Connecting non-quadratic variational models and MRFs

Spatially-discrete Markov random fields (MRFs) and spatially-continuous variational approaches are ubiquitous in low-level vision, including image restoration, segmentation, optical flow, and stereo. Even though both families of approaches are fairly similar on an intuitive level, they are frequently seen as being technically rather distinct since they operate on different domains. In this paper we explore their connections and develop a direct, rigorous link with a particular emphasis on first-order regularizers. By representing spatially-continuous functions as linear combinations of finite elements with local support and performing explicit integration of the variational objective, we derive MRF potentials that make the resulting MRF energy equivalent to the variational energy functional. In contrast to previous attempts, we provide an explicit connection for modern non-quadratic regularizers and also integrate the data term. The established connection opens certain classes of MRFs to spatially-continuous interpretations and variational formulations to a broad range of probabilistic learning and inference algorithms.