Topological phase estimation method for reparameterized periodic functions

TL;DR

This paper introduces a topological method using persistent homology to estimate the reparametrisation of noisy periodic signals, especially when the underlying function is unknown, with applications in vehicle positioning.

Contribution

It presents a novel shape-based estimation approach leveraging persistent homology to determine the number of periods and reparametrisation in unknown periodic signals.

Findings

Effective in estimating the number of periods from noisy signals

Successfully applied to vehicle positioning problem

Outperforms traditional methods in unknown function scenarios

Abstract

We consider a signal composed of several periods of a periodic function, of which we observe a noisy reparametrisation. The phase estimation problem consists of finding that reparametrisation, and, in particular, the number of observed periods. Existing methods are well-suited to the setting where the periodic function is known, or at least, simple. We consider the case when it is unknown and we propose an estimation method based on the shape of the signal. We use the persistent homology of sublevel sets of the signal to capture the temporal structure of its local extrema. We infer the number of periods in the signal by counting points in the persistence diagram and their multiplicities. Using the estimated number of periods, we construct an estimator of the reparametrisation. It is based on counting the number of sufficiently prominent local minima in the signal. This work is motivated by a vehicle positioning problem, on which we evaluated the proposed method.