step is chosen as for the SGH model, the AFOD filter, and the proposed algorithm [19], because these diffusion methods are highly sensitive to the time step. In the PM model and the YK model, the time step is chosen as . For the proposed algorithm, to evaluate the effect of the edge threshold parameter , Figures 5 and 6 present the denoising results under varying values of on the Lena image corrupted by Gaussian noise with the standard deviation of 10 and 15, respectively. The denoising results in Figures 5 and 6 are obtained at a fixed number of iterations and , respectively. Figures 5(b)5(d) show that is the optimal parameter value that gives the best visual diffusion result when , while is the optimal parameter value when , as can be seen from Figures 6(b)6(d). The other two ( and 4) are either too small or too large, which will lead to leaving noise in the denoised image or oversmoothing the image details. Therefore, the edge threshold needs to be increased for the larger standard deviation of noise. The number of iterations has been hand-tuned to produce the good result for different images.

Figure 5: Comparison of denoising results from the proposed algorithm under varying values of : (a) noisy image with the standard deviation of 10, (b) , (c) , and (d) .
Figure 6: Comparison of denoising results from the proposed algorithm under varying values of : (a) noisy image with the standard deviation of 15, (b) , (c) , and (d) .
5.1. Image Quality Measures

Let be the original image and let be the denoised image, and assume to be the size of . The MAE and PSNR are, respectively, defined as

Here, the smaller the MAE value is, the better the denoising effect will be. And the larger the PSNR value is, the less the image distortion will be.

Apart from the above two measures, the MSSIM index [23] is used to evaluate the overall image quality, defined aswhere and denote the content of the th local window in original and denoised images, respectively, is the number of local windows of the image, and the structural similarity is given bywhere and denote the mean value of and , respectively, and denote the variance of and , respectively, is the covariance of and , and , , where is the dynamic range of pixels values (255 for 8-bit grayscale images) and and are constants.

Here, the MSSIM can measure the similarity of two images including the luminance, contrast, and structure and a larger MSSIM value indicates a better image structure-preserving capacity.

5.2. Comparative Results

Figure 7(a) is the original Lena image and Figure 7(b) presents the noisy image corrupted by Gaussian noise with the standard deviation of . The edge threshold is set to 2 in all compared methods. The results of various denoising methods and the corresponding number of iterations are shown in Figures 7(c)7(g). In order to clearly display the denoising effects of the related methods, Figure 8 presents the enlarged portions of the images shown in Figure 7. From Figure 8(c), it can be observed that the PM model creates evident staircase effect in the denoised image. Although the YK model suppresses staircase effect, this model creates isolated speckles, as seen in Figure 8(d). Figures 8(e) and 8(l) illustrate that the SGH model loses some details, in spite of the good denoising ability. From Figure 8(f), we can observe that the denoised image obtained by the AFOD filter has a little staircase effect in the flat region because of the anisotropic diffusion at noisy pixels. It is clearly demonstrated that the proposed algorithm overcomes the deficiency of the SGH model and the AFOD filter, respectively, and can not only avoid staircase and speckle artifacts while smoothing noise, but also preserve details and edges perfectly, as can be perceived through Figures 8(g) and 8(n).

Figure 7: Comparison of the denoising results from various methods: (a) original Lena image, (b) noisy Lena image with the noise standard deviation of 10, (c) PM model (70 steps), (d) YK model (180 steps), (e) SGH model (200 steps), (f) AFOD filter (300 steps), and (g) proposed algorithm (330 steps).
Figure 8: The enlarged portions of the images shown in Figure 7: (a) and (h) corresponding to Figure 7(a), (b) and (i) corresponding to Figure 7(b), (c) and (j) corresponding to Figure 7(c), (d) and (k) corresponding to Figure 7(d), (e) and (l) corresponding to Figure 7(e), (f) and (m) corresponding to Figure 7(f), and (g) and (n) corresponding to Figure 7(g).

For further verifying the validity and versatility of the proposed algorithm, the original Barbara image, the noisy image with the standard deviation of , the denoised images from various methods, and the corresponding number of iterations are presented in Figures 9(a)9(g). All compared methods are carried out by setting . At the same time, the enlarged portions (homogeneous and textured regions) of the Barbara images after applying various denoising methods are presented in Figure 10 to have a clearer view. It is evident that the PM model, the YK model, and the AFOD filter cannot remove noise thoroughly in view of the fact that staircase artifacts or speckle artifacts exist in the flat regions of the denoised images, as seen in Figures 10(c), 10(d), and 10(f). From Figure 10(l), we can observe that a great number of texture details are smoothed by the SGH model, because the gradient modulus cannot effectively distinguish between details and noise. Figures 10(g) and 10(n) illustrate that the denoising and edge-preserving capacity of the proposed algorithm are better than those of the other methods.

Figure 9: Comparison of the denoising results from various methods: (a) original Barbara image, (b) noisy Barbara image with the noise standard deviation of 15, (c) PM model (80 steps), (d) YK model (200 steps), (e) SGH model (160 steps), (f) AFOD filter (260 steps), and (g) proposed algorithm (500 steps).
Figure 10: The enlarged portions of the images shown in Figure 9: (a) and (h) corresponding to Figure 9(a), (b) and (i) corresponding to Figure 9(b), (c) and (j) corresponding to Figure 9(c), (d) and (k) corresponding to Figure 9(d), (e) and (l) corresponding to Figure 9(e), (f) and (m) corresponding to Figure 9(f), and (g) and (n) corresponding to Figure 9(g).

On the whole, compared with the results obtained from the related methods, the denoised image of the proposed algorithm has a better visual effect and is more similar to the original image, which demonstrates that the proposed algorithm is able to get the best denoising performance.

In order to quantitatively analyze the performance of our algorithm, the MAE, PSNR, and MSSIM values of the Lena and Barbara images processed by various denoising methods at different noise levels are summarized in Tables 2 and 3, respectively. According to the MAE, PSNR, and MSSIM values of the denoised images, it is obvious that the proposed algorithm outperforms the other methods. Especially for the Barbara image, the PSNR values obtained by the proposed algorithm are obviously higher than those obtained from the other methods, which further illustrates that the proposed algorithm has a better superiority when the noisy image contains more texture details. Based on the above visual and quantitative analysis, the experimental results demonstrate that the proposed algorithm, which is more suitable to deal with the noisy images with abundant details, can preserve the structural features of the original image and obtain the high-quality denoised image.

Table 2: The MAE, PSNR, and MSSIM values of various methods for Lena image at different noise levels.
Table 3: The MAE, PSNR, and MSSIM values of various methods for Barbara image at different noise levels.
5.3. X-Ray Image Denoising

In the field of X-ray nondestructive testing, X-ray image denoising is a very critical premise for precisely recognizing defects. Figures 11 and 12 show the denoising results of two printed circuit board (PCB) X-ray images. For Figure 11(a), the parameters of the proposed algorithm are as follows: 150 iterations, and . And Figure 12(b) is obtained with and after 450 iterations. Figures 11(b) and 12(b) illustrate that the proposed algorithm can well remove random noise without damaging edges and details. Therefore, the proposed algorithm has a promising application in the X-ray image denoising.

Figure 11: Denoising results of a PCB X-ray image: (a) noisy image and (b) denoised image.
Figure 12: Denoising results of a PCB X-ray image: (a) noisy image and (b) denoised image.

6. Conclusions

In this paper, on the basis of analyzing the shortages of the PM model, the YK model, the SGH model, and the AFOD filter, we have proposed a new fourth-order partial differential equation for image denoising. In the proposed algorithm, the patch similarity modulus which can suppress noise and the difference curvature which can preserve edges are used to determine the diffusion coefficient of edges, details, and noise. And the denoising theory of the proposed algorithm has been described in detail. Besides, the accuracy and versatility of the proposed algorithm have been verified by variedly noisy images. Experimental results have demonstrated that our algorithm can avoid artifacts while removing noise and well preserve the structure information of the image. For our algorithm, because the calculation complexity of the patch similarity modulus is higher than those of the other edge indicators, the future work will be concentrated on reducing the computational time by the GPU (graphics processing unit) [24] without compromising its denoising performance.

Conflict of Interests

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

Acknowledgments

The authors would like to sincerely thank the editor and the reviewers for their constructive comments and suggestions. This work was supported by the National Natural Science Foundation of China under Grant 61271357, the International Cooperation Projects of Shanxi under Grant 2013081035, Outstanding Graduate Innovation Project of Shanxi Province Province under Grant 20143019, Graduate Students Science and Technology Foundation of North University of China under Grant 20131035, and Science Foundation of North University of China and the Opening Project of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology) under Grant KFJJ13-11M.

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