Proceedings of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013) Difference Curvature Driven Anisotropic Diffusion for Image Denoising Using Laplacian Kernel Yiyan WANG1,2 Zhuoer WANG3 1. Department of Physics and Engineering Technology Sichuan University of Arts and Science Dazhou, China 2. Laboratory of Image Science and Technology School of Computer Science and Engineering Southeast University Nanjing, China E-mail: [email protected] 3. Library of Sichuan University of Arts and Science Dazhou, China E-mail: [email protected] Abstract—Image noise removal forms a significant preliminary signal as an initial value. However, this model is referred as step in many machine vision tasks, such as object detection and isotropic diffusion. The disadvantage of isotropic diffusion pattern recognition. The original anisotropic diffusion is that it is symmetric and orientation insensitive, leading denoising methods based on partial differential equation often into blurred edges. Perona and Malik(PM)[4] developed an suffer the staircase effect and the loss of edge details when the anisotropic diffusion process as a nonlinear image noise image contains a high level of noise. Because its controlling removal method, which analogized heat diffusion to function is based on gradient, which is sensitive to noise. To adaptively remove the noise of the images. The main idea of alleviate this drawback, a novel anisotropic diffusion algorithm anisotropic diffusion is that it encourages intra-region is proposed. Firstly, we present a new controlling function smoothing and discourages inter-region at the edges[5]. The based on Laplacian kernel, then making use of the local analysis of an image, we propose a difference curvature driven decision on local smoothing is based on a diffusion to describe the intensity variations in images. Experimental coefficient, which is a function of the local image results on several natural and medical images show that the gradient. When the gradient is low, the smoothing takes new method has better performance in the staircase alleviation place. Meanwhile, the smoothing is suppressed where the and details preserving than the other anisotropic diffusions. gradient is high and an edge exists. However, it is found that the anisotropic diffusion is sensitive to noise as the whole Keywords-image denoising; partial differential equation; model is gradient-driven[6], which often suffers the difference curvature; Laplacian kernel staircase effect and the loss of edge details when the image contains a high level of noise[7]. To overcome this I. INTRODUCTION shortcoming, we propose a difference curvature driven anisotropic diffusion to smooth noisy images in this paper. The degradation of an image is usually unavoidable Firstly, we present a new controlling function based on during its acquisition and transmission[1]. It is necessary Laplacian kernel, then we employ the difference curvature to apply an efficient denosing technique to compensate for instead of gradient to control the diffusion coefficient, image corruption[2]. Denoising algorithm performance which plays an important role in filtering. Comparative mainly depends on a suitable representation to describe the experimental results on both natural and medical images original image information. Image noise removal remains a demonstrate that the proposed method can improve the challenge since it introduces artifacts and causes blurring. staircase effect and yields better PSNR and RMSE than the One of the major concerns in image denoising methods is other diffusions. their edge preservation capability. Among a variety of the The outline of this paper is as follows. In Section II, develop denoising techniques[3], partial differential at first, a simple introduction of the anisotropic diffusion is equation(PDE) based on models[4-10] have been widely given, then we propose a difference curvature driven used over the past few decades, due to its great advantage anisotropic diffusion scheme based on Laplacian Kernel. that it can preserve image edges while reducing noise. The Section III shows the extensive experiments results and basic idea is to deform a curve, a surface or an image using comparison. Finally, our conclusion is drawn in Section IV. a partial differential equation, and to obtain the desired results as the solution of this equation with the noisy image II. PROPOSED METHOD as initial conditions. Witkin[11] first found that the convolution of a signal with Gaussians as each scale was A. Anisotropic diffusion equivalent to solving a heat diffusion equation with the The application of partial differential equations(PDEs) Published by Atlantis Press, Paris, France. © the authors 2117 Proceedings of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013) in image process[10] has grown greatly in the past years. Δ=∂∂22 +∂∂ 22 uuxuy) in the function ⋅ instead of the The original PDE filtering model, proposed by Witkin[11], c() which found that the convolution of a signal with Gaussians gradient image of u , and proposed the fourth-order PDE as each scale was equivalent to solving a heat diffusion model where the controlling speed function c()⋅ was equation with the signal as an initial value. With regard to defined the following form[9] : an image u 0 , this process can be described with the 1 cu()Δ= following partial differential equation: 2 （7） 1+Δ()uK ∂u =ΔM uxt(,) （1） Although the fourth-order PDE can reduce the staircase ∂t in the denoised image, they have the disadvantage of Where is the diffusion conductance, and Δ is the M blurring edges. Moreover, the mathematical problems is Laplaican operator, the original image u0 is taken as the much challenging. initial state of the differential equation in (1). This model is To overcome this shortcoming, we propose a difference referred as linear heat equation that diffuses in all directions curvature method in the following. and destroys edges. Pernal and Malik(PM) were the first to B. Difference curvature diffusion try such an approach through controlling the speed of the diffusion and proposed a nonlinear adaptive diffusion Firstly, we present a new controlling function based on process[4], termed as anisotropic diffusion. The PM Laplacian kernel as: ∇= −∇ nonlinear diffusion equation is of the form: cuL ()exp() uK （8） ∂ u =∇∇() It is compared with two controlling functions of PM div c u u （2） ∂t diffusion from Fig1. Where u is the evolving image derived from the original ∇ image u0 at t time, and “ ” and “ div ” are the gradient and divergence operators, respectively, i.e. ∇=∂∂uuxiuyj() +∂∂() （3） div() c∇∇=∂∂ u u () x() c ∇∂∂+∂∂ u u x () y() c ∇∂∂ u u y （4） The function cu()∇ controls the speed of the diffusion. It is a nonnegative monotonically decreasing function of the gradient. A desirable characteristic of the conductance Fig1. Comparison of three controlling functions As shown in Fig1, compared with the two controlling function is that the diffusion speed at edges (where ∇u is functions of PM diffusion, the diffusion speed of Laplacian large) is low and the edges within the image are kept intact, kernel is between them at edge (where ∇u is large), and it is while the diffusion is encouraged within flat regions ∇u (where ∇u is small). Using gradient as an edge indicator, it smaller than them within flat regions (where is small). ()∇ ∇→∞ Meanwhile, unlike gradient and divergence, we define a requires that cu goes to zero when u and goes to new edge indicator as: 1 when ∇→u 0. The commonly used types of controlling =− Suττ uζζ （9） speed function with the above quality include: 1 cu()∇= Where uττ and uζζ represent the second derivatives in 2 （5） 1+∇()uK the direction of the gradient ∇u and in the direction 2 perpendicular to∇u , respectively. cu()∇=exp( −∇() uK) （6） uu22++2 uuu uu = xxx xyxy yyy uττ （10） Where K is the conductance parameter that influences uu22+ the diffusion process. In the PM model, diffusion takes xy uu22−+2 uuu uu ⋅ = yxx xyxy xyy place according to the controlling function c() to reduce uζζ （11） uu22+ the smoothing effect near edges, however, this controlling xy function depends on the gradient. It is well known that the In this paper, we call the proposed edge indicator PM diffusion often suffers the staircase effect and the loss difference curvature. It reflects the intensity variation from of edge details especially when noise contained in the image the standpoint of difference curvature, and it will be large in is rather large, because the gradient indicator is sensitive to the region of edge while small in flat and ramp regions[12], noise. Moreover, noise will cause infinity of gradient value so edges can be distinguished from flat and ramp regions in theory. As such, it will result in discounted denoising based on the value of difference curvature. Using this new performance of the anisotropic diffusion. concept to the whole image ,we can obtain the image’s You and Kaveh[8] improved the function c()⋅ by using corresponding difference curvature space. Applying this Δ difference curvature space and controlling function based on the Laplacian image of u (i.e., u , where Laplacian kernel, we improve the original anisotropic diffusion as follows: Published by Atlantis Press, Paris, France. © the authors 2118 Proceedings of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013) ∂u =∇div[] c() S u （12） ∂t L ⋅ S Where cL () is the Laplacian kernel function and is difference curvature, which have been defined in (8) and (9) above. (a) (b) (c) (d) Compared to the original anisotropic diffusion’s flaw of Fig3. Denoising results of barbara image with different methods. inaccurate estimation of the edge, especially when the image (a) Noisy image(σ =20 ), (b)PM, (c)Four-PDE, (d)PA contains a high level of noise, the difference curvature diffusion for images is more efficient than the gradient diffusion, because the difference curvature is more apt than the first order derivative of gradient to extract the intensity oscillations, which is common in an image.