Closed-Form Solution to Non-rigid 3D Surface Registration

We present a closed-form solution to the problem of recovering the 3D shape of a non-rigid inelastic surface from 3D-to-2D correspondences. This lets us detect and reconstruct such a surface by matching individual images against a reference configuration, which is in contrast to all existing approaches that require initial shape estimates and track deformations from image to image. We represent the surface as a mesh, and write the constraints provided by the correspondences as a linear system whose solution we express as a weighted sum of eigenvectors. Obtaining the weights then amounts to solving a set of quadratic equations accounting for inextensibility constraints between neighboring mesh vertices. Since available closed-form solutions to quadratic systems fail when there are too many variables, we reduce the number of unknowns by expressing the deformations as a linear combination of modes. The overall closed-form solution then becomes tractable even for complex deformations that require many modes.