Topological Data Analysis for True Step Detection in Piecewise Constant Signals

Abstract: This paper introduces a simple yet powerful approach based on topological data analysis (TDA) for detecting the true steps in a piecewise constant (PWC) signal. The signal is a two-state square wave with randomly varying in-between-pulse spacing, and subject to spurious steps at the rising or falling edges which we refer to as digital ringing. We use persistent homology to derive mathematical guarantees for the resulting change detection which enables accurate identification and counting of the true pulses. The approach is described and tested using both synthetic and experimental data obtained using an engine lathe instrumented with a laser tachometer. The described algorithm enables the accurate calculation of the spindle speed with the appropriate error bounds. The results of the described approach are compared to the frequency domain approach via Fourier transform. It is found that both our approach and the Fourier analysis yield comparable results for numerical and experimental pulses with regular spacing and digital ringing. However, the described approach significantly outperforms Fourier analysis when the spacing between the peaks is varied. We also generalize the approach to higher dimensional PWC signals, although utilizing this extension remains an interesting question for future research.