Provably scale-covariant continuous hierarchical networks based on scale-normalized differential expressions coupled in cascade

TL;DR

A theory for constructing hierarchical networks in such a way that the networks are guaranteed to be provably scale covariant and it is demonstrated that a simplified mean-reduced representation of the resulting QuasiQuadNet leads to promising experimental results on three texture datasets.

Abstract

This article presents a theory for constructing hierarchical networks in such a way that the networks are guaranteed to be provably scale covariant. We first present a general sufficiency argument for obtaining scale covariance, which holds for a wide class of networks defined from linear and non-linear differential expressions expressed in terms of scale-normalized scale-space derivatives. Then, we present a more detailed development of one example of such a network constructed from 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 we give explicit proofs of how the resulting representation allows for 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 (16)

• Figure 1: Commutative diagram for a scale-covariant hierarchical network constructed according to the presented sufficiency result. Provided that the individual differential operators ${\@fontswitch\mathcal{D}}_{k,s_k}$ between adjacent layers are scale covariant, which for example holds for the class of homogeneous differential expressions of the form (\ref{['eq-sc-cov-hom-diff-expr-general-gamma']}) as well as self-similar compositions of such operations that additionally satisfy corresponding homogeneity requirements, it follows that it will be possible to perfectly match the corresponding layers $F_k$ and $F_k'$ under a scaling transformations of the underlying image domain $f'(x') = f(x)$ for $x' = Sx$, provided that the scale parameter $s_k$ in layer $k$ is proportional to the scale parameter $s_1$ in the first layer, $s_k = r_k^2 \, s_1$, for some scalar constants $r_k$. For such a network constructed from scale-space operations based on the Gaussian scale-space theory framework, the scale parameters in the two domains should be related according to $s_k' = S^2 s_k$.
• Figure 2: A hierarchical network defined by coupling scale-covariant differential expressions formulated within the continuous scale-space framework will be guaranteed to be provably scale covariant provided that the scale parameters in higher layers $s_k$ for $k \geq 2$ are proportional to the scale parameter $s_1$ in the first layer. If the scale normalization parameter $\gamma$ in the scale-normalized derivative expressions is equal to one, then general differential expressions in terms of such derivatives can be used based on the transformation property (\ref{['eq-sc-transf-gammanorm-ders-gamma-eq-1']}). If the scale normalization parameter $\gamma$ is not equal to one, then one can take homogeneous polynomial differential expressions of the form (\ref{['eq-sc-cov-hom-diff-expr-general-gamma']}) as well as self-similar transformations of such expressions. (In this schematic illustration, the arguments of the layers $F_1$, $F_2$, $F_{k-1}$ and $F_k$, which should be $F_ 1(\cdot;\; s_1)$, $F_ 2(\cdot;\; s_1, s_2)$, $F_{k-1}(\cdot;\; s_1, s_2, \dots, s_{k-1})$ and $F_k(\cdot;\; s_1, s_2, \dots, s_{k-1}, s_k)$, respectively, have been suppressed to simplify the notation. The argument of the input data $f$ should be $f(\cdot)$.)
• Figure 3: 1-D Gaussian derivatives up to orders 0, 1 and 2 for $s_0 = 1$ with the corresponding 1-D quasi quadrature measures (\ref{['eq-quasi-quad-1D']}) computed from them at scale $s = 1$ for $C = 8/11$. (Horizontal axis: $x \in [-5, 5]$.)
• Figure 4: 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 5: 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.