Inextensible Non-Rigid Shape-From-Motion by Second-Order Cone Programming

We present a global and convex formulation for template-less 3D reconstruction of a deforming object with the perspective camera. We show for the first time how to construct a Second-Order Cone Programming (SOCP) problem for Non-Rigid Shape-from-Motion (NRSfM) using the Maximum-Depth Heuristic (MDH). In this regard, we deviate strongly from the general trend of using affine cameras and factorization-based methods to solve NRSfM. In MDH, the points' depths are maximized so that the distance between neighbouring points in camera space are upper bounded by the geodesic distance. In NRSfM both geodesic and camera space distances are unknown. We show that, nonetheless, given point correspondences and the camera's intrinsics the whole problem is convex and solvable with SOCP. We show with extensive experiments that our method accurately reconstructs quasi-isometric surfaces from partial views under articulated and strong deformations. It naturally handles missing correspondences, non-smooth objects and is very simple to implement compared to previous methods, with only one free parameter (the neighbourhood size).