A simple and local method for computing quasi-conformal map on 3D surfaces

Minqi Zhang, Ying He ∗
School of Computer Engineering, Nanyang Technological University, Singapore
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abstract
Quasi-conformal maps have bounded conformal distortion, and are the natural extension of the conformal maps. The existing techniques to compute the quasi-conformal map require a global coordinate system; thus, they are limited to models of simple topological types, such as genus-0 or 1 surfaces, for which one can obtain the global coordinates by the global parameterization. This paper presents a simple yet effective technique for computing a quasi-conformal map on surfaces of non-trivial topology. Our method extends the quasi-conformal iteration method (Lui et al., 2012) [8] from the complex plane to the manifold setting. It requires neither numerical solver nor the global coordinate system, thus, is easy to implement. Moreover, thanks to its simple and parallel structure, our method is well suited for parallel computing. Experimental results on 3D models of various topological types demonstrate the efficacy of our technique. © 2013 Elsevier Ltd. All rights reserved.
Keywords: Quasi-conformal mapping Beltrami coefficient Local coordinate chart Exponential map
1. Introduction Computing surface-to-plane and surface-to-surface maps plays a critical role in many graphics applications, such as texture mapping, surface remeshing, surface registration, spline construction, etc. Except the isometric maps, most maps from real-world models to their domains inevitably introduce either angle or area distortion. Thus, the methods for surface mapping are usually divided into two categories, angle preserving maps and area preserving maps. The last decade had witnessed a fast development of techniques of computing conformal mapping on 3D models. Representative work includes the least-squares conformal mapping [1], the holomorphic 1-form method [2], discrete surface Ricci flow [3], angle flattening [4], as-rigid-as-possible parameterization [5], circle pattern [6], discrete conformal equivalence [7], etc. It is well known that all genus-0 surfaces can be conformally mapped to the unit sphere S2 ; thus, they are conformally equivalent. But the conformal maps between surfaces of high genus usually do not exist. For example, two tori are conformally equivalent if and only if their ratios of the radius of the tube and the radius of the swept circle agree. In general, conformal maps are really rare for high genus models in real-world applications. A natural generalization of the conformal map is the quasi-conformal map, which has bounded conformality distortion. Intuitively speaking, a conformal map sends an infinitesimal circle to an infinitesimal circle, while a quasi-conformal map sends an infinitesimal circle to an infinitesimal ellipse with bounded eccentricity.
Computer-Aided Design 46 (2014) 192–199
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Computer-Aided Design
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A simple and local method for computing quasi-conformal map on 3D surfaces
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Corresponding author. Tel.: +65 6514 1008. E-mail addresses: mzhang1@.sg (MAmong all quasi-conformal maps, the extremal quasi-conformal map is of particular interest, since it minimizes the maximal conformal distortion. The extremal quasi-conformal map always exists but may not be unique. A Teichmüller map, or T-map, is a quasiconformal map associated with a holomorphic function and it is unique in its homotopic class. Lui et al. [8] formulated the T-map as a variational problem, which minimizes the conformality distortion and the harmonic energy of the magnitude of the Beltrami coefficient. They proposed the quasi-conformal iteration algorithm to compute the Beltrami coefficient and reconstruct the T-map by using the linear Beltrami solver. Due to the global coordinate system required in computing the Beltrami coefficient, their method is limited to models with simple topology, such as simply or multiply connected domains, genus-0 and genus-1 models, for which one can construct the global coordinate system by the global parameterization technique. In this paper, we propose a simple yet effective technique to compute the extremal quasi-conformal map on surfaces of nontrivial topology. Our method extends the quasi-conformal iteration from the complex plane to the manifold setting. For each mesh vertex, we construct a local coordinate chart by the exponential map, which is done efficiently in the pre-computation stage. Rather than the global method [8] which solves a global linear system, we iteratively compute the map in the local coordinate chart and then diffuse it through the entire mesh. Our method is linear, local, easy to implement and has a very natural parallel structure. It is also robust and insensitive to the initial mapping. For the initial map of poor quality (i.e., non-bijection with large number of overlapped/self-intersected regions), our method can efficiently diffuse it to a high quality quasi-conformal map. We have tested our method on both synthetic and real-world models of various