A Numerical Framework for Sobolev Metrics on the Space of Curves

Martin Bauer, Martins Bruveris, Philipp Harms, Jakob Møller-Andersen

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Abstract

Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics, but their discretization is still largely missing. In this paper, we present algorithms to numerically solve the geodesic initial and boundary value problems for these metrics. The combination of these algorithms enables one to compute Karcher means in a Riemannian gradient-based optimization scheme and perform principal component analysis and clustering. Our framework is sufficiently general to be applicable to a wide class of metrics. We demonstrate the effectiveness of our approach by analyzing a collection of shapes representing HeLa cell nuclei.
Original languageEnglish
JournalS I A M Journal on Imaging Sciences
Volume10
Issue number1
Pages (from-to)47-73
ISSN1936-4954
DOIs
Publication statusPublished - 2017

Keywords

  • Mathematics (all)
  • Applied Mathematics
  • B-splines
  • Geodesics
  • Karcher mean
  • Shape analysis
  • Shape registration
  • Sobolev metric
  • Boundary value problems
  • Geometry
  • Optimization
  • Sobolev spaces
  • B splines
  • Karcher means
  • Sobolev
  • Principal component analysis
  • COMPUTER
  • MATHEMATICS,
  • IMAGING
  • RIEMANNIAN-MANIFOLDS
  • SHAPE SPACES
  • STATISTICAL-ANALYSIS
  • PLANE-CURVES
  • HELA-CELLS
  • TRACKING
  • TRAJECTORIES
  • GEOMETRIES
  • shape analysis
  • shape registration
  • geodesics
  • Graphics techniques
  • Other topics in statistics
  • Combinatorial mathematics
  • Optimisation techniques
  • Interpolation and function approximation (numerical analysis)
  • computational geometry
  • curve fitting
  • optimisation
  • principal component analysis
  • set theory
  • Sobolev metrics
  • curve space
  • statistical shape analysis
  • Riemannian framework
  • Riemannian metric
  • unparametrized curves
  • Riemannian gradient-based optimization scheme

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