As-conformal-as-possible discrete volumetric mapping

In this paper, we tackle the problem of generalizing conformal maps to volumetric meshes. Current methods seek for harmonicity but unfortunately, no computational methods optimize conformality in the volumetric context. As it is proven that conformal maps do not exist for general volume transformations, we seek to optimize shape preservation with a generalization of the CauchyRiemann equations. Our algorithm is fast and easily adaptable to existing harmonic mapping methods. Compared to harmonic maps, results show improvements on both angular and volumetric energy measures at a cost below 1% of total computations. The method extends well in any dimension and several research areas could benefit from our derivations of volumetric conformal optimization. (a) Harmonic (b) ACAP (c) Uniform scale Figure 1: An identical planar cut through a sphere with a small bump that is parameterized with (a) the Laplace operator, (b) the our operator, and (c) the uniform scale operator (ω = 0.6125).