  Persistent homology is a tool from Topological Data Analysis (TDA) used to
summarize the topology underlying data. It can be conveniently represented
through persistence diagrams. Observing a noisy signal, common strategies to
infer its persistence diagram involve plug-in estimators, and convergence
properties are then derived from sup-norm stability. This dependence on the
sup-norm convergence of the preliminary estimator is restrictive, as it
essentially imposes to consider regular classes of signals. Departing from
these approaches, we design an estimator based on image persistence. In the
context of the Gaussian white noise model, and for large classes of
piecewise-H\"older signals, we prove that the proposed estimator is consistent
and achieves minimax rates. Notably, these rates coincide with the well known
minimax rates for H\"older continuous signals.
