Applications of variational models and partial differential equations in signal recovery and image restoration

Variational, level set and PDE based methods and their applications in digital image processing have been well developed and studied for the past twenty years. These methods were soon applied to image processing and signal reconstruction problems. This dissertation explores some aspects of compressive sensing, l1-minimizations and fast numerical solvers. Their applications in sparse signal denoising and toniographie reconstruction are also studied.;The first two chapters are devoted to the sparse reconstruction. An efficient method called the linearized Bregman iteration for numerically solving the l1-minimization problem was developed and analyzed in the first chapter. Besides the simplicity of the algorithm, the special structure of the iteration enables the so-called "kicking scheme" to accelerate the algorithm. This method is especially powerful when the involved linear matrix is the sub-matrix of Fourier transform or similar cases.;When the involved matrix is highly coherent, as in the sparse deconvolution problems, the standard Bregmanian methods appear to be less efficient due to the different properties of the matrix. To overcome this difficulty, a novel method utilizing the transport partial differential equation is proposed in the second chapter. The PDE can be incorporated with the Bregmanian methods and only slightly increases the complexity. Some properties of the PDE are proved. Numerical results demonstrate that this method is a significant improvement of the standard Bregmanian methods in terms of both convergence speed and reconstruction quality.;In recent years there are many works focusing on medical image reconstruction by means of nonlinear optimization. In the third chapter we proposed a new method for tomographic reconstruction which incorporate the technique of Equally-Sloped Tomography (EST). EST is a technique of tomographic acquisition and reconstruction for performing tomographic image reconstructions from a limited number of noisy projections in an accurate manner by avoiding direct interpolations. Using this technique the image reconstruction problem can be formulated as a constrained optimization and solved efficiently and accurately by iterative methods. The numerical experiment results indicate that the quality of the reconstructed image is significantly improved and the required radiation dose for achieving a desired resolution is largely reduced.