Convex Optimization for Multi-Class Image Labeling with a Novel Family of Total Variation Based Regularizers

We introduce a linearly weighted variant of the total variation for vector fields in order to formulate regularizers for multi-class labeling problems with non-trivial interclass distances. We characterize the possible distances, show that Euclidean distances can be exactly represented, and review some methods to approximate non-Euclidean distances in order to define novel total variation based regularizers. We show that the convex relaxed problem can be efficiently optimized to a prescribed accuracy with optimality certificates using Nesterov's method, and evaluate and compare our approach on several synthetical and real-world examples.