AMS Seminar: Misha Kazhdan (JHU- Computer Science) @ Whitehead 304

Title: Mobius Registration
Abstract: Conformal spherical parametrizations of genus-zero surfaces have been explored as a way to represent surfaces over a canonical domain. This, in turn, makes it possible to establish correspondences between pairs of shapes, enabling applications like interpolation and detail transfer. However, one challenge of using these parametrizations is that they are only unique up to Mobius transformation. As such, to use these parametrization for establishing correspondences between shapes, it is first necessary to register the two spherical parametrizations with respect to the Mobius transformations. That is, to find the Mobius transformation best aligning the two spherical maps.
In this talk we will address the problem of Mobius registration by expresing the space of Mobius transformations as the composition of inversions and rotations. We will show that these two classes of transformation are fundamentally different: Each spherical parametrization can be canonically normalized to remove inversion ambiguity. And, using existing techniques for fast correlation over SO(3), pairs of spherical parametrizations can be rotationally aligned. We will consider implications of inversion normalization for the calculation of spherical orbifolds and will conclude by discussing why Mobius registration may not be sufficient.