Variational PDE And Multifractal Based Image Modeling Theory, Algorithm And Application

From the machine vision and cognitive science, image processing is a basic tool used to reconstruct the geometrical relationship, topology structure, patterns and dynamics of the three-dimensional (3-D) world from two-dimensional (2-D) images or image sequences. Image restoration and image segmentation are two basic problems in the low-level vision processing. Belonging to the inverse problems, they are also two of the most challenging, active subjects in modern mathematics. However, since each image is a 2-D projection of a window of our 3-D world and is a matrix of positive integers which represents a pattern of radiant energy emitted by objects in space, it is difficult to handle with linear algorithms due to its non-Gaussian behavior and non-stationary property. Hence, it is desiderated to construct image models, which is allowed to deal with not only edge-like set, but also to other texture-like structure. As the general high-level priori knowledge, image models are very useful for designing image processors with good performance, and they are also helpful for implementing the human interaction in active vision through combining data constraints coming from image (bottom-up) with priori knowledge from high-level (top-down). Therefore the research of image modeling has great significance in the field of computer vision and image processing.This dissertation focuses on the research on image modeling and corresponding algorithms for some problems of the low-level vision process. We firstly research on the variational PDE image models for three major problems of image restoration, segmentation and image inpainting, in which the last problem is the research hotspot recently.Corresponding to the Image Intensified CCD (IICCD) camera, we then discuss themodeling and simulation of digital scene image synthesis based on the atmospheric propagation modeling theory. Finally, we focus on the image visual perceptual modeling and robust watermarking techniques for copyright protection problem of multimedia. The primary work and remarks of the paper are:(1) The paper takes a systematic research of the general ideas, basic concepts of image models and presents the state of arts on image modeling theory and approaches. The relationships among image models, low-level vision and image processing are established. A detailed analysis and comparative study of the models' representation, advantages and disadvantages is made and discussed.(2) Based on the theory of bounded variation functional space, the paper investigates the prime Total Variation (TV), prime-dual TV, extended TV models and related algorithms.We develop a new digital TV model for scalar-valued and vector-valued images, andprovide novel and simple proofs for its convex property and the digital Euler-Lagrange equation for minimizing the digital energy functional.Although total variation restoration model is good to de-noise non-texture images, it is not sufficient for real nature images since it removes small scale texture as well as noise. The drawbacks of total variation restoration model are analyzed in this paper. The paper investigates the possibility of extracting the texture with fractional order derivatives filter. And an improved total variation restoration model combined with fractional order derivatives singularities extracting is proposed.(3) This paper establishes the connections and unification among Bayesian inference, the variational PDE models and robust statistics theory. Starting from the half-quadratic regularization, the paper proposes the concept of transitional potential function in order to give a solution for the choice of the potential function in the variational restoration model. Investigating the properties of the transitional potential function, a series of valuable variational models are easily derived by half-quadratic theorem. Finally, the paper advances a class of generalizing and improved coupled "object-edge" models with variational formulation.(4) In order to solve the problem of the piece-wise smoothing Mumf...