| This dissertation proposes a joint minimization model for the blind deconvolution problem to recover the image and simultaneously identify the blur. This model implicitly defines a one-parameter family of blurred images and blurring functions, from which the user can decide, usually using additional information, which is the "best" restored image. To find a solution of the minimization problem, we use an alternating minimization implicit iterative scheme, in which we fix either the blur or the image and minimize respect to the other variable, each step of which is a standard non-blind deconvolution problem.; Since the joint minimization model is not convex and thus allows multiple solutions, we have found that the alternating minimization procedure always converges globally, but with the converged solution depending on the initial guess. We will give an analysis of the alternating minimization procedure which will explain the convergence behavior and the observed robustness of the method.; In PDE image restoration problems, one has to invert operators which is a sum of a blurring operator and an elliptic operator with highly varying coefficient. We present a preconditioner for operators of this kind which can be used in conjunction with the conjugate gradient method, and compare it with Vogel and Oman product preconditioner.; Finally, we apply the total variation regularization method to solve the multichannel (e.g. color) image deconvolution problem. Numerical experiments will demonstrate the effectiveness of the total variation regularization method. We will also present some results on multichannel blind deconvolution with total variation regularization. |
