DiscretePersistentHomologyTransform.jl
Persistent Homology Transform is produced and maintained by
Yossi Bokor and Katharine Turner
yossi.bokor@anu.edu.au and katharine.turner@anu.edu.au
This package provides an implementation of the Persistent Homology Transform, as defined in Persistent Homology Transform for Modeling Shapes and Surfaces. It also does Rank Functions of Persistence Diagrams, and implements Principal Component Analysis of Rank functions.
Installation
The best way to install DiscretePersistentHomologyTransform is to run the following in Julia
:
using Pkg
Pkg.add("DiscretePersistentHomologyTransform")
Functionality
- DiscretePersistentHomologyTransform computes the Persistent Homology Transform of simple, closed curves in $\mathbb{R}^2$.
- Rank functions of persistence diagrams.
- Principal Component Analysis of Rank Functions.
Persistent Homology Transform
Given an $m \times 2$ matrix of ordered points sampled from a simple, closed curve $C \subset \mathbb{R}^2$ (in either a clockwise or anti-clockwise direction), calculate the Persistent Homology Transform for a set of directions. You can either specify the directions explicity as a $n \times 2$ array (directions::Array{Float64}(n,2)
), or specify an integer (directions::Int64
) and then the directions used will be generated by
angles = [n*pi/(directions/2) for n in 1:directions]
directions = [[cos(x), sin(x)] for x in angles]
To perform the Persistent Homology Transform for the directions, run
PHT(points, directions)
This outputs an array of Eirene Persistence Diagrams, one for each direction.
Rank Functions
Given an Eirene Persistence Diagram $D$, DiscretePersistentHomologyTransform can calculate the Rank Function $r_D$ either exactly, or given a grid of points, calculate a discretised version. Recall that $D$ is an $n \times 2$ array of points, and hence the function Total_Rank_Exact
accepts an $n \times 2$ array of points, and returns a list of points critical points of the Rank function and the value at each of these points. Running
rk = Total_Rank_Exact(barcode)
we obtain the critical points via
rk[1]
which returns an array of points in $\mathbb{R}^2$, and the values through
rk[2]
wich returns an array of integers.
To obtain a discrete approximation of a Rank Function over a persistence diagram $D$, use Total_Rank_Grid
, which acceps as input an Eirene Persistence Diagram $D$, an increasing StepRange
for $x$-coordinates x_g
, and a decreasing StepRange
for $y$-coordinates y_g
. The StepRanges
are obtained by running
x_g = lb:delta:ub
x_g = ub:-delta:lb
with lb
being the lower bound so that $(lb, lb)$ is the lower left corner of the grid, and ub
being the upper bound so that $(ub,ub)$ is the top right corner of the grid, and $delta$ is the step size.
Finally, the rank is obtained by
rk = Total_Rank_Grid(D, x_g, y_g)
which returns an array or values.
PCA of Rank Functions
Given a set of rank functions, we can perform principal component analysis on them. The easiest way to do this is to use the wrapper function PCA
which has inputs an array of rank functions evaluated at the same points (best to use Total_Rank_Grid
to obtain them), an dimension $d$ and an array of weights weights
, where the weights correspond to the grid points used in Total_Rank_Grid
.
To perform Principal Component Analysis and obtain the scores run
scores = PCA(ranks, d, weights)
which returns the scores in $d$-dimensions.
Examples
Discrete Persistent Homology Transform
We will go through an example using a random shape and 20 directions. You can download the CSV file from here
To begin, load the CSV file into an array in Julia
Boundary = CSV.read("<path/to/file>")
Persistence_Diagrams = PHT(Boundary, 20)
You can then access the persistence diagram corresponding to the $i^{th}$ direction as
Persistence_Diagrams[i]