A Riemannian Framework for Intrinsic Comparison of Closed Genus-Zero Shapes View Full Text


Ontology type: schema:Chapter      Open Access: True


Chapter Info

DATE

2015

AUTHORS

Boris A. Gutman , P. Thomas Fletcher , M. Jorge Cardoso , Greg M. Fleishman , Marco Lorenzi , Paul M. Thompson , Sebastien Ourselin

ABSTRACT

We present a framework for intrinsic comparison of surface metric structures and curvatures. This work parallels the work of Kurtek et al. on parameterization-invariant comparison of genus zero shapes. Here, instead of comparing the embedding of spherically parameterized surfaces in space, we focus on the first fundamental form. To ensure that the distance on spherical metric tensor fields is invariant to parameterization, we apply the conjugation-invariant metric arising from the L2 norm on symmetric positive definite matrices. As a reparameterization changes the metric tensor by a congruent Jacobian transform, this metric perfectly suits our purpose. The result is an intrinsic comparison of shape metric structure that does not depend on the specifics of a spherical mapping. Further, when restricted to tensors of fixed volume form, the manifold of metric tensor fields and its quotient of the group of unitary diffeomorphisms becomes a proper metric manifold that is geodesically complete. Exploiting this fact, and augmenting the metric with analogous metrics on curvatures, we derive a complete Riemannian framework for shape comparison and reconstruction. A by-product of our framework is a near-isometric and curvature-preserving mapping between surfaces. The correspondence is optimized using the fast spherical fluid algorithm. We validate our framework using several subcortical boundary surface models from the ADNI dataset. More... »

PAGES

205-18

References to SciGraph publications

  • 2014-09. Overview of the Geometries of Shape Spaces and Diffeomorphism Groups in JOURNAL OF MATHEMATICAL IMAGING AND VISION
  • 2012-08. Computing quasiconformal maps using an auxiliary metric and discrete curvature flow in NUMERISCHE MATHEMATIK
  • 2013. A Family of Fast Spherical Registration Algorithms for Cortical Shapes in MULTIMODAL BRAIN IMAGE ANALYSIS
  • 2003. Discrete Differential-Geometry Operators for Triangulated 2-Manifolds in VISUALIZATION AND MATHEMATICS III
  • 2010-11. The metric geometry of the manifold of Riemannian metrics over a closed manifold in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
  • 2001-01. Group Actions, Homeomorphisms, and Matching: A General Framework in INTERNATIONAL JOURNAL OF COMPUTER VISION
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/978-3-319-19992-4_16

    DOI

    http://dx.doi.org/10.1007/978-3-319-19992-4_16

    DIMENSIONS

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    PUBMED

    https://www.ncbi.nlm.nih.gov/pubmed/26221675


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