The problem of comparing the shapes of surface arises in many fields,

including facial recognition, image processing, brain cortex analysis,

protein structure analysis and computer vision. It is referred to by

names such as surface registration, surface warping, best shape

analysis and geometric morphometrics. Hass and Koehl recently introduced a new

method to compare the shapes of two genus-zero surfaces (surfaces with no

holes) and to give an alighment between them. The method is based on a sequence of two

energy minimizations, first minimizing the Dirichlet energy to produce

a conformal map and then minimizing a newly defined symmetric distortion energy.

It produces a metric dsd, called the symmetric distortion metric,

on the space of piecewise-smooth genus-zero Riemannian

surfaces. In addition to giving a distance between any pair of such

surfaces, the method also produces an optimal correspondence

between them, a dieomorphism whose symmetric distortion energy is

minimized. The resulting theory has been implemented in software

and successfully tested in a variety of settings.

The theory is discussed in

J. Hass and P. Koehl, Comparing shapes of genus-zero surfaces,

Journal of Applied and Computational Topology, 1, (2017) 57–87.

Applications are given inP. Koehl and J. Hass, Landmark-free geometric methods in biological shape analysis,

Journal of The Royal Society Interface 12, (2015) 07--95.

Hass will talk on this at

Online Seminar GEOTOP-A, Next Talk

Date: August 23, 2019

An application to compare two brain cortices is shown in the figure