Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements View Full Text


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Article Info

DATE

2006-07

AUTHORS

Xavier Pennec

ABSTRACT

In medical image analysis and high level computer vision, there is an intensive use of geometric features like orientations, lines, and geometric transformations ranging from simple ones (orientations, lines, rigid body or affine transformations, etc.) to very complex ones like curves, surfaces, or general diffeomorphic transformations. The measurement of such geometric primitives is generally noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation), to compare observations, or to test hypotheses. Unfortunately, even simple geometric primitives often belong to manifolds that are not vector spaces. In previous works [1, 2], we investigated invariance requirements to build some statistical tools on transformation groups and homogeneous manifolds that avoids paradoxes. In this paper, we consider finite dimensional manifolds with a Riemannian metric as the basic structure. Based on this metric, we develop the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and χ2 law. We provide a new proof of the characterization of Riemannian centers of mass and an original gradient descent algorithm to efficiently compute them. The notion of Normal law we propose is based on the maximization of the entropy knowing the mean and covariance of the distribution. The resulting family of pdfs spans the whole range from uniform (on compact manifolds) to the point mass distribution. Moreover, we were able to provide tractable approximations (with their limits) for small variances which show that we can effectively implement and work with these definitions. More... »

PAGES

127

References to SciGraph publications

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  • 1998-07. Uniform Distribution, Distance and Expectation Problems for Geometric Features Processing in JOURNAL OF MATHEMATICAL IMAGING AND VISION
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  • 1997-12. A Framework for Uncertainty and Validation of 3-D Registration Methods Based on Points and Frames in INTERNATIONAL JOURNAL OF COMPUTER VISION
  • 1998. Feature-based registration of medical images: Estimation and validation of the pose accuracy in MEDICAL IMAGE COMPUTING AND COMPUTER-ASSISTED INTERVENTION — MICCAI’98
  • 1991. Theoretical and Distributional Aspects of Shape Analysis in PROBABILITY MEASURES ON GROUPS X
  • 1989. Stochastic Calculus in Manifolds in NONE
  • 2004. Principal Geodesic Analysis on Symmetric Spaces: Statistics of Diffusion Tensors in COMPUTER VISION AND MATHEMATICAL METHODS IN MEDICAL AND BIOMEDICAL IMAGE ANALYSIS
  • 2001-04. On Averaging Rotations in INTERNATIONAL JOURNAL OF COMPUTER VISION
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  • 2001. Rigid Point-Surface Registration Using an EM Variant of ICP for Computer Guided Oral Implantology in MEDICAL IMAGE COMPUTING AND COMPUTER-ASSISTED INTERVENTION – MICCAI 2001
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    http://scigraph.springernature.com/pub.10.1007/s10851-006-6228-4

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    http://dx.doi.org/10.1007/s10851-006-6228-4

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