Abstract
We build on the recently introduced PDE-G-CNN framework, which proposed the concept of non-linear morphological convolutions that are motivated by solving HJB-PDEs on lifted homogeneous spaces such as the homogeneous space of 2D positions and orientations isomorphic to \(G=SE(2)\). PDE-G-CNNs generalize G-CNNs and are provably equivariant to actions of the roto-translation group SE(2). Moreover, PDE-G-CNNs automate geometric image processing via orientation scores and allow for a meaningful geometric interpretation.
In this article, we show various functional properties of these networks:
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(1.)
PDE-G-CNNs satisfy crucial geometric and algebraic symmetries: they are semiring quasilinear, equivariant, invariant under time scaling, isometric, and are solved by semiring group convolutions.
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(2.)
PDE-G-CNNs exhibit a high degree of data efficiency: even under limited availability of training data they show a distinct gain in performance and generalize to unseen test cases from different datasets.
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(3.)
PDE-G-CNNs are extendable to well-known convolutional architectures. We explore a UNet variant of PDE-G-CNNs which has a new equivariant U-Net structure with PDE-based morphological convolutions.
We verify the properties and show favorable results on various datasets.
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Notes
- 1.
Convections and dilations/diffusion do not commute on the non-commutative G, and the desirable order is first convection and then dilation/diffusion [4, App. B, Prop. 3].
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Acknowledgement
We gratefully acknowledge the Dutch Foundation of Science NWO for funding of VICI 2020 Exact Sciences (Duits, Geometric learning for Image Analysis VI.C. 202-031). The git repository containing the vanilla PDE-G-CNN implementations can be found at: https://gitlab.com/bsmetsjr/lietorch.
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Pai, G., Bellaard, G., Smets, B.M.N., Duits, R. (2023). Functional Properties of PDE-Based Group Equivariant Convolutional Neural Networks. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14071. Springer, Cham. https://doi.org/10.1007/978-3-031-38271-0_7
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