Soliton Dynamics in Computational Anatomy

April 6, 2006
  • Soliton solutions
  • 35C08
Computational Anatomy (CA) introduces the idea that shapes may be transformed into each other by geodesic deformations on groups of diffeomorphisms. In particular, the template matching approach involves Riemannian metrics on the tangent space of the diffeomorphism group and employs their projections onto specific landmark shapes, or image spaces. A singular momentum map provides an isomorphism between landmarks (and outlines) for images and singular soliton solutions of the geodesic equation. This isomorphism suggests a new dynamical paradigm for CA, as well as a new data representation. The main references for this talk are Soliton Dynamics in Computational Anatomy, D. D. Holm, J. T. Ratnanather, A. Trouvé, L. Younes, Momentum Maps and Measure-valued Solutions for the EPDiff Equation, D. D. Holm and J. E. Marsden, In The Breadth of Symplectic and Poisson Geometry, A Festshrift for Alan Weinstein, 203-235,Progr. Math., 232, J.E. Marsden and T.S. Ratiu, Editors, Birkhäuser Boston, Boston, MA, 2004. Also at D. D. Holm and M. F. Staley, Interaction Dynamics of Singular Wave Fronts, at Martin Staley's website, under "Recent Papers" at