Image Denoising Based On Fourth-order Nonlinear Partial Differential Equation Models

Most of the digital image applications demand a high quality image.Unfortunately,images often are corrupted by noise during the process of collection,transmission,and recording.Hence,image denoising is an essential processing step preceding visual and automated analysis.It is the problem of finding a “clean”image,given a noisy one.It can also be defined as the process of recovering an image which has been contaminated by noise.In most cases,it is assumed that the noisy image is the sum of an underlying clean image and a noise component.Several approaches to address this issue have emerged: stochastic modeling,wavelet theory and partial differential equations(PDEs).In this dissertation,we focus on developing image denoising models based on fourth-order nonlinear PDEs.Under this,we aimed to establish appropriate definitions of the solutions,and determine existence of the solution,and if a solution exists,determine its uniqueness;lastly,the study aimed to carry out numerical experiments to demonstrate the effectiveness of the proposed models in denoising images corrupted by additive and multiplicative noise.The aim of noise removing models not only focuses on the denoising process,but it should also observe that no spurious details are created in the restored image.Most PDEs considered for this purpose are second-order equations.It is well known from the history of image denoising that the second-order non-linear methods like TV and PM lead to the formation of constant patches during the PDE evolution.Therefore,the filtered output appears blocky or staircased.In the recent decades,high-order PDEs(namely,fourthorder PDEs)have been introduced in image restoration,to overcome the staircase effects that caused by second-order equations.In response to targets set in this thesis,we proposed three adaptive fourth-order models to maintain the balance between noise removal,edge preservation and overcoming these problems.The first model was based on solving a fourth-order partial differential equation by defining its corresponding functional.Since the equation may degenerate,we transform the original equation by an invertible transformation and obtain the regularization equation.Then we proved,by use of Rothe's method,the existence of the weak solution of the regularization equation.At last,we obtain the existence and uniqueness of the solution of the original problem by a limit process.In chapter 3,using the fixed point theorem,the existence and uniqueness of the entropy solution of our second model were established.Although this technique is common in other areas of applied mathematics,it is rarely found in the image processing literature.This equation has the robust ability to remove noise while preserving semantically important features such as edges.To show the effectiveness of the suggested models in removing additive noise,we have discretized the equations and implemented on MATLAB program.The classical finite different method(Experiment I)and the Fast Explicit Diffusion(FED)method(Experiment ?)were used.Numerical results demonstrate the superiority of the models over some famous ones,such as LLT model,YK model and image curvature model.Multiplicative noise is quite different from additive Gaussian noise.In chapter 5,we focus on denoising images corrupted by multiplicative(speckle)noise.Furthermore,an adaptive fourth-order model have been suggested.For this model,relying on the results of chapter 2,the existence and uniqueness of the entropy solution have been established.To illustrate the effectiveness of the suggested method in denoising image contaminated by multiplicative Gamma noise.In Experiment ?I,we used three numerical schemes,classical finite different method,Fast Explicit Diffusion and Krylov Subspace Spectral(KSS).Numerical results demonstrate the superiority of the suggested model over some famous ones,such as AA model and SO model.