Numerical issues from inverse problems in image processing: Parameter estimation, and parallel algorithms for a high performance computing environment

Regularization techniques have been widely used in many applications for the solution of ill-posed inverse problems. Here the methods of Tikhonov and Total Variation regularization are studied. Applications and algorithms relevant to large scale image restoration and reconstruction are considered, including approaches to improve computational efficiency, regularization parameter selection and parallelization for graphical processing units (GPUs).;The parallelization of the least squares regularization problem by using multi-splitting is considered. The algorithm is composed of global and local iterations. The global iteration requires that solutions are obtained in the local iteration to solve the subproblem for multiple right hand sides, where each right hand side depends on the current global solution. The algorithm is made more efficient by updating the initial local Krylov subspaces per problem with minimal restarts. Also both the convergence and computational cost are studied. Numerical experiments validate the approach for maintaining convergence while reducing computational cost as compared to the global solution without splitting.;The choice of the regularization parameter plays a central role in the correct implementation of the regularization operator. The Unbiased Predictive Risk Estimator method is first reviewed as applied for Tikhonov regularization. Extension for Total Variation regularization is made difficult because of the nonlinearity of the operator. A linearization scheme and presented Krylov subspace approximation method bypass these difficulties. The feasibility and accuracy of the algorithm are presented for image restoration examples.;The GPU has shown its capability for large scale computations in high performance computing. For the solution of linear systems of equations, much effort has been devoted to efficient implementation of Krylov subspace-based solvers. Here the focus is to improve the computational efficiency of the projected CG algorithm utilizing the GPU. Although the current GPU architectures are better suited to single precision accuracy, the projected CG algorithm can still be used. A modification of the projected CG to use block operations, hence optimizing the usage of basic linear algebra subroutines of level 3 (BLAS3), further improves the GPU implementation. Numerical results using the GPU are provided to support the proposed algorithm.