Image Denoising Based On Some Novel Regularization PDE Models

During the past few decades noise removal, edge detection and enhancment, im-painting, segmentation, superresolution, among other image processing tasks, have been the subject of intense mathematical interest. Several approaches have emerged:stochas-tic modeling, wavelet theory and partial differential equations (PDE). In this dissertation, however, we consider only noise removal (denoising) using(PDE) techniques, and focus on additive and multiplicative (speckle) noise.Image denoising is the process by which an ideal image is recovered from an ob-served (non-ideal) image usually degraded or corrupted by some random noise. While the nature of noise is often unknown, in carrying out a denoising exercise it is, howev-er, important to approximate a noise model that is as close as possible to the reality of the noise in question. In this work we model noise after the Gaussian additive noise and Gamma distribution for speckle noise.The objective of image denoising exercise does not focus only on the removal of noise, but it also seeks a situation where:no new spurious details are introduced in the restored image; at each scale-space representation the boundaries/edges are sharp or p-reserved, and intra-region smoothing is preferred to inter-region smoothing at all scales. Hence, we propose minimization functionals of strict convexity, and work in the context of functions of boundary variation(BV). The BV space allows for discontinuous solu-tion, hence able to allow for edges and other discontinuities. The proposed functionals, therefore, lead to PDE’s which have the robust ability to remove or alleviate noise while p-reserving semantically important features such as edges, contour and texture. The models do not lead to illposedness arising from forward-backward diffusion.Additionally, apart from presenting the proofs of the existence and uniqueness of the minimization problems; existence and uniqueness of the solutions of the corresponding evolution problems and asymptotic stability of the solutions, we propose an alternative variational framework for image denoising. In this framework we consider the coefficient of the traditional total variation(TV) potential, as a function of the magnitude of gradient, in terms of linear, sublinear, and superlinear growth at infinity. We find that a coefficient that is of sublinear growth, and which then produces a potential that grows linearly, often gives a formulation, that does not only give good results, but which also submit progres-sively to mathematical analysis.Furthermore, to test the effectiveness of the proposed models, relative to some clas-sical models, we have discretized the evolution equations, carried out numerical experi-ments and presented our results and comparisons. Indeed, the experimental results show that the proposed models perform better in the real work of image denoising than models such as TV method, Perona and Malik(PM) method and even the more recent D-α-PM method.