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Local Statistics on Shape Diffeomorphisms Using a Depth Potential Function

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Mathematical Methods for Signal and Image Analysis and Representation

Part of the book series: Computational Imaging and Vision ((CIVI,volume 41))

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Abstract

Shape diffeomorphisms are used along with statistical models to localize shape differences within a set of similar surfaces. Diffeomorphisms in images are found by looking at similarities within the intensity function of the image. Surfaces do not enjoy the availability of such an intensity function and one therefore needs to choose which function will give the most accurate match between similar features. Thus, it is important to choose an intensity function that faithfully represents features of the surface as accurately as possible. In this paper, we present a fast method which uses a curvature scale space to capture surface features over a wide range of scales. We recall the relationship between Poisson equation and scale space representations, and we use this relationship to formulate a potential function which integrates curvature information over a wide range of scales. The resulting map is a global function of bending (as measured by curvature) that is used to find diffeomorphic maps between surfaces. The potential function showed to be more accurate than a depth map computed on the surface.

As an application of the depth potential function, we use it to analyze diffeomorphic maps of the human brain through a statistical analysis of surface deformation. We used surfaces extracted from 92 subjects affected with very mild to mild dementia and 97 healthy subjects. Using T 2-statistics on the diffeomorphic map between the groups of demented and non-demented subjects, we were able to detect the atrophy of the interior and exterior temporal lobe due to the presence of mild dementia.

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Authors and Affiliations

  1. School of Computer Science, McGill University, Montreal, Canada

    Maxime Boucher

  2. McConnell Brain Imaging Center, Montreal Neurological Institute, McGill University, Montreal, Canada

    Alan Evans

Authors
  1. Maxime Boucher
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  2. Alan Evans
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Correspondence to Maxime Boucher .

Editor information

Editors and Affiliations

  1. Dept. Mathematics & Computer Science, Eindhoven University of Technology, Eindhoven, 5600 MB, The Netherlands

    Luc Florack

  2. Dept. Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, Eindhoven, 5600 MB, The Netherlands

    Remco Duits

  3. Dept. Applied Mathematics, Delft University of Technology, Delft, 2628 CD, The Netherlands

    Geurt Jongbloed

  4. Probability & Stochastic Networks (PNA), Centrum Wiskunde & Informatica, Amsterdam, 1098 XG, The Netherlands

    Marie-Colette van Lieshout

  5. Fakultät für Mathematik, Universität Duisburg-Essen, Essen, 45141, Germany

    Laurie Davies

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Boucher, M., Evans, A. (2012). Local Statistics on Shape Diffeomorphisms Using a Depth Potential Function. In: Florack, L., Duits, R., Jongbloed, G., van Lieshout, MC., Davies, L. (eds) Mathematical Methods for Signal and Image Analysis and Representation. Computational Imaging and Vision, vol 41. Springer, London. https://doi.org/10.1007/978-1-4471-2353-8_11

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