QCMC - Quasi-conformal Parameterization for Multiply-connected Domains

Kin Tat Ho The Chinese University of Hong Kong Ronald Lok Ming Lui The Chinese University of Hong Kong

Computational Geometry mathscidoc:1609.09004

Advances in Computational Mathematics, 42, (2), 279–312, 2016.6
This paper presents a method to compute the quasi-conformal parameterization (QCMC) for a multiply-connected 2D domain or surface. QCMC computes a quasi-conformal map from a multiply-connected domain S onto a punctured disk D_S associated with a given Beltrami diff erential. The Beltrami diff erential, which measures the conformality distortion, is a complex-valued function μ: S→ℂ with supremum norm strictly less than 1. Every Beltrami diff erential gives a conformal structure of S. Hence, the conformal module of D_S, which are the radii and centers of the inner circles, can be fully determined by μ, up to a Möbius transformation. In this paper, we propose an iterative algorithm to simultaneously search for the conformal module and the optimal quasi-conformal parameterization. The key idea is to minimize the Beltrami energy subject to the boundary constraints. The optimal solution is our desired quasi-conformal parameterization onto a punctured disk. The parameterization of the multiply-connected domain simpli fies numerical computations and has important applications in various fields, such as in computer graphics and vision. Experiments have been carried out on synthetic data together with real multiply-connected Riemann surfaces. Results show that our proposed method can efficiently compute quasi-conformal parameterizations of multiply-connected domains and outperforms other state-of-the-art algorithms. Applications of the proposed parameterization technique have also been explored.
Quasi-conformal, parameterization, multiply-connected, Beltrami di fferential, conformal module, Beltrami energy
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  title={QCMC - Quasi-conformal Parameterization for Multiply-connected Domains},
  author={Kin Tat Ho, and Ronald Lok Ming Lui},
  booktitle={Advances in Computational Mathematics},
Kin Tat Ho, and Ronald Lok Ming Lui. QCMC - Quasi-conformal Parameterization for Multiply-connected Domains. 2016. Vol. 42. In Advances in Computational Mathematics. pp.279–312. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160905152343168367616.
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