Globally optimal affine epipolar geometry from apparent contours

We study the problem of estimating the epipolar geometry from apparent contours of smooth curved surfaces with affine camera models. Since apparent contours are viewpoint dependent, the only true image correspondences are projections of the frontier points, i.e., surface points whose tangent planes are also their epipolar planes. However, frontier points are unknown a priori and must be estimated simultaneously with epipolar geometry. Previous approaches to this problem adopt local greedy search methods which are sensitive to initialization, and may get trapped in local minima. We propose the first algorithm that guarantees global optimality for this problem. We first reformulate the problem using a separable form that allows us to search effectively in a 2D space, instead of on a 5D hypersphere in the classical formulation. Next, in a branch-and-bound algorithm we introduce a novel lower bounding function through interval matrix analysis. Experimental results on both synthetic and real scenes demonstrate that the proposed method is able to quickly obtain the optimal solution.