Orientation Scores should be a Piece of Cake
This work addresses the challenge of efficient and interpretable wavelet design for orientation scores in geometric deep learning, offering an incremental improvement by replacing trainable layers with pre-defined cake wavelets.
The paper tackled the problem of designing wavelets for orientation scores to minimize position-orientation uncertainty, showing that cake wavelets achieve an uncertainty gap less than 1.1 and approach the minimum of 1 in the limit. It also demonstrated that using these wavelets in (PDE-)G-CNNs reduces network complexity and improves interpretability with only a slight performance impact.
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.