Misha Kazhdan Johns Hopkins University In this talk we will look at the problem of finding a transformation that registers two genus-zero shapes. We will start by looking at cMCF – a method for conformally parametrizing surfaces over the sphere to obtain a representation of surfaces over a canonical domain. Next we will decompose the space of lowest-frequency transformations (i.e. Mobius transformations) into two parts, inversions and rotations. We will show that since inversions can be described by advecting along curl-free vector fields, inversion ambiguity can be canonically removed through normalization. In contrast, rotations, which can be described by advecting along divergence-free vector fields, require explicit pairwise registration (which can be done using fast signal processing). Finally we will investigate the use of optical flow for refining the registration obtained by matching over the group of Mobius transformations.