Skip to main content

Equivariant Deep Learning via Morphological and Linear Scale Space PDEs on the Space of Positions and Orientations

Part of the Lecture Notes in Computer Science book series (LNIP,volume 12679)

Abstract

We present PDE-based Group Convolutional Neural Networks (PDE-G-CNNs) that generalize Group equivariant Convolutional Neural Networks (G-CNNs). In PDE-G-CNNs a network layer is a set of PDE-solvers where geometrically meaningful PDE-coefficients become trainable weights. The underlying PDEs are morphological and linear scale space PDEs on the homogeneous space \(\mathbb {M}_d\) of positions and orientations. They provide an equivariant, geometrical PDE-design and model interpretability of the network.

The network is implemented by morphological convolutions with approximations to kernels solving morphological \(\alpha \)-scale-space PDEs, and to linear convolutions solving linear \(\alpha \)-scale-space PDEs. In the morphological setting, the parameter \(\alpha \) regulates soft max-pooling over balls, whereas in the linear setting the cases \(\alpha = 1/2\) and \(\alpha = 1\) correspond to Poisson and Gaussian scale spaces respectively.

We show that our analytic approximation kernels are accurate and practical. We build on techniques introduced by Weickert and Burgeth who revealed a key isomorphism between linear and morphological scale spaces via the Fourier-Cramér transform. It maps linear \(\alpha \)-stable Lévy processes to Bellman processes. We generalize this to \(\mathbb {M}_{d}\) and exploit this relation between linear and morphological scale-space kernels.

We present blood vessel segmentation experiments that show the benefits of PDE-G-CNNs compared to state-of-the-art G-CNNs: increase of performance along with a huge reduction in network parameters.

Keywords

  • Convolutional neural networks
  • Scale space theory
  • Cramér transform
  • Geometric deep learning
  • Morphological convolutions and PDEs

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-75549-2_3
  • Chapter length: 13 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   89.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-75549-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   119.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.

References

  1. Akian, M., Quadrat, J., Viot, M.: Bellman processes. LNCIS 199, 302–311 (1994)

    MATH  Google Scholar 

  2. Bardi, M., Capuzzo-Dolcetta, I.: Discontinuous viscosity solutions and applications. In: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Systems and Control: Foundations and Applications. Birkhäuser, Boston, MA (1997) https://doi.org/10.1007/978-0-8176-4755-1_5

  3. Bekkers, E.J., Lafarge, M.W., Veta, M., Eppenhof, K.A.J., Pluim, J.P.W., Duits, R.: Roto-translation covariant convolutional networks for medical image analysis. In: Frangi, A.F., Schnabel, J.A., Davatzikos, C., Alberola-López, C., Fichtinger, G. (eds.) MICCAI 2018. LNCS, vol. 11070, pp. 440–448. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-00928-1_50

    CrossRef  Google Scholar 

  4. Ben Arous, G.: Development asymptotique du noyau de la chaleur hypoelliptique sur la diagonale. In: Annales de l’institut Fourier, pp. 73–99 (1989)

    Google Scholar 

  5. Burgeth, B., Weickert, J.: An explanation for the logarithmic connection between linear and morphological systems. In: Proceedings 4th SSVM pp. 325–339 (2003)

    Google Scholar 

  6. Cohen, T., Welling, M.: Group equivariant convolutional networks. In: Proceedings of the 33rd International Conference on Machine Learning, pp. 2990–2999 (2016)

    Google Scholar 

  7. Duits, R., Bekkers, E.J., Mashtakov, A.: Fourier transform on \(\mathbb{M}_3\) for exact solutions to linear PDEs. Entropy (SI: 250 year of Fourier) 21(1), 1–38 (2019)

    Google Scholar 

  8. Duits, R., Franken, E.M.: Left invariant parabolic evolution equations on \({SE}(2)\) and contour enhancement via orientation scores. QAM-AMS 68, 255–331 (2010)

    MATH  Google Scholar 

  9. Duits, R., Meesters, S., Mirebeau, J.M., Portegies, J.M.: Optimal paths of the reeds-shepp car with applications in image analysis. JMIV 60(6), 816–848 (2018)

    CrossRef  Google Scholar 

  10. Elad, M.: Deep, Deep Trouble. SIAM-NEWS p. 12 (2017)

    Google Scholar 

  11. ter Elst, A.F.M., Robinson, D.W.: Weighted subcoercive operators on Lie groups. J. Funct. Anal. 157, 88–163 (1998)

    MathSciNet  CrossRef  Google Scholar 

  12. Evans, L.C.: Partial differential equations. AMS, Providence, R.I. (2010)

    MATH  Google Scholar 

  13. Finzi, M., Bondesan, R., Welling, M.: Probabilistic numeric convolutional neural networks. https://arxiv.org/pdf/2010.10876.pdf pp. 1–22 (2020)

  14. Garoni, T., Frankel, N.: Lévy flights: exact results and asymptotics beyond all orders. J. Math. Phys. 43(5), 2670–2689 (2002)

    MathSciNet  CrossRef  Google Scholar 

  15. Grigorian, A.: Heat Kernel and Analysis on Manifolds. Math. Dep, Bielefeld (2009)

    Google Scholar 

  16. LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nat. Res. 521(7553), 436–444 (2015)

    Google Scholar 

  17. Litjens, G., Bejnodri, B., van Ginneken, B., Sánchez, C.: A survey on deep learning in medical image analysis. Med. Image Anal. 42, 60–88 (2017)

    CrossRef  Google Scholar 

  18. Pauwels, E., van Gool, L., Fiddelaers, P., Moons, T.: An extended class of scale-invariant and recursive scale space filters. IEEE Trans. Pattern Anal Mach. Intell. 17(7), 691–701 (1995)

    CrossRef  Google Scholar 

  19. Schmidt, M., Weickert, J.: Morphological counterparts of linear shift-invariant scale-spaces. J. Math. Imag. Vision 56(2), 352–366 (2016)

    MathSciNet  CrossRef  Google Scholar 

  20. Siffre, L.: Rigid-motion scattering for image classification. Ph.D. thesis, Ecole Polyechnique, Paris (2014)

    Google Scholar 

  21. Smets, B., Portegies, J., Bekkers, E., Duits, R.: PDE-based group equivariant convolutional neural networks. arXiv preprint arXiv:2001.09046 (2020)

  22. Staal, J., Abramoff, M., Niemeijer, M., Viergever, M., van Ginneken, B.: Ridge-based vessel segmentation in images of the retina. IEEE TMI 23(4), 501–509 (2004)

    Google Scholar 

  23. Weiler, M., Geiger, M., Welling, M., Boomsma, W., Cohen, T.: 3D steerable CNNs: learning equivariant features in volumetric data. In: NeurIPS, pp. 1–12 (2018)

    Google Scholar 

  24. Yosida, K.: Functional Analysis. CM, vol. 123. Springer, Heidelberg (1995). https://doi.org/10.1007/978-3-642-61859-8

    CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bart Smets .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Duits, R., Smets, B., Bekkers, E., Portegies, J. (2021). Equivariant Deep Learning via Morphological and Linear Scale Space PDEs on the Space of Positions and Orientations. In: Elmoataz, A., Fadili, J., Quéau, Y., Rabin, J., Simon, L. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2021. Lecture Notes in Computer Science(), vol 12679. Springer, Cham. https://doi.org/10.1007/978-3-030-75549-2_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-75549-2_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-75548-5

  • Online ISBN: 978-3-030-75549-2

  • eBook Packages: Computer ScienceComputer Science (R0)