We consider the problem of non-rigid shape matching using the functional map framework. Specifically, we analyze a commonly used approach for regularizing functional maps, which consists in penalizing the failure of the unknown map to commute with the Laplace-Beltrami operators on the source and target shapes. We show that this approach has certain undesirable fundamental theoretical limitations, and can be undefined even for trivial maps in the smooth setting. Instead we propose a novel, theoretically well-justified approach for regularizing functional maps, by using the notion of the resolvent of the Laplacian operator. In addition, we provide a natural one-parameter family of regularizers, that can be easily tuned depending on the expected approximate isometry of the input shape pair. We show on a wide range of shape correspondence scenarios that our novel regularization leads to an improvement in the quality of the estimated functional, and ultimately pointwise correspondences before and after commonly-used refinement techniques.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): CRG2017, CRG2018-3730
Acknowledgements: The authors would like to thank the anonymous reviewers for their valuable comments and helpful suggestions. Parts of this work were supported by the KAUST OSR Awards No. CRG2017-3426 and CRG2018-3730, a gift from the NVIDIA Corporation, and the ERC Starting Grant No. 758800 (EXPROTEA).