QCMC: quasi-conformal parameterizations for multiply-connected domains
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QCMC: quasi-conformal parameterizations for multiply-connected domains Kin Tat Ho1 · Lok Ming Lui1
Received: 1 September 2014 / Accepted: 25 May 2015 © Springer Science+Business Media New York 2015
Abstract 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 DS associated with a given Beltrami differential. The Beltrami differential, which measures the conformality distortion, is a complex-valued function μ : S → C with supremum norm strictly less than 1. Every Beltrami differential gives a conformal structure of S. Hence, the conformal module of DS , which are the radii and centers of the inner circles, can be fully determined by μ, up to a M¨obius 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 with the conformal module of the parameter domain incorporated. The optimal solution is our desired quasi-conformal parameterization onto a punctured disk. The parameterization of the multiply-connected domain simplifies 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
Communicated by: A. Iserles Lok Ming Lui
[email protected] Kin Tat Ho [email protected] 1
Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong
K. T. Ho, L. M. Lui
of multiply-connected domains and outperforms other state-of-the-art algorithms. Applications of the proposed parameterization technique have also been explored. Keywords Quasi-conformal · Parameterization · Multiply-connected · Beltrami differential · Conformal module · Beltrami energy Mathematics Subject Classifications (2010) 37K25 · 68U05 · 68U10
1 Introduction Parameterization refers to the process of mapping a complicated domain one-to-one and onto a simple canonical domain. For example, according to the Riemann mapping theorem, a simply-connected open surface can be conformally mapped onto the unit disk D. The geometry of the canonical domain is usually much simpler than its original domain. Hence, by parameterizing a complicated domain onto its simple parameter domain, a lot of numerical computations can be simplified. Parameterizations have been extensively studied and various parameterization algorithms have been developed. In particular, conformal parameterizations have been widely used, since it preserves the local geometry well. For example, in computer graphics, conformal parameterizations of 3D surfaces onto 2D images have been applied for texture mapping [2]. While in medical imaging, conformal parameterizations have been used for obtaining sur
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