We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.
Modeling the Space of Point Landmark Constrained Diffeomorphisms
Surface registration plays a fundamental role in shape analysis and geometric processing. Generally, there are three criteria in evaluating a surface mapping result: diffeomorphism, small distortion, and feature alignment. To fulfill these requirements, this work proposes a
novel model of the space of point landmark constrained diffeomorphisms.
Based on Teichm¨uller theory, this mapping space is generated by the Beltrami coefficients, which are infinitesimally Teichm¨uller equivalent to 0. These Beltrami coefficients are the solutions to a linear equation group. By using this theoretic model, optimal registrations can be achieved by iterative optimization with linear constraints in the diffeomorphism space,
such as harmonic maps and Teichm¨uller maps, which minimize different types of distortion. The theoretical model is rigorous and has practical value. Our experimental results demonstrate the efficiency and efficacy of the proposed method.
- Publication Date:
- NSF-PAR ID:
- 10185279
- Journal Name:
- European Conference on Computer Vision (ECCV2020)
- Sponsoring Org:
- National Science Foundation
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- https://doi.org/10.1007/978-3-030-58577-8_22
