The following packages can be used to compute the persistent homology from a point cloud:
Our Field Filtration and Radial Filtration methods are available here:
- FieldFiltrations (uses CGAL and GUDHI)
In addition, you can use our fork of the HERA code to calculate the Frechet average and variances of persistence diagrams:
- HERA (use geom_matching/wasserstein/frechet_average after building)
In topology, homology provides us with a description of an object in terms of its holes. In three dimensions, we distinguish holes in dimensions 0, 1, and 2. A 0-dimensional hole is a gap separating different components. A 1-dimensional hole is a tunnel that can be encircled by a loop. A 2-dimensional hole is a void enclosed by a shell.
By computing the homology of a filtration (a sequence of growing/shrinking objects), we can study how the topology changes as we vary some filtration parameter α. For example, one can study how the superlevel set of a field evolves as one lowers the threshold from +∞ to –∞. In our paper (Elbers & van de Weygaert, 2018), we consider growing bubble networks modelled as balls in R3 with increasing radius.
In both examples above, we see the death of multiple gaps as components merge. In the top example, we also see the birth and death of a tunnel. In the bottom example, we see the birth of two tunnels in the final panel.
Recall that α is the filtration parameter. We assign every hole a birth scale αbirth and death scale αdeath at which the hole appears and disappears, respectively. The difference αdeath – αbirth is the persistence of the hole. We can plot the (birth, death)-pairs in a persistence diagram. The horizontal/vertical distance from the diagonal is the persistence.