Nonisometric Surface Registration via Conformal Laplace-Beltrami Basis Pursuit
This addresses the challenge of computing correspondences between intrinsically dissimilar surfaces in geometry processing, which is an incremental improvement over existing methods focused on nearly isometric cases.
The paper tackles the problem of surface registration for non-isometric shapes by proposing a variational model that aligns Laplace-Beltrami eigensystems via conformal deformations, enabling accurate point-to-point maps with robustness to noise and landmark errors.
Surface registration is one of the most fundamental problems in geometry processing. Many approaches have been developed to tackle this problem in cases where the surfaces are nearly isometric. However, it is much more challenging to compute correspondence between surfaces which are intrinsically less similar. In this paper, we propose a variational model to align the Laplace-Beltrami (LB) eigensytems of two non-isometric genus zero shapes via conformal deformations. This method enables us compute to geometric meaningful point-to-point maps between non-isometric shapes. Our model is based on a novel basis pursuit scheme whereby we simultaneously compute a conformal deformation of a 'target shape' and its deformed LB eigensytem. We solve the model using an proximal alternating minimization algorithm hybridized with the augmented Lagrangian method which produces accurate correspondences given only a few landmark points. We also propose a reinitialization scheme to overcome some of the difficulties caused by the non-convexity of the variational problem. Intensive numerical experiments illustrate the effectiveness and robustness of the proposed method to handle non-isometric surfaces with large deformation with respect to both noise on the underlying manifolds and errors within the given landmarks.