# What is the Space of Shapes and What Can We Do With It?

Wednesday, November 15, 2000 - 3:30pm - 4:30pm

Keller 3-180

David Mumford (Brown University)

Computer vision needs a quantitative representation of shape for object recognition. Subjectively, we have a clear idea of what shape means. But what is the right mathematical theory of shape? We propose that there is a hierarchy of nested shape spaces like the hierarchy of Sobolev/Ck spaces of functions which define subsets S in R2 or R3 (think of closed subsets bounded by piecewise smooth curves) with varying degrees of complexity. The advantage of having such spaces is that they give a setting for numerous questions:

a) What are the natural metrics and norms on these spaces and are they Banach manifolds, b) Define a natural cell decomposition of this non-linear space into local linear charts, c) Define a tangent space and Riemannian metric and find its geodesics and its curvature, d) Define a set of probability measures on these spaces, find their supports and relations.

Of course, all these questions have been partly addressed already. Thus the medial axis is a natural construction for constructing local linear charts, Riemannian metrics have been studied in the related question of the space of diffeomorphisms and probability measures have been introduced using stochastic differential equations or polygonal approximation. I will try to pull these ideas together and point out where work needs to be done. The case of R2 doesn't seem too hard but R3 is much more difficult.

a) What are the natural metrics and norms on these spaces and are they Banach manifolds, b) Define a natural cell decomposition of this non-linear space into local linear charts, c) Define a tangent space and Riemannian metric and find its geodesics and its curvature, d) Define a set of probability measures on these spaces, find their supports and relations.

Of course, all these questions have been partly addressed already. Thus the medial axis is a natural construction for constructing local linear charts, Riemannian metrics have been studied in the related question of the space of diffeomorphisms and probability measures have been introduced using stochastic differential equations or polygonal approximation. I will try to pull these ideas together and point out where work needs to be done. The case of R2 doesn't seem too hard but R3 is much more difficult.