Significant progress has been recently made in Non-Rigid Structure-from-Motion (NRSfM). However, existing methods do not handle poorly-textured surfaces that deform non-smoothly. These are nonetheless common occurrence in real-world applications. An important unanswered question is whether shading can be used to robustly handle these cases. Shading is complementary to motion because it constrains reconstruction densely at textureless regions, and has been used in several other reconstruction problems. The challenge we face is to simultaneously and densely estimate non-smooth, non-rigid shape from each image together with non-smooth, spatially-varying surface albedo (which is required to use shading). We tackle this using an energy-based formulation that combines a physical, discontinuity-preserving deformation prior with motion, shading and contour information. This is a large-scale, highly non-convex optimization problem, and we propose a cascaded optimization that converges well without an initial estimate. Our approach works on both unorganized and organized small-sized image sets, and has been empirically validated on four real-world datasets for which all state-of-the-art approaches fail.

Shape-from-Template (SfT) is the problem of inferring the shape of a deformable object as observed in an image using a shape template. We call 2DSfT the `usual' instance of SfT where the shape is a surface embedded in 3D and the image a 2D projection. We introduce 1DSfT, a novel instance of SfT where the shape is a curve embedded in 2D and the image a 1D projection. We focus on isometric deformations, for which 2DSfT is a well-posed problem, and admits an analytical local solution which may be used to initialize nonconvex refinement. Through a complete theoretical study of 1DSfT with perspective projection, we show that it is related to 2DSfT, but may have very different properties: (i) 1DSfT cannot be exactly solved locally and (ii) 1DSfT cannot be solved uniquely, as it has a discrete amount of at least two solutions. We then propose two convex initialization algorithms, a local analytical one based on infinitesimal planarity and a global one based on inextensibility. We show how nonconvex refinement can be implemented where, contrarily to current 2DSfT methods, one may enforce isometry exactly using a novel angle-based parameterization. Finally, our method is tested with simulated and real data.