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

## TL;DR

This paper develops a theoretical framework for constructing hierarchical networks that are provably scale covariant using scale-normalized differential expressions, with a practical application to texture analysis demonstrating promising results.

## Contribution

It introduces a general sufficiency condition for scale covariance in hierarchical networks based on differential expressions and develops a biologically inspired model called QuasiQuadNet.

## Key findings

- Proves scale and rotation covariance of the proposed network.
- Demonstrates promising texture analysis results on three datasets.
- Provides a theoretical foundation for scale-covariant hierarchical networks.

## 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.