An Inverse Problem Of Image Processing In A Degradation Of Diffusion

As a branch of signal processing, digital image processing have important applications in the field of applied science of astronomy, medicine, remote sensing, nanotechnology, security checks, as well as the visual psychology and so on. Generally speaking, the main content of image processing include image noise reduction, image fuzzy, image restoration and image segmentation. Traditional image restoration method almost based on the method of linear sys-tem,Such as Wiener deconvolution filter method, wavelet analysis, least square method with the constraint and so on, the core of these methods is to construct a proper linear filter. From the 1990s, Partial differential equation method was applied to image processing, and quickly developed into an interdisciplinary branch of emerging.This paper consider a class inverse problem of image processing in a degradation of diffu-sion, and research the reconstruction of clear images of the original using fuzzy images. Differ-ent from the heat conduction diffusion based on isotropic, the diffusion coefficient is changing, and allow the degradation is zero at the regional boundaries. This allows us to consider both the two circumstances of anisotropic diffusion and slow diffusion, but also bring the essential difficulties for the theory analysis and numerical simulation.This article can be divided into the following five chapters:The first chapter is the introduction of the article, and it briefly introduced some research background of the inverse problem of partial differential equations and valuable research dy-namic at home and abroad, especially the application of inverse problem in image processing in recent years made a detailed introduction.The second chapter, First, the article gives the image removing the fuzzy model equation: f=u+η, Second, we deduce a nonlinear elliptic equations using the method of regularization; Finally, in order to facilitate analysis, we turn it into a parabolic equation with time.The third chapter gives a degenerate parabolic equation and discuss it by the optimal con-trol method. The problem P is proposed in Section 3.1. In Section 3.2, through the analysis of the problem, the problem is turned into an optimal control problem. We prove the existence of the minimum for the control problem in Section 3.3. In Section 3.4, we get the necessary condition which satisfies the optimal solution. In Section 3.5, according to the necessary con-ditions, we prove the two important properties of the optimal solution. Summarize this chapter in Section 3.6.The fourth chapter mainly do numerical simulation on the basis of theoretical analysis of the issues. In Section 4.1, the finite difference scheme of parabolic equations is established. Landweber iteration method is put forward and write the iterative steps in Section 4.2. In Section 4.3, we realize numerical calculation from the original image to the blurred image to reconstruct images using MATLAB, and analyzes the effect of different reconstruction under the different parameters. The Section 4.4 make a summary.The fifth chapter is summary and outlook. For problem P, theoretical analysis is done, and looking for the best iterative method to achieve the best effect of refactoring is our direction in the future. Of course, in this paper, we study on gray image, the research of complex image will be the main follow-up work, so the difficulty of the theoretical analysis of the inverse problem is increased.