Nonisometric Surface Registration via Conformal Laplace–Beltrami Basis Pursuit

Stefan C. Schonsheck, Michael M. Bronstein, Rongjie Lai*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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 to compute geometrically 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 eigensystem. We solve the model using a proximal alternating minimization algorithm hybridized with the augmented Lagrangian method which produces accurate correspondences given only a few landmark points. We also propose a re-initialization 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 noises on the underlying manifolds and errors within the given landmarks or feature functions.

Original languageEnglish
Article number30
JournalJournal of Scientific Computing
Volume86
Issue number3
DOIs
StatePublished - Mar 2021
Externally publishedYes

Funding

FundersFunder number
NSF DMS1522645
National Science FoundationDMS-1522645, DMS–1752934
Center for Selective C-H Functionalization, National Science Foundation
Center for Hierarchical Manufacturing, National Science Foundation
Horizon 2020 Framework Programme724228
Engineering Research Centers
European Research Council

    Keywords

    • Conformal deformation
    • Laplace–Beltrami eigensystem
    • Nonisometric manifold matching
    • Shape analysis

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