Reconstructing sharply folding surfaces: A convex formulation

In recent years, 3D deformable surface reconstruction from single images has attracted renewed interest. It has been shown that preventing the surface from either shrinking or stretching is an effective way to resolve the ambiguities inherent to this problem. However, while the geodesic distances on the surface may not change, the Euclidean ones decrease when folds appear. Therefore, when applied to discrete surface representations, such constant-distance constraints are only effective for smoothly deforming surfaces, and become inaccurate for more flexible ones that can exhibit sharp folds. In such cases, surface points must be allowed to come closer to each other. In this paper, we show that replacing the equality constraints of earlier approaches by inequality constraints that let the mesh representation of the surface shrink but not expand yields not only a more faithful representation, but also a convex formulation of the reconstruction problem. As a result, we can accurately reconstruct surfaces undergoing complex deformations that include sharp folds from individual images.