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

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1 Citaat (Scopus)

Samenvatting

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.
Originele taal-2Engels
Pagina's27-39
Aantal pagina's13
DOI's
StatusGepubliceerd - 30 apr 2021
Evenement8th International Conference on Scale Space and Variational Methods in Computer Vision -
Duur: 16 mei 202020 mei 2021
https://link.springer.com/book/10.1007/978-3-030-75549-2

Congres

Congres8th International Conference on Scale Space and Variational Methods in Computer Vision
Verkorte titelSSVM 2021
Periode16/05/2020/05/21
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