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

arXiv: https://arxiv.org/abs/1905.13555 · HTML: https://mlsanity.com/paper/1905.13555/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 May 2019.

## Built on by (most cited first, 9 of 9)

- [Scale-invariant scale-channel networks: Deep networks that generalise to previously unseen scales](https://mlsanity.com/paper/2106.06418/) (2021)
- [Covariance properties under natural image transformations for the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2303.09803/) (2023)
- [Riesz networks: scale invariant neural networks in a single forward pass](https://mlsanity.com/paper/2305.04665/) (2023)
- [Unified theory for joint covariance properties under geometric image transformations for spatio-temporal receptive fields according to the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2311.10543/) (2023)
- [Riesz feature representation: scale equivariant scattering network for classification tasks](https://mlsanity.com/paper/2307.08467/) (2023)
- [Approximation properties relative to continuous scale space for hybrid discretizations of Gaussian derivative operators](https://mlsanity.com/paper/2405.05095/) (2024)
- [Scale generalisation properties of extended scale-covariant and scale-invariant Gaussian derivative networks on image datasets with spatial scaling variations](https://mlsanity.com/paper/2409.11140/) (2024)
- [Hybrid Lie semi-group and cascade structures for the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2509.15748/) (2025)
- [Scale-invariant Gaussian derivative residual networks](https://mlsanity.com/paper/2603.02843/) (2026)

## Abstract (arXiv)

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.