Harmonic Shape Interpolation on Multiply-connected Planar Domains

Abstract

Shape interpolation is a fundamental problem in computer graphics. Recently, there have been some interpolation methods developed which guarantee that the results are of bounded amount of geometric distortion, hence ensure high quality interpolation. However, none of these methods is applicable to shapes within the multiply-connected domains. In this work, we develop an interpolation scheme for harmonic mappings, that specifically addresses this limitation. We opt to interpolate the pullback metric of the input harmonic maps as proposed by Chen et al. [CWKBC13]. However, the interpolated metric does not correspond to any planar mapping, which is the main challenge in the interpolation problem for multiply-connected domains. We propose to solve this by projecting the interpolated metric into the planar harmonic mapping space. Specifically, we develop a Newton iteration to minimize the isometric distortion of the intermediate mapping, with respect to the interpolated metric. For more efficient Newton iteration, we further derived a simple analytic formula for the positive semidefinite (PSD) projection of the Hessian matrix of our distortion energy. Through extensive experiments and comparisons with the state-of-the-art, we demonstrate the efficacy and robustness of our method for various inputs.