# A time-causal and time-recursive scale-covariant scale-space representation of temporal signals and past time

**Authors:** Tony Lindeberg  
**arXiv:** [2202.09209](https://arxiv.org/abs/2202.09209)  
**Date:** 2022-02-18  
**Summary page:** https://www.summarizepaper.com/en/paper/2202.09209v4/

## Key points

- Tony Lindeberg presents a theory for performing temporal smoothing on temporal signals
- The theory guarantees that temporally smoothed signals at coarser temporal scales constitute simplifications of corresponding temporally smoothed signals at any finer temporal scale, including the original signal
- The temporal smoothing process is both time-causal and time-recursive, meaning it does not require access to future information and can be performed with no other temporal memory buffer of the past than the resulting smoothed temporal scale-space representations themselves
- Specific subsets of parameter settings for linear and shift-invariant temporal smoothing operators can obtain temporal scale covariance
- The resulting time-causal limit kernel that obeys this property constitutes a canonical temporal kernel for processing real-time scenarios when the regular Gaussian kernel cannot be used because of its non-causal access to information from the future
- This theory has general applicability for both modeling continuous phenomena over multiple scales and digital processing measured signals in real-time scenarios
- Potential applications for this theory include fields such as neuroscience, computer vision, and robotics.

## Summary

In this article, Tony Lindeberg presents a theory for performing temporal smoothing on temporal signals in a way that guarantees temporally smoothed signals at coarser temporal scales constitute simplifications of corresponding temporally smoothed signals at any finer temporal scale, including the original signal. The temporal smoothing process is both time-causal and time-recursive, meaning it does not require access to future information and can be performed with no other temporal memory buffer of the past than the resulting smoothed temporal scale-space representations themselves. Lindeberg shows how specific subsets of parameter settings for linear and shift-invariant temporal smoothing operators can additionally obtain temporal scale covariance. This guarantee means that if the input signal is rescaled by a uniform scaling factor, then the resulting temporal scale-space representations of the rescaled signal will constitute mere rescalings of the original input signal's representations, complemented by a shift along the temporal scale dimension. The resulting time-causal limit kernel that obeys this property constitutes a canonical temporal kernel for processing real-time scenarios when the regular Gaussian kernel cannot be used because of its non-causal access to information from the future. Moreover, we cannot require additional memory beyond what is contained in the smoothing process itself which also serves as a multi-scale temporal memory of past events. This theory has general applicability for both modeling continuous phenomena over multiple scales and digital processing measured signals in real-time scenarios. The article concludes with an overview of potential applications for this theory in fields such as neuroscience, computer vision and robotics. Overall, this work provides valuable insights into how to perform effective and efficient smoothing on complex time series data while maintaining important properties such as causality and scalability across different resolutions.