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A linear formulation for disk conformal parameterization of simply-connected open surfaces

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Abstract

Surface parameterization is widely used in computer graphics and geometry processing. It simplifies challenging tasks such as surface registrations, morphing, remeshing and texture mapping. In this paper, we present an efficient algorithm for computing the disk conformal parameterization of simply-connected open surfaces. A double covering technique is used to turn a simply-connected open surface into a genus-0 closed surface, and then a fast algorithm for parameterization of genus-0 closed surfaces can be applied. The symmetry of the double covered surface preserves the efficiency of the computation. A planar parameterization can then be obtained with the aid of a Möbius transformation and the stereographic projection. After that, a normalization step is applied to guarantee the circular boundary. Finally, we achieve a bijective disk conformal parameterization by a composition of quasi-conformal mappings. Experimental results demonstrate a significant improvement in the computational time by over 60%. At the same time, our proposed method retains comparable accuracy, bijectivity and robustness when compared with the state-of-the-art approaches. Applications to texture mapping are presented for illustrating the effectiveness of our proposed algorithm.

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Acknowledgment

Lok Ming Lui is supported by RGC GRF (Project ID: 404612).

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Authors and Affiliations

  1. John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

    Gary Pui-Tung Choi

  2. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, Hong Kong

    Lok Ming Lui

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  1. Gary Pui-Tung Choi
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  2. Lok Ming Lui
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Correspondence to Lok Ming Lui.

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Communicated by: Yang Wang

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Choi, G.PT., Lui, L.M. A linear formulation for disk conformal parameterization of simply-connected open surfaces. Adv Comput Math 44, 87–114 (2018). https://doi.org/10.1007/s10444-017-9536-x

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  • DOI: https://doi.org/10.1007/s10444-017-9536-x

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