Numerical Assessment of Anisotropic Diffusion Equation for Image Blurring Using SOR Iteration Nurul Afiqah Basran 1* , Jeng Hong Eng 1 , Azali Saudi 2 , Jumat Sulaiman 1 1 Faculty of Science and Natural Resources, Universiti Malaysia Sabah 2 Knowledge Technology Research Unit, Faculty of Computing and Informatics, Universiti Malaysia Sabah, Kota Kinabalu, Malaysia * Corresponding author email: nurulafiqahbasran@yahoo.com Abstract: Blurring the image while preserving the important features such as edges is a crucial study in computer vision. This paper presents the results of applying three iterative methods which are Jacobi, Gauss Seidel and Successive Overrelaxation (SOR) to solve anisotropic diffusion equation for image blurring, where the output image of Jacobi is used as a control image. The number of iterations and computational time required to solve the anisotropic diffusion equation are used to measure the performance of the considered iterative methods. The findings show that SOR method is more efficient to smooth the inner region of an image compared to Jacobi and Gauss-Seidel methods in which the SOR required the least number of iterations and computational time. Keywords: diffusion equation, partial difference equation, image blurring, SOR method. 1. Introduction The application of mathematical models in image processing and analysis has begun since early 1960 [1], where the field was highly occupied in study of computer science and engineering. Numerical method is one of the mathematical tools used to solve image processing problems especially using techniques of functional analysis and the theory of partial differential equations (PDEs). Nowadays, many researchers employ the diffusion-wave equation [2], heat equation [3], Poisson Equation [4] and Laplace equation as PDE-based image processing techniques for image segmentation [5], image restoration-denoising and deblurring[6], edge detection and enhancement [7] purposes. In image processing, image blurring is also known as the process of image denoising, smoothing or edge detection. The finite difference method (FDM) has been applied for the solution of PDEs by approximating any partial derivatives. There are three standard types of PDEs that consist of elliptic, hyperbolic and parabolic. The two-dimensional heat or diffusion equation applied on image blurring techniques is one of parabolic PDEs [8]. The approximation equations derived after discretization process can be made by using explicit, implicit, Crank-Nicholson or other schemes. The solution of the generated system of linear equations can be solved by using direct or iterative methods. Furthermore, diffusion or heat equation applied in image processing and analysis can also be referred as the scale space. The theory of scale space provided a framework to undergo various image processing techniques across multiple scales [9]. In this study, three iterative methods were applied to solve an anisotropic diffusion equation for image blurring. The approximate equation is derived using implicit scheme to discretize diffusion equation in which this approximation equation can be used to construct the generated system of linear equations. Then, this linear system can be solved iteratively by using Jacobi, Gauss-Seidel and SOR methods. In previous studies, it was proven that SOR method was the most suitable way to solve image blending problem [10]. 2. Related Work The PDEs techniques had been widely used in image processing problems and their techniques also have been used to construct the reliable and fast algorithm that is numerically efficient to solve the problem. Image processing problem based on diffusion equation or also known as the scale space had been discussed by Weickert et al. [9]. This equation has been widely applied for image filtering including image denoising, segmentation and edge detection. It has also been used in other fields of study such as biomedical by providing the important information from the image effectively. For example, the study conducted by Yilmaz et al. [11] applied an adaptive anisotropic diffusion to filter out unnecessary noise occurred on cone beam computed tomography (CBCT) images in order to identify the region of interest (ROI) in the diagnosis process. An improved nonlinear diffusion algorithm had been developed [12] for image denoising problem. Noise is a random signal that appears as random speckles which significantly corrupt the image quality. Therefore, the new method had been verified as an efficient method to reduce image noise while maintaining important detail using wavelet coefficient. A recent study conducted [13] in image denoising problem, proved that a combination of classical additive operator splitting and a nonlinear relaxation algorithm are able to produce an accurate image restoration which is also able to control the problem of stability. Unlike the research conducted by Atlas et al. [14], more focusis on reducing the phenomenon of an image called staircase effect by proposing efficient tools through interpolation of two classical models which are Perona-Malik Equation (PME) and −Laplacian with → ∞. Meanwhile, the new diffusion coefficient has also been proposed [15] earlier for image smoothing method. They suggested a time-dependent anisotropic diffusion by investigating the relation between Gaussian scale and gradient Journal of Computer Science & Computational Mathematics, Volume 8, Issue 4, December 2018 DOI: 10.20967/jcscm.2018.04.009