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We present nonlinear formulations to Shape-from-Template (SfT) and Non-Rigid Structure-from-Motion (NRSfM) faithfully exploiting the isometric, conformal and equiareal deformation models. Existing work uses relaxations such as inextensibility or requires knowing the optic flow field around the correspondences, an impractical assumption. In contrast, the proposed formulations only require point correspondences and resolve all ambiguities using the notions of maximal depth and maximal isometry heuristics. We propose solution methods using Semi-Definite Programming (SDP) for all formulations. We show that straightforward SDP models conflict with the usual maximal depth heuristic and propose an adapted opposite-depth parameterisation demonstrating a lesser relaxation gap. Experimental results on many real-world benchmark datasets demonstrate superior accuracy over existing methods.
Camera pose is a very natural concept in 3D vision in the rigid setting. It is however much more difficult
to work with in deformable settings. Consequently, numerous deformable reconstruction methods simply ignore
camera pose. We analyse the concept of pose in deformable settings and prove that it is unconstrained with the
existing formulations, properly justifying the existing pose-less methods reconstructing structure only. We explain
this result intuitively by the impossibility to define an intrinsic coordinate frame to a general deforming object. The
proposed analysis uses the isometric deformation model and extends to the weaker models including conformality
and equiareality. We propose a novel prior to rescue camera pose estimation in deformable settings, which attributes
the deforming object’s dominant rigid-body motion to the camera. We show that adding this prior to any existing
formulation fully constrains camera pose and leads to elegant two-step solution methods, involving deformable
structure reconstruction using a base method in the first step, and absolute orientation or Procrustes analysis in
the second step. We derive the proposed approach for the template-based and template-less settings, respectively
implemented using Shape-from-Template (SfT) and Non-Rigid Structure-from-Motion (NRSfM) as base methods,
and validate them experimentally, showing that the computed pose is qualitatively and quantitatively plausible.