Metric Spaces on the space of "Shape"

David Mumford
Brown University
Applied Mathematics

The space of 2D and 3D shapes is an inherently non-linear infinite dimensional object,
but it is a manifold, that is, the set of infinitesimal deformations of any specific shape is a
vector space. Thus one can compare the challenge of putting a metric on the space of
shapes to the metrics developed by functional analysts on spaces of functions. In
particular, one can classify them by (a) how many derivatives they control and (b) first,
second, pth powers of or sups of the deformation are bounded. In the first part of the
talk, we will sketch this general perspective and give examples. In the second part of the
talk, I want to focus on one metric, the "Weil-Petersen metric" on 2D shapes. This metric
is defined using complex analysis and has the truly remarkable property that it makes
the space of 2D shapes into a homogeneous metric metric under the transitive action of
Diff(S1), the group of diffeomorphisms of the circle. We will explain this metric, also with
examples and explain how it is related both to the deformation metrics of Miller et al and
to the medial axis.

Presentation (PDF File)

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