Time-causal and time-recursive wavelets

This study 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”—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 the decomposition of the signal into different components at different scales while adhering to temporal causality. The study establishes theoretical foundations for these time-causal wavelet representations and maps structural relationships to the non-causal Ricker/Mexican hat wavelets. We also describe how efficient discrete approximations of the theory presented can be performed in terms of first-order recursive filters coupled in cascade, enabling 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. We propose this notion of time-causal wavelet analysis as a generic multi-purpose tool for signal processing tasks, where streams of signals are to be processed in real time—specifically for signals that may contain local variations over a rich span of temporal scales or more generally for analyzing physical or biophysical temporal phenomena, where a fully time-causal analysis needs to be physically realistic, by not accessing data from the future, as is often done for prerecorded data.