In this thesis, we present an algorithm that computes the one-dimensional persistent
homology in a geodesic metric. The use of a geodesic metric allows us to approximate
the shortest homology basis of the underlying space.
First, we present the theoretical background of one-dimensional persistent homology
of geodesic spaces. The main result of this section is the connection between
the critical points of persistent homology and geodesic loops in the space, which
form the shortest basis of the first homology group. Next, we present a new algorithm
based on the thoery. This algorithm approximates the shortest basis of the
first homology group. Finally, we present some results that were computed using
our implementation of the algorithm in the Julia programming language. We test
the algorithm on a few synthetic data set and one, larger, more realistic data set.