Topological phase estimation method for reparameterized periodic functions

We consider a signal composed of several periods of a periodic function, of which we observe a noisy reparametrization. The phase estimation problem consists of finding that reparametrization 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 reparametrization. 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.