Optimization of the Non-Linear Diffussion Equations

Rukia Fwamba, Isaac Chepkwony, W. Fwamba
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Abstract

Partial Differential Equations are used in smoothening of images. Under partial differential equations an image is termed as a function; f(x, y), XÎR2. The pixel flux is referred to as an edge stopping function since it ensures that diffusion occurs within the image region but zero at the boundaries; ux(0, y, t) = ux(p, y, t) = uy(x, 0, t) = uy(x, q, t). Nonlinear PDEs tend to adjust the quality of the image, thus giving images desirable outlooks. In the digital world there is need for images to be smoothened for broadcast purposes, medical display of internal organs i.e MRI (Magnetic Resonance Imaging), study of the galaxy, CCTV (Closed Circuit Television) among others. This model inputs optimization in the smoothening of images. The solutions of the diffusion equations were obtained using iterative algorithms i.e. Alternating Direction Implicit (ADI) method, Two-point Explicit Group Successive Over-Relaxation (2-EGSOR) and a successive implementation of these two approaches. These schemes were executed in MATLAB (Matrix Laboratory) subject to an initial condition of a noisy images characterized by pepper noise, Gaussian noise, Brownian noise, Poisson noise etc. As the algorithms were implemented in MATLAB, the smoothing effect reduced at places with possibilities of being boundaries, the parameters Cv (pixel flux), Cf (coefficient of the forcing term), b (the threshold parameter) alongside time t were estimated through optimization. Parameter b maintained the highest value, while Cv exhibited the lowest value implying that diffusion of pixels within the various images i.e. CCTV, MRI & Galaxy was limited to enhance smoothening. On the other hand the threshold parameter (b) took an escalated value across the images translating to a high level of the force responsible for smoothening.
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非线性扩散方程的优化
偏微分方程用于平滑图像。在偏微分方程中,图像被称为函数;f(x, y), XÎR2。像素通量被称为边缘停止函数,因为它确保扩散发生在图像区域内,而在边界为零;ux(0, y, t) = ux(p, y, t) = uy(x, 0, t) = uy(x, q, t)。非线性 PDE 往往会调整图像的质量,从而使图像呈现出理想的外观。在数字世界中,需要对图像进行平滑处理,以用于广播目的、内部器官的医学显示(即 MRI(磁共振成像))、星系研究、闭路电视(CCTV)等。该模型为图像平滑化提供了优化方案。利用迭代算法,即交替方向隐含法(ADI)、两点显式组连续超松弛法(2-EGSOR)以及这两种方法的连续实施,获得了扩散方程的解。这些方案是在 MATLAB(矩阵实验室)中执行的,初始条件是以胡椒噪声、高斯噪声、布朗噪声、泊松噪声等为特征的噪声图像。由于算法在 MATLAB 中执行,平滑效果在有可能成为边界的地方有所降低,参数 Cv(像素通量)、Cf(强制项系数)、b(阈值参数)和时间 t 都是通过优化估算出来的。参数 b 保持最高值,而 Cv 显示最低值,这意味着 CCTV、MRI 和 Galaxy 等不同图像中像素的扩散受到限制,从而增强了平滑化效果。另一方面,阈值参数(b)在所有图像中的值都在上升,这意味着平滑化的作用力很大。
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