Abstract

In regularization, edge-preserving constraints have received considerable attention due to the need for reconstructing high-quality images with sharp edges. The use of the $\ell_q$-norm in the gradient of the image in the regularization term has shown potential for preserving edges in reconstructions. One typically replaces the $\ell_q$-norm term with a sequence of $\ell_2$-norm weighted gradient terms with the weights determined from the current solution estimate. To overcome the large dimensionality, (hybrid) Krylov subspace methods can be employed to solve the 2-norm regularized problems. One disadvantage, however, is the need to generate a new Krylov subspace from scratch for every new two-norm regularized problem. The majorization-minimization Krylov subspace method (MMGKS) combines norm reweighting with generalized Krylov subspaces (GKS) to solve the reweighted problem. After projecting the problem using a small dimensional subspace that expands each iteration, the regularization parameter is selected. Basis expansion repeats until a sufficiently accurate solution is found. Nevertheless, for large-scale problems that require many expansion steps to converge, storage and the cost of repeated orthogonalizations may present overwhelming memory and computational requirements. In this talk we discuss a new method, RMMGKS , that keeps the memory requirements bounded through recycling the solution subspace by alternating between enlarging and compressing the GKS subspace. Numerical examples from dynamic photoacoustic tomography and streaming X-ray CT imaging are used to illustrate the effectiveness of the described methods.   This is joint work with Mirjeta Pasha and Eric de Sturler.