The Applications Of Geometric Variational Theory In Image Processing

This paper is based on variational method and partial differential equations. We focus on some common problems in image processing, such as denoising, segmentation, deblurring and image enhancement. The mean idea is to deal with an energy minimization problem by looking for the solution of the related Euler-Lagrange equation. We also give some fast algorithms which are stable and efficient to solve the equations. The mean results are as follows:1. We study the Nonlocal-BV function space and give some properties. Then the nonlocal-ROF model is discussed about the following topics:existence and uniqueness of the solution in nonlocal-BV space, existence and uniqueness of the solution to the related heat equation. We give the definition of the nonlocal quantum BV function, and show the applications in image segmentation numerically.2. We discuss the frame-based image delurring, including different choice of the regu-larization term from space domain and frequency domain. In order to compare the choice, we give three different models, then we explain them in detail including the numerical implement.3. We introduce some applications of Tensor Voting method by combining it with clas-sical models. we modify Geodesic Active Contour model by adding influence of Tensor Voting in the stopping function during the detecting procedure, and we im-prove ROF model with the same idea. Numerical results show the effectiveness.4. Based on the Retinex theory, we give a new model to decompose the image into illu-mination function and reflection function. We also give some existence and conver-gence theory about the new model. Some fast algorithms help us get good numerical results.