  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.
