Provably scale-covariant networks from oriented quasi quadrature measures in cascade

TL;DR

A continuous model for hierarchical networks based on a combination of mathematically derived models of receptive fields and biologically inspired computations based on an oriented quasi quadrature combination of first- and second-order directional Gaussian derivatives is presented.

Abstract

This article presents a continuous model for hierarchical networks based on a combination of mathematically derived models of receptive fields and biologically inspired computations. Based on a functional model of complex cells in terms of an oriented quasi quadrature combination of first- and second-order directional Gaussian derivatives, we couple such primitive computations in cascade over combinatorial expansions over image orientations. Scale-space properties of the computational primitives are analysed and it is shown that the resulting representation allows for provable scale and rotation covariance. A prototype application to texture analysis is developed and it is demonstrated that a simplified mean-reduced representation of the resulting QuasiQuadNet leads to promising experimental results on three texture datasets.

Figures (3)

• Figure 1: 1-D Gaussian derivatives up to orders 0, 1 and 2 for $s_0 = 1$ with the corresponding 1-D quasi quadrature measures computed from them at scale $s = 1$ for $C = 8/11$. (Horizontal axis: $x \in [-5, 5]$.)
• Figure 2: Example of a colour-opponent receptive field profile for a double-opponent simple cell in the primary visual cortex (V1) as measured by Johnson et al.JohHawSha08-JNeuroSci. (left) Responses to L-cones corresponding to long wavelength red cones, with positive weights represented by red and negative weights by blue. (middle) Responses to M-cones corresponding to medium wavelength green cones, with positive weights represented by red and negative weights by blue. (right) Idealized model of the receptive field from a first-order directional derivative of an affine Gaussian kernel $\partial_{\varphi}g(x, y;\; \Sigma)$ according to (\ref{['eq-spat-RF-model']}) for $\sigma_1 = \sqrt{\lambda_1} = 0.6$, $\sigma_2 = \sqrt{\lambda_2} = 0.2$ in units of degrees of visual angle, $\alpha = 157~\hbox{degrees}$ and with positive weights for the red-green colour-opponent channel $U = R-G$ with positive values represented by red and negative values by green.
• Figure 3: Significant eigenvectors of a complex cell in the cat primary visual cortex, as determined by Touryan et al.TouFelDan05-Neuron from the response properties of the cell to a set of natural image stimuli, using a spike-triggered covariance method (STC) that computes the eigenvalues and the eigenvectors of a second-order Wiener kernel using three different parameter settings (cutoff frequencies) in the system identification method (from left to right). Qualitatively, these kernel shapes agree well with the shapes of first- and second-order affine Gaussian derivatives.