Conformal Surface Alignment With Optimal Mobius 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 en- ables this task to be acccomplished in a canonical 2D do- main, wherein the surfaces can be aligned using a M obius transformation. Current algorithms for estimating M obius transformations, however, often cannot provide satisfactory alignment or are computationally too costly. This paper in- troduces a novel globally optimal algorithm for estimating Mobius transformations to align surfaces that are topologi- cal discs. Unlike previous methods, the proposed algorithm deterministically calculates the best transformation, with- out 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.