Crate alphalpha

This crate implements the alpha and weak-alpha filtration of a set of points in the plane. Both are filtrations of the Delaunay triangulation. With the optional lophat feature, the crate also provides a function for computing the boundary matrix of the filtration.

• The alpha filtration is constructed similarly to the Čech filtration. Grow balls of radius r around each point and intersect each ball with the corresponding Voronoi cell. This nerve of this collection of open sets is the alpha filtration at radius r.
• The weak alpha filtration is a sub-filtration of the Vietoris-Rips filtration. Namely, at each filtration value r, the weak alpha filtration is equal to the Vietoris-Rips filtration intersected with the Delaunay triangulation.

Filtrations of the Delauny triangulation are implemented as a DelaunayTriangulation from the spade crate. The filtration value is stored in the data() struct associated to each vertex, undirected edge and inner face.

WARNING: To avoid unecessary square roots, the filtration times are squared from their theoretical value.

Structs

• FilteredEdge
  Represents an edge in ℝ^2 that enters the filtration at the specified time.
• FilteredFace
  Represents a face in ℝ^2 that enters the filtration at the specified time.
• FilteredPoint2
  Represents a point in ℝ^2 that enters the filtration at the specified time.

Functions

• alpha_filtration
  Computes the non-truncated alpha filtration of a set of a points in the plane. Filtration times are assigned according to the algorithm described in the Gudhi documentation.
• sparsify
  Extracts the sparse boundary matrix of the filtration of the Delaunay triangulation. Additionally returns the times at which each column enters the filtration. Columns are sorted by dimension and then filtration time.
• weak_alpha_filtration
  Computes the non-truncated weak-alpha filtration of a set of a points in the plane. As described by Gitto, all simplices are given the same filtration value as they would be given in the Vietoris-Rips filtration.

Type Definitions

• DelaunayFiltration
  A Delaunay triangulation of a set of points in the plane, in which each point, edge and triangle additionaly carries the time at which it enters the filtration.