Persistent homology based on Alpha complex on MATLAB

Sharing a naive implementation of persistent homology calculation from data on MATLAB

PersistentHomologyOnMATLAB

Overview

This is a naive implimetation of persistent homology calculation and visualization by persistent diagram. My implimentation is based on [1] for the reduction of boundary oparator matrix, and [2] for calculate minimum bounding sphere to yield Alpha filtration.

[1] Edelsbrunner, H., & Harer, J. L. (2022). Computational topology: an introduction. American Mathematical Society.

[2] Welzl, E. (1991). Smallest enclosing disks (balls and ellipsoids). In H. Maurer (Ed.), New Results and New Trends in Computer Science (pp. 359–370). Springer. https://doi.org/10.1007/BFb0038202

Usage

Here is an example file for short introduction.

close all; clc; clear;

% Example 
dat = load("../Dat/Lorenz-chaos-dt0.001-T1000.mat");
ndata = 1e2;
data = dat.x(1000:50:(1000+ndata*50), :);
[PD, Rinfs] = get_PD_H012_from_3Ddata(data);

% Visualization
figure;
scatter3(data(:, 1), data(:, 2), data(:, 3), 'filled'); hold on
VisualizePersistentDiagram(PD, Rinfs);

Practical functions are following:

[PD, Rinfs] = get_PD_H0...(d-1)_from_dDdata(data)

This calculates persistent diagram for <math-renderer class="js-inline-math" style="display: inline" data-static-url="https://github.githubassets.com/static" data-run-id="22411487ac6f18a08d83595eb03a3d30">$p$</math-renderer>-th persistent homology groups <math-renderer class="js-inline-math" style="display: inline" data-static-url="https://github.githubassets.com/static" data-run-id="22411487ac6f18a08d83595eb03a3d30">$(p = 0,\dots, d-1)$</math-renderer>, where <math-renderer class="js-inline-math" style="display: inline" data-static-url="https://github.githubassets.com/static" data-run-id="22411487ac6f18a08d83595eb03a3d30">$d$</math-renderer> is the dimension of input data. Output PD is data points of persistent diagram for each dimensions, and Rinfs recomended dummy values to show a persistent pair at infinity.

VisualizePersistentDiagram(PD, Rinfs);

This shows plots of persistent diagram based on inputs PD and Rinfs.