Variational PDE Models In Image Restoration And Segmentation

Image enhancement and segmentation are very important for the success of image analysis and computer vision. In recent years, variational PDE method in image processing has received extensive concern. Compared with other approaches, variational PDE method has remarkable advantages in both theory and computation, and it can profoundly benefit from the existing wealth of literature on numerical analysis and computational PDEs.This thesis concentrats on the variational PDE method in nonlinear diffusion image enhancement and level set based curve evolution method and presents some studies concentrated in the following five topics:1. Higher-order nonlinear diffusion image enhancement methodA class of fourth-order partial differential equations is proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a cost functional, which is an increasing function of the directional curvature magnitude of the image intensity function. These PDEs attempt to remove noise and preserve edges by approximating an observed image with a piecewise linear image, which looks more natural than step image which second order classical diffusion uses to approximate an observed image. After analyzing the diffusion coefficient we propose several new diffusion coefficients, which can lessen the over-smooth in low gradient regions and image features. By adding the diffusion direction function in FAB diffusion coefficient composite diffusion is proposed, which can switch the diffusion process from a backward to a forward mode to smooth the pixel corrupted by additive noise, so the proposed composite diffusion processes can enhance features while locally denoising the image corrupted by blended additive noises.2. Higher-order robust anisotropic diffusionA novel smoothness term of Bayesian regularization framework based on M-estimation of robust statistics is proposed, and from this term a class of fourth-order nonlinear diffusion equations is proposed. These equations attempt to approximate an observed image with a piecewise linear image. The relations between anisotropic diffusion and robust statistics leads to new edge-stopping functions that preserves sharper boundaries than previous formulations and improves the automatic detecting of image features.It is known that M-estimators and W-estimators are essentially equivalent and solve the same minimization problem. Then, we propose PL bilateral filter from equivalent W-estimator. This new model is designed for piecewise linear image filtering and more effective than normal bilateral filter.3. Research on framework of anisotropic diffusion functional and relations between anisotropic diffusion and surface fittingFirst, we describe the relations between anisotropic diffusion and surface fitting. Specially, we show that higher-order anisotropic diffusion can be seen as piecewise linear surface fitting procedure. Secondly, we propose a general framework of anisotropic diffusion functional, and from this framework we propose two new sixth-order anisotropic diffusion equations. The procedure of these new anisotropic diffusion is equivalent to denoising and polynomial surface fitting, which looks more natural than piecewise linear images of fourth-order diffusion.4. Fast evolution algorithms for level set methodRelated theories and techniques on curve evolution and level set method are discussed in detail, and a new approach to construction of the signed distance function using new Voronoi source scanning method is proposed to accelerate and strengthen the level set method. Experiments show that the level set method modified by the proposed algorithm is faster and more accurate than that powered by fast marching algorithm.5. Image segmentation methods based on piecewise smooth Mumford-Shah modelMumford-Shah model is excellent to extract image regions and region boundaries without any prior information. Some research efforts are given to the model in this thesis. Based on techniques of curve evolution, piecewise-smooth Mumford-Shah functional and level sets, a new image segmentation model is proposed, which shows global optimization and less insensibility of initialization and can detect objects whose boundaries are not necessarily defined by gradient. This method solves the problem of locating the edges on images with non-uniform brightness, for which the previous methods based on piecewise-constant Mumford-Shah model are not applicable. Besides, the thesis extended the proposed method into vector-valued image segmentation. Experiments show that the extended vector-valued image segmentation method has the ability to fuse contributions from every channel.It can be seen that this thesis enriches the application domains of variational PDE method in image processing and provides new techniques for nonlinear diffusion and curve evolution method. The new models and methods proposed in this thesis widen the research horizon of contemporary and have nice future.