Orientation Scores should be a Piece of Cake

2025-04-01

AI Summary

Conventional geometric deep learning struggles with efficient and interpretable extraction of orientation-sensitive features due to fundamental limitations in directional representation. Method: We propose a position-orientation jointly optimal wavelet construction grounded in the uncertainty principle, introducing a novel PDE-G-CNN paradigm that eliminates trainable lifting layers. Our approach leverages orientation score theory and SE(2) group-theoretic analysis to enable explicit, wavelet-driven feature encoding. Contribution/Results: Theoretically, we establish the first rigorous proof that cake wavelets asymptotically achieve the Heisenberg-type position-orientation uncertainty bound in the SE(2) fractional Fourier domain (uncertainty gap <1.1, approaching the theoretical limit of 1). Empirically, our method preserves near-lossless accuracy while substantially reducing model parameters and computational cost. Moreover, it enhances architectural transparency and feature interpretability—enabling direct correspondence between learned filters and analytically derived wavelets.

Technology Category

• Computer Vision: Other Foundations of Computer Vision
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We axiomatically derive a family of wavelets for an orientation score, lifting from position space $mathbb{R}^2$ to position and orientation space $mathbb{R}^2 imes S^1$, 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.

Problem

• Derive wavelets for orientation scores with fast reconstruction
• Prove cake wavelets minimize position-orientation uncertainty
• Use cake wavelets to simplify and interpret (PDE-)G-CNNs

Innovation

• Wavelets lift to position-orientation space
• Cake wavelets minimize uncertainty gap
• Untrained lifting layer reduces network complexity