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Mathematical Methods for Signal and Image Analysis and Representation pp 159–171Cite as

Scale Space Representations Locally Adapted to the Geometry of Base and Target Manifold

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

Abstract

We generalize the Gaussian multi-resolution image paradigm for a Euclidean domain to general Riemannian base manifolds and also account for the codomain by considering the extension into a fibre bundle structure. We elaborate on aspects of parametrization and gauge, as these are important in practical applications. We subsequently scrutinize two examples that are of interest in bio-mathematical modeling, viz. scale space on the unit sphere, used among others for codomain regularization in the context of high angular resolution diffusion imaging (HARDI), and retino-cortical scale space, proposed as a biologically plausible model of the human visual pathway from retina to striate cortex.

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Notes

  1. 1.

    The operator exp(tΔ g) is bounded and defines a strongly continuous semigroup for t∈ℝ+∪{0}.

  2. 2.

    Additional Γ-corrections will be needed for non-scalar signals, recall Sect. 9.2.1, e.g. \(D_{\mu}^{A,\varGamma}v^{\nu}(x) = (\delta^{\nu}_{\rho }(\partial_{\mu}+A_{\mu}(x)) + \varGamma_{\rho\mu}^{\nu}(x))v^{\rho}(x)\).

  3. 3.

    In physics, however, one typically considers Hermitean operators i∂ μ, respectively i∂ μ+A μ.

  4. 4.

    This definition differs from the one proposed by Georgiev, which fails to be self-dual [184].

  5. 5.

    At this level of rigor we ignore the singularity at the origin, but cf. [153].

  6. 6.

    Homegeneity in fact holds almost everywhere in the sense that space remains flat except at the foveal centre, at which the Ricci curvature tensor degenerates. This singularity can be removed at the expense of introducing global curvature with appreciable magnitude in the fovea centralis [153].

  7. 7.

    Cf. functions.wolfram.com for further properties of \(Y^{m}_{\ell}\) and \(P^{m}_{\ell}\).

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Acknowledgements

The Netherlands Organisation for Scientific Research (NWO) is gratefully acknowledged for financial support.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science & Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, 5600 MB, Eindhoven, The Netherlands

    Luc Florack

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  1. Luc Florack
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Correspondence to Luc Florack .

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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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Florack, L. (2012). Scale Space Representations Locally Adapted to the Geometry of Base and Target Manifold. 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_9

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