Planar Shape Interpolation Based On Teichmüller Mapping

• Published in: Computer graphics forum Vol. 35; no. 7; pp. 43 - 56
• Main Authors: Nian, Xianshun, Chen, Falai
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.10.2016
• Subjects: Algorithms; Analysis; Beltrami coefficient; Categories and Subject Descriptors (according to ACM CCS); Computer graphics; conformal distortion; I.3.3 [Computer Graphics]: Picture/Image Generation-Display Algorithms; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism-Animation; Interpolation; shape interpolation; Studies; Teichmüller mapping
• ISSN: 0167-7055; 1467-8659
• DOI: 10.1111/cgf.13002

Shape interpolation is a classical problem in computer graphics and has been widely investigated in the past two decades. Ideal shape interpolation should be natural and smooth which have good properties such as affine and conformal reproduction, bounded distortion, no fold‐overs, etc. In this paper, we present a new approach for planar shape interpolation based on Teichmüller maps ‐ a special type of maps in the class of quasi‐conformal maps. The algorithm consists of two steps. In the first step, a Teichmüller map is computed from the source shape to the target shape, and then the Beltrami coefficient is interpolated such that the conformal distortion is linear with respect to the time variable. In the second step, the intermediate shape is reconstructed by solving the Beltrami equation locally over each triangle and then stitching the mapped triangles by conformal transformations. The new approach preserves all the good properties mentioned above and produces more natural and more uniform intermediate shapes than the start‐of‐the‐art methods. Especially, the conformal distortion changes linearly with respect to the time variable. Experiment results show that our method can produce appealing results regardless of interpolating between the same or different objects.