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

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.