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'

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... 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.