Optimization of the Non-Linear Diffussion Equations

Abstract

Partial Differential Equations are used in smoothening of images. Under partial differential equations an image is termed as a function; f(x, y), XÎR<sup>2</sup>. 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; u<sub>x</sub>(0, y, t) = u<sub>x</sub>(p, y, t) = u<sub>y</sub>(x, 0, t) = u<sub>y</sub>(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 C<sub>v</sub> (pixel flux), C<sub>f</sub> (coefficient of the forcing term), b (the threshold parameter) alongside time t were estimated through optimization. Parameter b maintained the highest value, while C<sub>v</sub> 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.