Conformal Surface Alignment with Optimal Möbius Search

Deformations of surfaces with the same intrinsic shape can often be described accurately by a conformal model. A major focus of computational conformal geometry is the estimation of the conformal mapping that aligns a given pair of object surfaces. The uniformization theorem enables this task to be acccomplished in a canonical 2D domain, wherein the surfaces can be aligned using a Möbius transformation. Current algorithms for estimating Möbius transformations, however, often cannot provide satisfactory alignment or are computationally too costly. This paper introduces a novel globally optimal algorithm for estimating Möbius transformations to align surfaces that are topological discs. Unlike previous methods, the proposed algorithm deterministically calculates the best transformation, without requiring good initializations. Further, our algorithm is also much faster than previous techniques in practice. We demonstrate the efficacy of our algorithm on data commonly used in computational conformal geometry.