Time-causal and time-recursive wavelets

Abstract

This paper presents a framework for time-causal wavelet analysis. It targets real-time processing of temporal signals, where data from the future are not available. The study builds upon temporal scale-space theory, originating from a complete classification of temporal smoothing kernels that guarantee non-creation of new structures from finer to coarser temporal scale levels. We construct temporal wavelets from the temporal derivatives of a special time-causal smoothing kernel, referred to as the time-causal limit kernel, as arising from the classification of variation-diminishing smoothing transformations with the complementary requirement of temporal scale covariance, to guarantee self-similar handling of structures in the input signal at different temporal scales. This enables decomposition of the signal into different components at different scales, while adhering to temporal causality. The paper establishes theoretical foundations for these time-causal wavelet representations, and maps structural relationships to the non-causal Ricker or Mexican hat wavelets. We also describe how efficient discrete approximations of the presented theory can be performed in terms of first-order recursive filters coupled in cascade, which enables numerically well-conditioned real-time processing with low resource usage. We characterize and quantify how the continuous scaling properties transfer to the discrete implementation, demonstrating how the proposed time-causal wavelet representation can reflect the duration of locally dominant temporal structures in the input signal.