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