Reference Work Entry

Handbook of Mathematical Methods in Imaging

pp 1363-1401

Variational Methods in Shape Analysis

  • Martin RumpfAffiliated withComputational Science Center, University of ViennaBonn University
  • , Benedikt WirthAffiliated withRICAM, Austrian Academy of SciencesBonn University

Abstract

The concept of a shape space is linked both to concepts from geometry and from physics. On one hand, a path-based viscous flow approach leads to Riemannian distances between shapes, where shapes are boundaries of objects that mainly behave like fluids. On the other hand, a state-based elasticity approach induces a (by construction) non-Riemannian dissimilarity measure between shapes, which is given by the stored elastic energy of deformations matching the corresponding objects. The two approaches are both based on variational principles. They are analyzed with regard to different applications, and a detailed comparison is given.