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The Hessian of Axially Symmetric Functions on SE(3) and Application in 3D Image Analysis

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Scale Space and Variational Methods in Computer Vision (SSVM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10302))

Abstract

We propose a method for computation of the Hessian of axially symmetric functions on the roto-translation group SE(3). Eigendecomposition of the resulting Hessian is then used for curvature estimation of tubular structures, similar to how the Hessian matrix of 2D or 3D image data can be used for orientation estimation. This paper focuses on a new implementation of a Gaussian regularized Hessian on the roto-translation group. Furthermore we show how eigenanalysis of this Hessian gives rise to exponential curve fits on data on position and orientation (e.g. orientation scores), whose spatial projections provide local fits in 3D data. We quantitatively validate our exponential curve fits by comparing the curvature of the spatially projected fitted curve to ground truth curvature of artificial 3D data. We also show first results on real MRA data. Implementations are available at: http://lieanalysis.nl/orientationscores.html.

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Notes

  1. 1.

    In general the Hessian depends on the imposed connection on the tangent bundle T(SE(3)), where \(\mathbf {H}= (\nabla _{\mathcal {A}_i}^* d \tilde{U}) (\mathcal {A}_j)\). Here we follow [6, App. 4] and choose \(\nabla \) as the left Cartan connection, since it is the correct connection for left-invariant processing.

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Acknowledgements

The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant Lie Analysis, agr. nr. 335555. Tom Dela Haije gratefully acknowledges the Netherlands Organization for Scientific Research (NWO, No. 617.001.202) for financial support.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands

    Michiel H. J. Janssen, Tom C. J. Dela Haije, Frank C. Martin, Erik J. Bekkers & Remco Duits

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  1. Michiel H. J. Janssen
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  2. Tom C. J. Dela Haije
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  3. Frank C. Martin
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  4. Erik J. Bekkers
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  5. Remco Duits
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Corresponding author

Correspondence to Michiel H. J. Janssen .

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Editors and Affiliations

  1. University of Copenhagen , Copenhagen, Denmark

    François Lauze

  2. Technical University of Denmark , Kongens Lyngby, Denmark

    Yiqiu Dong

  3. Technical University of Denmark , Kongens Lyngby, Denmark

    Anders Bjorholm Dahl

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Janssen, M.H.J., Dela Haije, T.C.J., Martin, F.C., Bekkers, E.J., Duits, R. (2017). The Hessian of Axially Symmetric Functions on SE(3) and Application in 3D Image Analysis. In: Lauze, F., Dong, Y., Dahl, A. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2017. Lecture Notes in Computer Science(), vol 10302. Springer, Cham. https://doi.org/10.1007/978-3-319-58771-4_51

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  • DOI: https://doi.org/10.1007/978-3-319-58771-4_51

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