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

Research output: Contribution to conferencePaperAcademic

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 Md 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 α -scale-space PDEs, and to linear convolutions solving linear α -scale-space PDEs. In the morphological setting, the parameter α regulates soft max-pooling over balls, whereas in the linear setting the cases α=1/2 and α=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 α -stable Lévy processes to Bellman processes. We generalize this to Md 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.
Original languageEnglish
Pages27-39
DOIs
Publication statusPublished - 30 Apr 2021
Event8th International Conference on Scale Space and Variational Methods in Computer Vision -
Duration: 16 May 202020 May 2021
https://link.springer.com/book/10.1007/978-3-030-75549-2

Conference

Conference8th International Conference on Scale Space and Variational Methods in Computer Vision
Abbreviated titleSSVM 2021
Period16/05/2020/05/21
Internet address

Keywords

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

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