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Positive Signal Spaces and the Mehler-Fock Transform

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Geometric Science of Information (GSI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10589))

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Abstract

Eigenvector expansions and perspective projections are used to decompose a space of positive functions into a product of a half-axis and a solid unit ball. This is then used to construct a conical coordinate system where one component measures the distance to the origin, a radial measure of the distance to the axis and a unit vector describing the position on the surface of the ball. A Lorentz group is selected as symmetry group of the unit ball which leads to the Mehler-Fock transform as the Fourier transform of functions depending an the radial coordinate only. The theoretical results are used to study statistical properties of edge magnitudes computed from databases of image patches. The constructed radial values are independent of the orientation of the incoming light distribution (since edge-magnitudes are used), they are independent of global intensity changes (because of the perspective projection) and they characterize the second order statistical moment properties of the image patches. Using a large database of images of natural scenes it is shown that the generalized extreme value distribution provides a good statistical model of the radial components. Finally, the visual properties of textures are characterized using the Mehler-Fock transform of the probability density function of the generalized extreme value distribution.

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Notes

  1. 1.

    Available at http://tofu.psych.upenn.edu/~upennidb.

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Acknowledgements

The support of the Swedish Research Council through a framework grant for the project “Energy Minimization for Computational Cameras” (2014-6227) is gratefully acknowledged.

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  1. Linköping University, 58183, Linköping, Sweden

    Reiner Lenz

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  1. Reiner Lenz
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Correspondence to Reiner Lenz .

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Editors and Affiliations

  1. Ecole Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land and Air Systems, Limours, France

    Frédéric Barbaresco

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Lenz, R. (2017). Positive Signal Spaces and the Mehler-Fock Transform. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_86

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  • DOI: https://doi.org/10.1007/978-3-319-68445-1_86

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  • Publisher Name: Springer, Cham

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