Hierarchies and Ranks for Persistence Pairs

In this paper, which is available as a preprint on my personal homepage, we developed a novel hierarchy, the interlevel set persistence hierarchy that relates the critical pairs stored in a persistence diagram with each other.

Aleph ships with a tool called interlevel_set_persistence_hierarchy that permits the calculation of this hierarchy for 1D data and 2D VTK data. The tool supports the calculation of both superlevel and sublevel sets as explained in the paper.

Simple 1D example

As a first simple 1D example, let us compare the two functions described in the paper. We first need the input data. Put the following two lines in a file 1D.txt:

3 1 6 5 8 2 7 4
3 1 8 2 7 5 6 4

Each line represents a function, or rather, its image. Notice that the domain of the function does not matter for topological calculations.

New, we call the tool and take a look at the hierarchy based on sublevel sets:

$ ./interlevel_set_persistence_hierarchy 1D.txt

Among others, the output should contain the following:

0: 1    inf
1: 5    6
2: 2    8
3: 4    7

0 -- 1
2 -- 3
0 -- 2


0: 1    inf
1: 2    8
2: 5    6
3: 4    7

3 -- 2
1 -- 3
0 -- 1

The mapping shows the index that was assigned to a given critical point. The remaining lines indicate how those points are connected. Individual hierarchies are separated by two newlines, so they can be parsed easily later on.

Putting the first hierarchy into a file 01.txt and the second hierarchy into a file 02.txt, we can use another script to compare them:

$ ./compare_interlevel_set_persistence_hierarchies.py 01.txt 02.txt

The output of this script is unfortunately still somewhat rough (if you want to help improve the script, please consider contributing to Aleph). The file /tmp/M.txt should now contain the following output:

0.000000000000000000e+00 2.000000000000000000e+03
2.000000000000000000e+03 0.000000000000000000e+00

The off-diagonal values indicate that the two hierarchies are different.

A 2D VTK example

The tool is also capable of reading VTK files—as long as they describe structured grids. To reproduce the 2D examples from the paper, we first need to download the two files stored in this gist.

To calculate the interlevel set persistence hierarchy for the superlevel sets of these data, we have to use the following command:

$ ./interlevel_set_persistence_hierarchy -S 3_peaks_symmetrical.vtk 3_peaks_ridge.vtk

This should result in the following output:

0: 0.757965     0.556043
1: 1.00489      inf
2: 0.50947      0.435131

1 -- 0
1 -- 2


0: 1.00334      inf
1: 0.76121      0.554819
2: 0.50635      0.389117

0 -- 1
1 -- 2

We may now compare the hierarchies with each other again using the Python script shown above.

Conclusion

This concludes the brief demonstration of interlevel set persistence hierarchies. Please read the preprint for more information, and stay tuned for the forthcoming paper (which is still under review).