The mechanics of CNN filtering with rectification

Abstract

This thesis is the first investigation of the mechanical properties of convolutional filtering with rectification through the lens of symmetric and antisymmetric convolutional kernels. Symmetric kernels cause image content to diffuse isotropically with no net displacement, whereas antisymmetric kernels cause directional displacement and attenuation effects when sequentially convolved at varying orientations. The speed of information displacement is linearly related to the ratio of antisymmetric vs total kernel energy. Symmetry properties are analyzed in the spectral domain via the discrete cosine transform (DCT), where the structure of small convolutional filters (e.g. 3 × 3 pixels) is dominated by low-frequency bases, specifically the DC Σ and gradient components ∇, which define the fundamental modes of information propagation. Applying this analysis to trained CNN filters, we find that low-order frequency components (specifically DC and gradient bases) dominate, accounting for over 92% of classification performance in popular models such as VGG16 and ResNet50. The symmetry of kernels evolves with depth: early layers are predominantly antisymmetric, emphasizing oriented gradients, whereas deeper layers become increasingly symmetric, promoting diffusion. Furthermore, we observe that filters organize into correlated bipolar orientation structures across channels and layers, maintaining directional alignment between consecutive kernels while suppressing orthogonal activations. This work provides a systematic framework for analyzing, interpreting, and potentially guiding the design of improved architectures and learning algorithms through their geometric and spectral structure.

Date	29 Dec 2025
Original language	American English
Awarding Institution	École de technologie supérieure
Supervisor	Matthew Toews (Supervisor)