# Temporal scale selection in time-causal scale space

## TL;DR

This paper develops a theoretical framework for temporal scale selection in time-causal scale space, enabling real-time analysis of signals like video or audio streams with properties similar to non-causal Gaussian scale space.

## Contribution

It introduces a novel theoretical approach for temporal scale selection in time-causal, recursive scale space, achieving true temporal scale invariance with discrete scale levels.

## Key findings

- Achieves scale invariance in time-causal scale space
- Transfers properties from Gaussian to time-causal scale space
- Enables real-time signal analysis with characteristic scales

## Abstract

When designing and developing scale selection mechanisms for generating hypotheses about characteristic scales in signals, it is essential that the selected scale levels reflect the extent of the underlying structures in the signal. This paper presents a theory and in-depth theoretical analysis about the scale selection properties of methods for automatically selecting local temporal scales in time-dependent signals based on local extrema over temporal scales of scale-normalized temporal derivative responses. Specifically, this paper develops a novel theoretical framework for performing such temporal scale selection over a time-causal and time-recursive temporal domain as is necessary when processing continuous video or audio streams in real time or when modelling biological perception. For a recently developed time-causal and time-recursive scale-space concept defined by convolution with a scale-invariant limit kernel, we show that it is possible to transfer a large number of the desirable scale selection properties that hold for the Gaussian scale-space concept over a non-causal temporal domain to this temporal scale-space concept over a truly time-causal domain. Specifically, we show that for this temporal scale-space concept, it is possible to achieve true temporal scale invariance although the temporal scale levels have to be discrete, which is a novel theoretical construction.