Image Enhancing And Denoising Models Based On PDEs And Curvature-Driven Flows

PDE-based methods and mean curvature flow have shown their effectiveness in image processing and computer vision. Their basic idea is to deform a given curve,surface or image with a PDE, and obtain the desired results as the solution of the equation. And in the case of color images, with a system of coupled PDEs. The art of this technique is in the design and analysis of these PDEs.Partial differential equations can be obtained from variational problems. Assume that a varia-tional approach to an image processing problem is characterized bywhere E denotes a given energy functional computed over the image u. Let F(φ) denote the first variation of E. Under general assumptions, a necessary condition for u to be a minimizer of u is that F{u) = 0. In order to compute a solution numerically, the condition is preferred to be embedded into a dynamical scheme, that is, the (local) minima may be computed by means of the steady-state solution of the equationu_t = F(u),where t is an artificial time-marching parameter.Another possibility is to work on the PDEs directly without thinking of any energy, which is known as PDE-based methods. These models can be formally written in the following form:where u(x, t) denotes the restored version of the initial degenerated data u₀(x), and Du, D²u represents the gradient and the Hessian Matrix of u with respect to the spatial variable x,respectively. Since Koenderink and Witkin rigorously introduced the notion of scale space independently, which meansthe representation of images simultaneously at multiple scales[44] [89], PDE-based methods in image processing have been developed rapidly in the past few years. We in our paper shall only touch upon some aspects of the improvement of the Koenderink-Witkin's theory,including the well-known Perona-Malik model, Rudin and Osher's shock filter. We shall also introduce the anisotropic diffusion model developed by Alvarez,Lions,Morel and the Mean curvature flow by Sochen,Kimmel,Malladi in our paper.We can see that using PDE and curve/surface based flows in image analysis can not only lead to modelling images in a simplified formalism in a continuous domain, but also bring on achieving high speed,accuracy,and stability with the help of the extensive available research on numerical analysis, such as ENO scheme,WENO scheme and Level Set scheme,AOS scheme and so on. Moreover, the theory of VISCOSITY SOLUTIONS and BV SOLUTIONS provide frameworks for the possibility of providing not only successful algorithms but also useful theoretical results such as existence and uniqueness of solutions.Our paper is organized as follows. In the first chapter, we firstly introduce the general degenerate image model,and then introduce the energy method and the PDE-based method for image restoration. The former is mainly on ROF model,Strong-Chan model and Aubert-Vese model and so on, the later is on Alvarez-Lions-Morel selective smoothing model and Sochen-Kimmel-Malladi mean curvature flow and Osher-Rudin's shock filters. In Chapter 2, we generalize Sochen-Kimmel-Malladi model and verify the existence of BV solutions to it. We also verify the existence and uniqueness of CLASSICAL SOLUTIONS to the corresponding regularization problem there. In Chapter 3,we first introduce our work on the improvement of Barcelos-Chen's model, present a framework of existence and uniqueness of viscosity solution to a class of initial value problem coupled with operators. And then, by combining our model (PII) with a modified shock filter, we present our parabolic-hyperbolic model for image deconvolution and denoising.