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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)


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.


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

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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.

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