  We start with a simple introduction to topological data analysis where the
most popular tool is called a persistent diagram. Briefly, a persistent diagram
is a multiset of points in the plane describing the persistence of topological
features of a compact set when a scale parameter varies. Since statistical
methods are difficult to apply directly on persistence diagrams, various
alternative functional summary statistics have been suggested, but either they
do not contain the full information of the persistence diagram or they are
two-dimensional functions. We suggest a new functional summary statistic that
is one-dimensional and hence easier to handle, and which under mild conditions
contains the full information of the persistence diagram. Its usefulness is
illustrated in statistical settings concerned with point clouds and brain
artery trees. The appendix includes additional methods and examples, together
with technical details. The R-code used for all examples is available at
http://people.math.aau.dk/~christophe/Rcode.zip.
