Efficient Loopy Belief Propagation Using the Four Color Theorem

Abstract

Summary. Recent work on early vision such as image segmentation, image denoising, stereo matching, and optical flow uses Markov Random Fields. Although this formulation yields an NP-hard energy minimization problem, good heuristics have been developed based on graph cuts and belief propagation. Nevertheless both approaches still require tens of seconds to solve stereo problems on recent PCs. Such running times are impractical for optical flow and many image segmentation and denoising problems and we review recent techniques for speeding them up. Moreover, we show how to reduce the computational complexity of belief propagation by applying the Four Color Theorem to limit the maximum number of labels in the underlying image segmentation to at most four. We show that this provides substantial speed improvements for large inputs, and this for a variety of vision problems, while maintaining competitive result quality.

New Measure of Economic Development Based on the Four-Colour Theorem

“…Secondly, from the point of view of other sciences, this theorem is still only a mathematical curiosity, because non-mathematical sciences lack its significant applications, i.e., those that would solve a major problem. So far, only a few attempts have been made to apply this theorem in various fields of science, but these endeavours have not led to significant progress [ 41 , 42 , 43 , 44 , 45 , 46 ]. We decided to break this deadlock.…”