# Time-causal and time-recursive spatio-temporal receptive fields

arXiv: https://arxiv.org/abs/1504.02648 · HTML: https://mlsanity.com/paper/1504.02648/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Apr 2015.

## Built on by (most cited first, 7 of 7)

- [Covariance properties under natural image transformations for the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2303.09803/) (2023)
- [DeepSITH: Efficient Learning via Decomposition of What and When Across Time Scales](https://mlsanity.com/paper/2104.04646/) (2021)
- [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)
- [Covariant spatio-temporal receptive fields for spiking neural networks](https://mlsanity.com/paper/2405.00318/) (2024)
- [A precortical module for robust CNNs to light variations](https://mlsanity.com/paper/2202.07432/) (2022)
- [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-covariant spiking wavelets](https://mlsanity.com/paper/2602.02020/) (2026)

## Abstract (arXiv)

We present an improved model and theory for time-causal and time-recursive spatio-temporal receptive fields, based on a combination of Gaussian receptive fields over the spatial domain and first-order integrators or equivalently truncated exponential filters coupled in cascade over the temporal domain. Compared to previous spatio-temporal scale-space formulations in terms of non-enhancement of local extrema or scale invariance, these receptive fields are based on different scale-space axiomatics over time by ensuring non-creation of new local extrema or zero-crossings with increasing temporal scale. Specifically, extensions are presented about (i) parameterizing the intermediate temporal scale levels, (ii) analysing the resulting temporal dynamics, (iii) transferring the theory to a discrete implementation, (iv) computing scale-normalized spatio-temporal derivative expressions for spatio-temporal feature detection and (v) computational modelling of receptive fields in the lateral geniculate nucleus (LGN) and the primary visual cortex (V1) in biological vision. We show that by distributing the intermediate temporal scale levels according to a logarithmic distribution, we obtain much faster temporal response properties (shorter temporal delays) compared to a uniform distribution. Specifically, these kernels converge very rapidly to a limit kernel possessing true self-similar scale-invariant properties over temporal scales, thereby allowing for true scale invariance over variations in the temporal scale, although the underlying temporal scale-space representation is based on a discretized temporal scale parameter. We show how scale-normalized temporal derivatives can be defined for these time-causal scale-space kernels and how the composed theory can be used for computing basic types of scale-normalized spatio-temporal derivative expressions in a computationally efficient manner.