On the Topology and Geometry of Spaces of Affine Shapes

  • Authors:

  • David Groisser;Hemant D. Tagare

  • Affiliations:

  • Department of Mathematics, University of Florida, Gainesville, USA 32611-8105;Department of Diagnostic Radiology, Department of Electrical Engineering, Yale University, New Haven, USA 06520

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 2009

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Abstract

We define the space of affine shapes of k points in R n to be the topological quotient of (R n ) k modulo the natural action of the affine group of R n . These spaces arise naturally in many image-processing applications, and despite having poor separation properties, have some topological and geometric properties reminiscent of the more familiar Procrustes shape spaces Σ n k in which one identifies configurations related by an orientation-preserving Euclidean similarity transformation. We examine the topology of the connected, non-Hausdorff spaces Sh n k in detail. Each Sh n k is a disjoint union of naturally ordered strata, each of which is homeomorphic in the relative topology to a Grassmannian, and we show how the strata are attached to each other. The top stratum carries a natural Riemannian metric, which we compute explicitly for kn, expressing the metric purely in terms of "pre-shape" data, i.e. configurations of k points in R n .