Mathematical Methods In Image Denoising And Image Segmentation

Partial differential equations (PDEs) and variational methods have been increas-ingly used as powerful methods for image processing. In PDE formulation, image restoration and image segmentation are achieved by constructing PDEs directly,while in variational formulation, image restoration and image segmentation are achieved by solving an energy minimization problem,which allows conveniently integrating a va-riety of useful information, such as shape prior and constraints on the regularity of object boundaries. In this thesis,we show some efficient models,pay more atten-tion to the application of fast algorithms and also present theoretical analysis of some models.Main research results are as follows:Part one: Mathematical methods in image denoising1, A fractional differential fidelity-based PDE model for image denoisingWe introduce a new fidelity term based on the combination of fractional order fidelity term and global fidelity term to measure the similarity in the variation of im-ages, which can prevent the staircase effect and simultaneously enhance the image's texture details. Numerical results are presented in the end to demonstrate that with respect to image denoising capability, our fractional fidelity-based model outperforms the gradient fidelity-based model.2, Image denoising via time-delay regularization coupled nonlinear diffusion e-quationsWe propose a methodology for image denoising that is based on time-delay reg-ularization coupled nonlinear diffusion equations. Utilizing aspects of time-delay reg-ularization and anisotropic nonlinear diffusion, the proposed denoising model is very effective in tackling noisy images and images with fine structures. Since time-delay regularization incorporates into the filtering process information obtained from the im-ages at each step of the iteration process, time-delay regularization is a better alterna-tive to Gaussian filtering of pre-smoothing in the construction of diffusion coefficients from the image itself. By Galerkin method, we present a detailed mathematical anal-ysis of the proposed model in the form of the proof of existence and uniqueness of weak solutions of the model.Part two: Mathematical methods in image segmentation1, A variational model for joint restoration and segmentation based on the Mum-ford Shah modelWe propose a variational model for joint restoration and segmentation based on the Mumford-Shah model. Utilizing aspects of image restoration, the proposed model can effectively tackle images with a high level of blurriness or noise. The energy can be minimized efficiently using the alternating minimization (AM) algorithm, mainly because the AM algorithm is equiped with the convergence analysis and is easy to calculation. We also show the existence and uniqueness of minimizer for the proposed energy minimization problem.2, Image segmentation via mean curvature regularized Mumford-Shah model and thresholdingWe propose a variational model by combining the Mumford-Shah model with the mean curvature of the image surface. Specifically, we first solve the mean curvature regularized Mumford-Shah model to get the smooth solution, and then the segmen-tation is obtained by the thresholding procedure. The variational model can be mini-mized efficiently using the augmented lagrangian algorithm. The experimental results show that the proposed method is not only able to preserve the geometry of objec-t shapes, especially object corners, but it is also more accurate than state-of-the-art methods.