Isometric Non-rigid Shape-from-Motion in Linear Time

We study Isometric Non-Rigid Shape-from-Motion (Iso-NRSfM): given multiple intrinsically calibrated monocular images, we want to reconstruct the time-varying 3D shape of an object undergoing isometric deformations. We show that Iso-NRSfM is solvable from the warps (the inter-image geometric transformations). We propose a new theoretical framework based on Riemmanian manifolds to represent the unknown 3D surfaces, as embeddings of the camera's retinal planes. This allows us to use the manifolds' metric tensor and Christoffel Symbol fields, which we prove are related across images by simple rules depending only on the warps. This forms a set of important theoretical results. Using the infinitesimal planarity formulation, it then allows us to derive a system of two quartics in two variables for each image pair. The sum-of-squares of these polynomials is independent of the number of images and can be solved globally, forming a well-posed problem for N ≥ 3 images, whose solution directly leads to the surface's normal field. The proposed method outperforms existing work in terms of accuracy and computation cost on synthetic and real datasets.