Numerical LDDMM of $m$-currents with finite elements

Please always quote using this URN: urn:nbn:de:0296-matheon-10063
  • We present a unified computational framework for matching 3d geometric objects (points, lines, surfaces, volumes) of highly varying shape. Our approach is based on the Large Deformation Diffeomorphic Metric Mapping (LDDMM) method acting on $m$-currents. After stating an optimization algorithm in the function space of admissible morph generating velocity fields, two innovative aspects in this framework are presented: First, we spatially discretize the velocity field with conforming adaptive finite elements and discuss advantages of this new approach. Secondly, we directly compute the temporal evolution of discrete $m$-current attributes. Several numerical experiments demonstrate the effectiveness of this approach.

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Metadaten
Author:Andreas Günther, Hans Lamecker, Martin Weiser
URN:urn:nbn:de:0296-matheon-10063
Referee:Peter Deuflhard
Language:English
Date of first Publication:2012/01/23
Institute:Zuse Institute Berlin (ZIB)
MSC-Classification:37-XX DYNAMICAL SYSTEMS AND ERGODIC THEORY [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX] / 37Exx Low-dimensional dynamical systems / 37E30 Homeomorphisms and diffeomorphisms of planes and surfaces
49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Jxx Existence theories / 49J20 Optimal control problems involving partial differential equations
58-XX GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx](For geometric integration theory, see 49Q15) / 58Axx General theory of differentiable manifolds [See also 32Cxx] / 58A25 Currents [See also 32C30, 53C65]
58-XX GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx](For geometric integration theory, see 49Q15) / 58Jxx Partial differential equations on manifolds; differential operators [See also 32Wxx, 35-XX, 53Cxx] / 58J72 Correspondences and other transformation methods (e.g. Lie- Bäcklund) [See also 35A22]
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Preprint Number:866