Specular surface reconstruction from sparse reflection correspondences

We present a practical approach for surface reconstruction of smooth mirror-like objects using sparse reflection correspondences (RCs). Assuming finite object motion with a fixed camera and un-calibrated environment, we derive the relationship between RC and the surface shape. We show that by locally modeling the surface as a quadric, the relationship between the RCs and unknown surface parameters becomes linear. We develop a simple surface reconstruction algorithm that amounts to solving either an eigenvalue problem or a second order cone program (SOCP). Ours is the first method that allows for reconstruction of mirror surfaces from sparse RCs, obtained from standard algorithms such as SIFT. Our approach overcomes the practical issues in shape from specular flow (SFSF) such as
the requirement of dense optical flow and undefined/infinite flow at parabolic points. We also show how to incorporate auxiliary information such as sparse surface normals into our framework. Experiments, both real and synthetic are shown that validate the theory presented.