Optimization of the Non-Linear Diffussion Equations
No Thumbnail Available
Date
2024-10
Journal Title
Journal ISSN
Volume Title
Publisher
Science Publishing Group
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
2
. 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.
Description
Research Article
Keywords
Citation
Fwamba, R. N., Chepkwony, I., & Fwamba, W. S. (2024). Optimization of the Non-Linear Diffussion Equations. Science Journal of Applied Mathematics and Statistics, 12(1), 13–19. https://doi.org/10.11648/j.sjams.20241201.12