On the Convexity and Feasibility of the Bounded Distortion Harmonic Mapping Problem

Computation of mappings is a central building block in many geometry processing and graphics applications. The pursuit to compute mappings that are injective and have a controllable amount of conformal and isometric distortion is a long endeavor which has received significant attention by the scientific community in recent years. The difficulty of the problem stems from the fact that the space of bounded distortion mappings is nonconvex. In this paper, we consider the special case of harmonic mappings which have been used extensively in many graphics applications. We show that, somewhat surprisingly, the space of locally injective planar harmonic mappings with bounded conformal and isometric distortion has a convex characterization. We describe several projection operators that, given an arbitrary input mapping, are guaranteed to output a bounded distortion locally injective harmonic mapping that is closest to the input mapping in some special sense. In contrast to alternative approaches, the optimization problems that correspond to our projection operators are shown to be always feasible for any choice of distortion bounds. We use the boundary element method (BEM) to discretize the space of planar harmonic mappings and demonstrate the effectiveness of our approach through the application of planar shape deformation.