Communications in Mathematical Sciences

Volume 15 (2017)

Number 7

Optimized quasiconformal parameterization with user-defined area distortions

Pages: 2027 – 2054



Ka Chun Lam (Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong)

Lok Ming Lui (Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong)


Parameterization, a process of mapping a complicated domain onto a simple canonical domain, is crucial in different areas such as computer graphics, medical imaging and scientific computing. Conformal parameterization has been widely used since it preserves the local geometry well. However, a major drawback is the area distortion introduced by the conformal parameterization, causing inconvenience in many applications such as texture mapping in computer graphics or visualization in medical imaging. This work proposes a remedy to construct a parameterization that balances between conformality and area distortions. We present a variational algorithm to compute the optimized quasiconformal parameterization with controllable area distortions. The distribution of the area distortion can be prescribed by users according to the application. The main strategy is to minimize a combined energy functional consisting of an area mismatching term and a regularization term involving the Beltrami coefficient of the map. The Beltrami coefficient controls the conformality of the parameterization. Landmark constraints can be incorporated into the model to obtain landmark-aligned parameterization. Experiments have been carried out on both synthetic and real data. Results demonstrate the efficacy of the proposed algorithm to compute the optimized parameterization with controllable area distortion while preserving the local geometry as well as possible.


area-preserving mapping, Beltrami coefficient, conformality distortion, parameterization, texture mapping

2010 Mathematics Subject Classification

65D18, 65K10, 92C55

Received 18 August 2016

Accepted 27 June 2017

Published 16 October 2017