Browsing by Author "Fwamba, Rukia Nasimiyu"

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    Optimization of the Non-Linear Diffussion Equations
    (Science Publishing Group, 2024-10) Fwamba, Rukia Nasimiyu; Chepkwony, isaac; Wekulo, Saidi Fwamba
    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.
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    Toning up Images by Smoothening Edges
    (IOSR-JM, 2024) Gatoto, James; Fwamba, Rukia Nasimiyu
    Background: Under Partial Differential Equations, an image is a function 2 f x y X ( , ),  . These equations are used in smoothening of images. The necessity to have an excellent image quality is increasingly required in the current world. Most of the obtained images are not as smooth as we would want hence they are blurred. Use of the nonlinear diffusion equation is essential in the current day smoothening of images. This model inputs smoothness in the denoising process. This research improved the quality of images through the use of nonlinear PDEs of parabolic nature i.e. the heat equation. Materials and Methods: Numerical schemes of ADI (Alternating Direction Implicit method) and 2-EGSOR(2- Point Explicit Group Successive Over Relaxation) were used to solve the equations in MATLAB subject to an initial condition of a noisy image, generating various smoothened images. Results: Comparatively output from ADI is very close to the original image in terms of alignment, smoothness i.e. refined texture, well outlined contours and overall detail without stair casing. .This method is characterized with a smaller residue and much time lapse. An error analysis too was carried out using Root Mean Square Method. ADI & the blocked (ADI and EGSOR) register a comparatively lower RMSE. Conclusion: The most suitable algorithm for image smoothening is Alternating Direction Implicit Method