Diffeomorphic Matching and Dynamic Deformable Surfaces in 3d Medical Imaging

R. Azencott, R. Azencott, R. Glowinski, J. He, A. Jajoo, Y. Li, A. Martynenko, R. H.W. Hoppe, R. H.W. Hoppe, S. Benzekry, S. H. Little, W. A. Zoghbi

Research output: Contribution to journalArticle

12 Scopus citations

Abstract

We consider optimal matching of submanifolds such as curves and surfaces by a variational approach based on Hilbert spaces of diffeomorphic transformations. In an abstract setting, the optimal matching is formulated as a minimization problem involving actions of diffeomorphisms on regular Borel measures considered as supporting measures of the reference and the target submanifolds. The objective functional consists of two parts measuring the elastic energy of the dynamically deformed surfaces and the quality of the matching. To make the problem computationally accessible, we use reproducing kernel Hilbert spaces with radial kernels and weighted sums of Dirac measures which gives rise to diffeomorphic point matching and amounts to the solution of a finite dimensional minimization problem. We present a matching algorithm based on the first order necessary optimality conditions which include an initial-value problem for a dynamical system in the trajectories describing the deformation of the surfaces and a final-time problem associated with the adjoint equations. The performance of the algorithm is illustrated by numerical results for examples from medical image analysis.

Original languageEnglish (US)
Pages (from-to)235-274
Number of pages40
JournalComputational Methods in Applied Mathematics
Volume10
Issue number3
DOIs
StatePublished - 2010

Keywords

  • Dirac measures
  • deformable surfaces
  • diffeomorphic image matching
  • gradient method
  • medical image analysis
  • reproducing kernel Hilbert spaces

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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