We show that by extending the Laplacian formalism, which was first introduced in the Graphics community to regularize 3D meshes, we can turn the monocular 3D shape reconstruction of a deformable surface given correspondences with a reference image into a much better-posed problem. This allows us to quickly and reliably eliminate outliers by simply solving a linear least squares problem. This yields an initial 3D shape estimate, which is not necessarily accurate, but whose 2D projections are. The initial shape is then refined by a constrained optimization problem to output the final surface reconstruction. Our approach allows us to reduce the dimensionality of the surface reconstruction problem without sacrificing accuracy, thus allowing for real-time implementations.

Recent works have shown that 3D shape of non-rigid surfaces can be accurately retrieved from a single image given a set of 3D-to-2D correspondences between that image and another one for which the shape is known. However, existing approaches assume that such correspondences can be readily established, which is not necessarily true when large deformations produce significant appearance changes between the input and the reference images. Furthermore, it is either assumed that the pose of the camera is known, or the estimated solution is pose-ambiguous. In this paper we relax all these assumptions and, given a set of 3D and 2D unmatched points, we present an approach to simultaneously solve their correspondences, compute the camera pose and retrieve the shape of the surface in the input image. This is achieved by introducing weak priors on the pose and shape that we model as Gaussian Mixtures. By combining them into a Kalman filter we can progressively reduce the number of 2D candidates that can be potentially matched to each 3D point, while pose and shape are refined. This lets us to perform a complete and efficient exploration of the solution space and retain the best solution.