Robust Zero-crossings Detection in Noisy Signals using Topological Signal Processing

TL;DR

This work introduces a robust zero-crossings detection method for noisy 1D signals using 0-dimensional persistent homology. It constructs sign-based point clouds, computes persistence diagrams, and thresholds high-persistence gaps with a parameter $\mu$ to produce crossing intervals, with root estimates obtained by averaging interval bounds. Automatic $\mu$ selection is provided via statistical outlier methods and Isolation Forest. Compared to a state-of-the-art Lipschitz-based zero finder, the 0D persistence approach generally converges faster, yields lower error, and recovers all zeros within an interval, demonstrating robustness across a range of sampling frequencies and noise levels. The technique has practical implications for engineering and signal-processing tasks requiring reliable multi-root detection in time series.

Abstract

We explore a novel application of zero-dimensional persistent homology from Topological Data Analysis (TDA) for bracketing zero-crossings of both one-dimensional continuous functions, and uniformly sampled time series. We present an algorithm and show its robustness in the presence of noise for a range of sampling frequencies. In comparison to state-of-the-art software-based methods for finding zeros of a time series, our method generally converges faster, provides higher accuracy, and is capable of finding all the roots in a given interval instead of converging only to one of them. We also present and compare options for automatically setting the persistence threshold parameter that influences the accurate bracketing of the roots.

Figures (31)

• Figure 1: An example point cloud in $\mathbb{R}$ is given at the top of the figure. The persistence diagram points $(a_i-a_{i-1})$ are given at the bottom left, drawn at the location of the $a_{i-1}$ value. Note that high value points in this diagram occur at splits in the point cloud. Then at right we show the histogram of (death-time) points in the persistence diagram.
• Figure 2: A pulse signal showing the $p$ and $q$ notation. The $P$ points are those on the top row; the $Q$ on the bottom. For a threshold $\mu = 3\Delta t$, we see entries $(p_{i-1},p_i)$ from $\operatorname{dgm} P$ and $(q_{j-1},q_j)$ from $\operatorname{dgm} Q$.
• Figure 3: An example using $x(t)=\sin (t)+\sin (10 / 3 t)$. The intervals returned by the algorithm are $[r_1,r_2]$ and $[r_5,r_6]$ (without a uncertainty flag); and $[r_3,r_4]$ and $[r_7,r_8]$ (with an uncertainty flag). On the right, the persistence diagram is shown. The threshold was chosen to be $\mu = 0.4$, but different choices of $\mu$ will result in different zero-crossing intervals.
• Figure 4: Sorted persistence points for $x_{9}$ (using SNR $=15$ dB) with statistical measures for outlier detection plotted.
• Figure 5: Application of Isolation Forest on sorted persistence points of $x_{4}$.

Theorems & Definitions (2)

• Proposition 3.1
• proof