Orientation Scores should be a Piece of Cake
Finn M. Sherry, Chase van de Geijn, Erik J. Bekkers, Remco Duits
TL;DR
The paper tackles interpretability and robustness gaps in convolutional architectures by fixing the lifting from $\mathbb{R}^2$ to oriented space $SE(2)$ via an orientation-score transform. It introduces four axioms—polar separability, fast reconstruction, directionality, and minimum $SE(2)$-uncertainty—to derive a family of lifting wavelets, showing they are well-approximated by cake wavelets. The angular profiles of cake wavelets achieve an uncertainty gap $UG$ approaching the theoretical minimum $1$ as angular resolution increases, and trained liftings can be represented as left-invariant convolutions of cake-wavelet liftings, enabling fixed lifting with comparable performance. Empirical results on retinal-vessel segmentation demonstrate substantial parameter reductions for PDE-G-CNNs with fixed lifting and only minimal performance loss, while improving interpretability and alignment with neurogeometry.
Abstract
We axiomatically derive a family of wavelets for an orientation score, lifting from position space $\mathbb{R}^2$ to position and orientation space $\mathbb{R}^2\times S^1$, with fast reconstruction property, that minimise position-orientation uncertainty. We subsequently show that these minimum uncertainty states are well-approximated by cake wavelets: for standard parameters, the uncertainty gap of cake wavelets is less than 1.1, and in the limit, we prove the uncertainty gap tends to the minimum of 1. Next, we complete a previous theoretical argument that one does not have to train the lifting layer in (PDE-)G-CNNs, but can instead use cake wavelets. Finally, we show experimentally that in this way we can reduce the network complexity and improve the interpretability of (PDE-)G-CNNs, with only a slight impact on the model's performance.