  In many applications concerning the comparison of data expressed by
$\mathbb{R}^m$-valued functions defined on a topological space $X$, the
invariance with respect to a given group $G$ of self-homeomorphisms of $X$ is
required. While persistent homology is quite efficient in the topological and
qualitative comparison of this kind of data when the invariance group $G$ is
the group $\mathrm{Homeo}(X)$ of all self-homeomorphisms of $X$, this theory is
not tailored to manage the case in which $G$ is a proper subgroup of
$\mathrm{Homeo}(X)$, and its invariance appears too general for several tasks.
This paper proposes a way to adapt persistent homology in order to get
invariance just with respect to a given group of self-homeomorphisms of $X$.
The main idea consists in a dual approach, based on considering the set of all
$G$-invariant non-expanding operators defined on the space of the admissible
filtering functions on $X$. Some theoretical results concerning this approach
are proven and two experiments are presented. An experiment illustrates the
application of the proposed technique to compare 1D-signals, when the
invariance is expressed by the group of affinities, the group of
orientation-preserving affinities, the group of isometries, the group of
translations and the identity group. Another experiment shows how our technique
can be used for image comparison.
