Total Generalized Variation on a Tree

SIAM Journal on Imaging Sciences, Volume 17, Issue 2, Page 1040-1077, June 2024.
Abstract.We consider a class of optimization problems defined over trees with unary cost terms and shifted pairwise cost terms. These problems arise when considering block coordinate descent (BCD) approaches for solving inverse problems with total generalized variation (TGV) regularizers or their nonconvex generalizations. We introduce a linear-time reduction that transforms the shifted problems into their nonshifted counterparts. However, combining existing continuous dynamic programming (DP) algorithms with the reduction does not lead to BCD iterations that compute TGV-like solutions. This problem can be overcome by considering a box-constrained modification of the subproblems or smoothing the cost terms of the TGV regularized problem. The former leads to shifted and box-constrained subproblems, for which we propose a linear-time reduction to their unconstrained counterpart. The latter naturally leads to problems with smooth unary and pairwise cost terms. With this in mind, we propose two novel continuous DP algorithms that can solve (convex and nonconvex) problems with piecewise quadratic unary and pairwise cost terms. We prove that the algorithm for the convex case has quadratic worst-case time and memory complexity, while the algorithm for the nonconvex case has exponential time and memory complexity, but works well in practice for smooth truncated total variation pairwise costs. Finally, we demonstrate the applicability of the proposed algorithms for solving inverse problems with first-order and higher-order regularizers.