Nonisometric Surface Registration via Conformal Laplace–Beltrami Basis Pursuit

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.

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Availability of data and material

The datasets generated during and/or analysed during the current study are available in the FAUST repository (http://faust.is.tue.mpg.de).

Notes

  1. 1.

    In fact, we do not need to require that the surfaces have the same number of points, but doing so for now will allow for more convenient notation.

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Funding

The research of S. Schonsheck and R. Lai is supported in part by NSF DMS–1522645 and an NSF Career Award DMS–1752934. M. Bronstein is partially supported by ERC Consolidator grant No. 724228 (LEMAN).

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R. Lai and M. Bronstein proposed the research. R. Lai and S. Schonsheck performed the research and wrote the paper.

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Correspondence to Rongjie Lai.

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Schonsheck, S.C., Bronstein, M.M. & Lai, R. Nonisometric Surface Registration via Conformal Laplace–Beltrami Basis Pursuit. J Sci Comput 86, 30 (2021). https://doi.org/10.1007/s10915-020-01390-y

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Keywords

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