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Erschienen in:
2021 | OriginalPaper | Buchkapitel
Equivariant Deep Learning via Morphological and Linear Scale Space PDEs on the Space of Positions and Orientations
verfasst von : Remco Duits, Bart Smets, Erik Bekkers, Jim Portegies
Erschienen in: Scale Space and Variational Methods in Computer Vision
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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.