Speaker
Description
Reconstructing high-quality images with sharp edges requires edge-preserving regularization, often imposed using the $\ell_1$-norm of the gradient. To get a computationally tractable problem, the $\ell_1$-norm term is typically replaced with a sequence of $\ell_2$-norm weighted gradient terms with the weights determined from the current solution estimate. The majorization-minimization generalized Krylov subspace method (MM-GKS) has the advantage of combining the updating of the regularization operator with generalized Krylov subspaces (GKS). Unfortunately, the storage and the cost of repeated orthogonalization can present overwhelming memory requirements and computational costs.
We present a variant of MM-GKS that provably converges to the minimum of the smoothed functional even if the solution search space dimension remains very small. This substantially improves theoretical results for MM-GKS where the convergence proof relies on (eventually) spanning the full problem space. Using this result, we develop a new method that solves the minimization/imaging problem by alternatingly compressing and expanding the search space while maintaining strict monotonic convergence. Our method can solve large-scale problems efficiently both in terms of computational complexity and memory requirements. We further generalize our proposed method to handle streaming problems where the data is either not all available simultaneously or the size of the problem demands it be treated as such. We demonstrate the utility of our approach on several image reconstruction and restoration problems. This is joint work with Mirjeta Pasha (VT) and Eric de Sturler (VT).