Time-causal and time-recursive wavelets

TL;DR

Time-causal wavelet analysis addresses real-time signal processing by enforcing causality and recursivity through the time-causal limit kernel, built as a cascade of truncated exponentials. It merges temporal scale-space with wavelet theory to produce time-causal, scale-covariant mother wavelets via derivatives of the limit kernel and enables exact bandpass representations and straightforward reconstruction. The paper provides continuous and discrete formulations, including differential equations, discretization via recursive filters, and normalization schemes, and demonstrates with experiments that the method captures structures across a rich span of temporal scales with reduced delay compared to non-causal methods. This framework offers a principled, computationally efficient toolkit for real-time monitoring, time-series analysis, and modeling biological/physical temporal phenomena in a fully time-causal setting.

Abstract

When to apply wavelet analysis to real-time temporal signals, where the future cannot be accessed, it is essential to base all the steps in the signal processing pipeline on computational mechanisms that are truly time-causal. This paper describes how a time-causal wavelet analysis can be performed based on concepts developed in the area of 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. By necessity, convolution with truncated exponential kernels in cascade constitutes the only permissable class of kernels, as well as their temporal derivatives as a natural complement to fulfil the admissibility conditions of wavelet representations. For a particular way of choosing the time constants in the resulting infinite convolution of truncated exponential kernels, to ensure temporal scale covariance and thus self-similarity over temporal scales, we describe how mother wavelets can be chosen as temporal derivatives of the resulting time-causal limit kernel. By developing connections between wavelet theory and scale-space theory, 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 signals. We propose that this notion of time-causal wavelet analysis could be a valuable 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 analysing physical or biophysical temporal phenomena, where a fully time-causal analysis is called for to be physically realistic.

Figures (11)

• Figure 1: The time-causal limit kernel $\Psi(t;\; \tau, c)$ according to (\ref{['eq-def-time-caus-lim-kern-inf-conv']}), truncated after the 8 truncated exponential kernels having the longest time constants, with its scale-normalized temporal derivatives $\Psi_{\zeta^n}(t;\; \tau, c) = \tau^{n \, \gamma/2} \, \Psi_{t^n}(t;\; \tau, c)$ up to order $n = 2$, with the scale normalization power $\gamma = 1$ corresponding to $L_1$-normalization across scales, for different combinations of the temporal scales $\sigma = \sqrt{\tau} \in \{ 1, 2 \}$ and different values to the distribution parameter $c \in \{ \sqrt{2}, 2 \}$. (Horizontal axes: time $t \in [0, 12]$. Vertical axes: kernel values with different ranges $[0, 0.50]$, $[-1.0, 1.0]$ or $[-4.0, 4.0]$, depending on the order $n$ of temporal differentiation.)
• Figure 2: The Gaussian kernel $g(t;\; \tau)$ according to (\ref{['eq-1D-gauss-kernel']}) with its scale-normalized temporal derivatives $g_{\zeta^n}(t;\; \tau) = \tau^{n \, \gamma/2} \, g_{t^n}(t;\; \tau)$ up to order $n = 2$, with the scale normalization power $\gamma = 1$ corresponding to $L_1$-normalization across scales, for different temporal scales $\sigma = \sqrt{\tau} \in \{ 1, 2 \}$. (Horizontal axes: time $t \in [-10, 10]$. Vertical axes: kernel values with different ranges, depending on the order $n$ of temporal differentiation.)
• Figure 3: Equivalent discrete approximations of the time-causal limit kernel $\Psi(t;\; \tau, c)$ as obtained from a set of first-order recursive filters coupled in cascade, with the time constants determined according to (\ref{['eq-disc-time-constant']}), and with the infinite cascade truncated after the 8 recursive filters having the longest time constants, together with discrete approximations of its scale-normalized temporal derivatives $\Psi_{\zeta^n}(t;\; \tau, c) = \tau^{n \, \gamma/2} \, \Psi_{t^n}(t;\; \tau, c)$ up to order $n = 2$, obtained by applying first- and second-order temporal difference operators $\delta_t$ and $\delta_{tt}$ according to (\ref{['eq-temp-der-approx-molecules']}) to the discrete approximations of time-causal limit kernel, with the scale normalization power $\gamma = 1$ corresponding to $L_1$-normalization across scales, for different combinations of the temporal scales $\sigma = \sqrt{\tau} \in \{ 1, 2, 4 \}$ and different values to the distribution parameter $c \in \{ \sqrt{2}, 2 \}$. (Horizontal axes: time $t \in [0, 25]$. Vertical axes: kernel values with different ranges $[0, 0.50]$, $[-0.40, 0.40]$ or $[-1.0, 1.0]$, depending on the order $n$ of temporal differentiation.)
• Figure 4: Graphs over the dependency on the scale parameter $\sigma = \sqrt{\tau}$ for the discrete $l_1$-norms according to (\ref{['eq-lpnorms-limitkernders']}) of the equivalent discrete kernels illustrated in Figure \ref{['fig-disc-limitkern-graphs']}, that correspond to the discrete approximations of the derivatives of the time-causal limit kernel, as obtained with the discretization methodology described in Section \ref{['sec-disc-approx']} for the choice of the scale normalization power $\gamma = 1$. For reference, the graphs also show the evolution properties over scales of the $L_1$-norms of the temporal derivatives of the continuous time-causal limit kernel, according to (\ref{['l1-norm-psi-t-c2']}), (\ref{['l1-norm-psi-tt-c2']}), (\ref{['l1-norm-psi-t-csqrt2']}) and (\ref{['l1-norm-psi-tt-csqrt2']}), which are constant over scales for $\gamma = 1$. As can be seen from these graphs, there are some transient phenomena at finer levels of scales, where the influence of the discretization effects is strongest. Towards coarser levels of scales, however, the discrete $l_p$-norms asymptotically approach constant values. When the distribution parameter $c = \sqrt{2}$, there are very good matches between the asymptotic values and the corresponding expressions for the fully continuous theory. When the distribution parameter $c =2$, there are, however, certain deviations. (Horizontal axes: temporal scale parameter in units of $\sigma = \sqrt{\tau}$. Vertical values: magnitudes of the discrete $l_p$-norms, with red curves showing the results for the distribution parameter $c = 2$ and blue curves showing the results for the distribution parameter $c = \sqrt{2}$.)
• Figure 5: Illustration of the conceptual stages involved in characterizing the scaling properties of the scale-normalized temporal derivatives of the time-causal limit kernel. The left column show two discrete blob-like model signals based on the discrete analogue of the Gaussian kernel according to (\ref{['eq-blob-model']}) for two different reference scales $\sigma_{\text{ref}} = \sqrt{\tau_{\text{ref}}} \in \{1, 4 \}$, together with the scale-normalized derivatives of order 2 for the scale normalization power $\gamma = 3/4$ over the scale range $\sigma = \sqrt{\tau} \in [1/8, 64]$. The right column shows two edge-like model signals according to (\ref{['eq-edge-model']}) together with the scale-normalized derivatives of order 1 for $\gamma = 1/2$ over the same scale ranges. All results have been computed with the distribution parameter $c = \sqrt{2}$ for the discrete analogue of the time-causal limit kernel. (The reason why the slopes of the red and blue stripes are not vertical, but oblique, is because of the different amounts of temporal delay at different temporal scales.) (Horizontal axes: time $t \in [-17, 17]$. Vertical axes in the temporal scale-space representations: Effective scale = $\log_2 \sqrt{\tau}$.) (In the time-causal wavelet representations, red denotes positive values and blue denotes negative values.)