Metric Spaces on the space of Shape
David Mumford
Brown University
Applied Mathematics
The space of two- and three-dimensional shapes is an inherently nonlinear, infinite-dimensional object, but it is a manifold; that is, the set of infinitesimal deformations of any specific shape forms a vector space. Thus, the challenge of defining a metric on the space of shapes can be compared to the development of metrics by functional analysts on spaces of functions. In particular, such metrics may be classified according to (a) how many derivatives they control, and (b) whether first, second, or pth powers—or supremum norms—of the deformation are bounded.
In the first part of the talk, we outline this general perspective and provide examples. In the second part, we focus on a specific metric: the Weil–Petersson metric on two-dimensional shapes. This metric is defined using complex analysis and has the remarkable property of making the space of two-dimensional shapes into a homogeneous metric space under the transitive action of Diff(S¹), the group of diffeomorphisms of the circle.
We explain this metric with illustrative examples and describe its connections to the deformation metrics of Miller et al. and to the medial axis framework.
