  In this article, we study Euler characteristic techniques in topological data
analysis. Pointwise computing the Euler characteristic of a family of
simplicial complexes built from data gives rise to the so-called Euler
characteristic profile. We show that this simple descriptor achieve
state-of-the-art performance in supervised tasks at a very low computational
cost. Inspired by signal analysis, we compute hybrid transforms of Euler
characteristic profiles. These integral transforms mix Euler characteristic
techniques with Lebesgue integration to provide highly efficient compressors of
topological signals. As a consequence, they show remarkable performances in
unsupervised settings. On the qualitative side, we provide numerous heuristics
on the topological and geometric information captured by Euler profiles and
their hybrid transforms. Finally, we prove stability results for these
descriptors as well as asymptotic guarantees in random settings.
