Globally Optimal Bilinear Programming for Computer Vision Applications

Abstract

We present a practical algorithm that provably achieves the global optimum for a class of bilinear programs commonly arising in computer vision applications. Our approach relies on constructing tight convex relaxations of the objective function and minimizing it in a branch and bound framework. A key contribution of the paper is a novel, provably convergent branching strategy that allows us to solve large-scale problems by restricting the branching dimensions to just one set of variables constituting the bilinearity. Experiments with synthetic and real data validate our claims of optimality, speed and convergence. We contrast the optimality of our solutions with those obtained by a traditional singular value decomposition approach. Among several potential applications, we discuss two: exemplar-based face reconstruction and non-rigid structure from motion. In both cases, we compute the best bilinear fit that represents a shape, observed in a single image from an arbitrary viewpoint, as a combination of the elements of a basis.