Distribution Matching with the Bhattacharyya Similarity: A Bound Optimization Framework

We present efficient graph cut algorithms for three problems: (1) finding a region in an image, so that the histogram (or distribution) of an image feature within the region most closely matches a given model; (2) co-segmentation of image pairs and (3) interactive image segmentation with a user-provided bounding box. Each algorithm seeks the optimum of a global cost function based on the Bhattacharyya measure, a convenient alternative to other matching measures such as the Kullback–Leibler divergence. Our functionals are not directly amenable to graph cut optimization as they contain non-linear functions of
fractional
terms, which make the ensuing optimization problems challenging. We first derive a family of parametric bounds of the Bhattacharyya measure by introducing an auxiliary labeling. Then, we show that these bounds are
auxiliary functions
of the Bhattacharyya measure, a result which allows us to solve each problem efficiently via graph cuts. We show that the proposed optimization procedures converge within very few graph cut iterations. Comprehensive and various experiments, including quantitative and comparative evaluations over two databases, demonstrate the advantages of the proposed algorithms over related works in regard to optimality, computational load, accuracy and flexibility.