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Free-Boundary Conformal Parameterization of Point Clouds

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Abstract

With the advancement in 3D scanning technology, there has been a surge of interest in the use of point clouds in science and engineering. To facilitate the computations and analyses of point clouds, prior works have considered parameterizing them onto some simple planar domains with a fixed boundary shape such as a unit circle or a rectangle. However, the geometry of the fixed shape may lead to some undesirable distortion in the parameterization. It is therefore more natural to consider free-boundary conformal parameterizations of point clouds, which minimize the local geometric distortion of the mapping without constraining the overall shape. In this work, we develop a free-boundary conformal parameterization method for disk-type point clouds, which involves a novel approximation scheme of the point cloud Laplacian with accumulated cotangent weights together with a special treatment at the boundary points. With the aid of the free-boundary conformal parameterization, high-quality point cloud meshing can be easily achieved. Furthermore, we show that using the idea of conformal welding in complex analysis, the point cloud conformal parameterization can be computed in a divide-and-conquer manner. Experimental results are presented to demonstrate the effectiveness of the proposed method.

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Acknowledgements

This work was supported in part by the National Science Foundation under Grant No. DMS-2002103 (to Gary P. T. Choi), and HKRGC GRF under Project ID 2130549 (to Lok Ming Lui).

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Gary P. T. Choi

  2. Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD, USA

    Yechen Liu

  3. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Lok Ming Lui

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  1. Gary P. T. Choi
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  2. Yechen Liu
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Correspondence to Lok Ming Lui.

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Choi, G.P.T., Liu, Y. & Lui, L.M. Free-Boundary Conformal Parameterization of Point Clouds. J Sci Comput 90, 14 (2022). https://doi.org/10.1007/s10915-021-01641-6

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  • DOI: https://doi.org/10.1007/s10915-021-01641-6

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