A Cutting-Plane Method for Sublabel-Accurate Relaxation of Problems with Product Label Spaces

Many problems in imaging and low-level vision can be formulated as nonconvex variational problems. A promising class of approaches to tackle such problems are convex relaxation methods, which consider a lifting of the energy functional to a higher-dimensional space. However, they come with increased memory requirements due to the lifting. The present paper is an extended version of the earlier conference paper by Ye et al. (in: DAGM German conference on pattern recognition (GCPR), 2021) which combined two recent approaches to make lifting more scalable: product-space relaxation and sublabel-accurate discretization. Furthermore, it is shown that a simple cutting-plane method can be used to solve the resulting semi-infinite optimization problem. This journal version extends the previous conference work with additional experiments, a more detailed outline of the complete algorithm and a user-friendly introduction to functional lifting methods.