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Delaunay-Rips persistence

Delaunay-Rips persistence screenshot

Pipeline description

This example illustrates the use of the TTKDelaunayRipsPersistenceDiagram and TTKDelaunayRipsPersistenceGenerators filters to compute persistence diagrams (and associated persistent generators) of the Delaunay-Rips filtration of an input point set.

Three examples are provided. The first two are point sets in 3D (hence, with easily visualizable persistence generators) and the last is a point set in 5D given as a CSV ASCII file (one line per point, one column per dimension).

The dataset 67P-Churyumov-Gerasimenko.vtp consists of a 3D mesh that is a scan of Comet 67P/Churyumov–Gerasimenko, as captured by the Rosetta mission. As this comet is a contact binary, the point set features two significant 2-dimensional persistent pairs (one very persistent, corresponding to the cavity enclosed by the surface, the other less persistent, corresponding to one of the lobes). A generator for each of them is shown (one color per generator in the above screenshot, top left view)

The dataset K4.csv contains points noisily sampled next to the edges of a regular tetrahedron embedded in 3D. It features three significant 1-dimensional persistent pairs (cyan bars in the diagram). For each of them, a persistent generator is shown (one color per generator in the above screenshot, top center view).

The dataset hypersphere5.csv contains 10,000 points in 5D sampled next to a 4-sphere (some Gaussian noise is applied). This point set features one significant 4-dimensional persistent pair (red bar in the diagram).

For each of these datasets, its persistence diagrams are shown in the corresponding bottom views in the above screenshot.

ParaView

To reproduce the above screenshot, go to your ttk-data directory and enter the following command: 

paraview states/delaunayRipsPersistence.pvsm

Python code

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#!/usr/bin/env python

from paraview.simple import *

comet67PChuryumovGerasimenkovtp = XMLPolyDataReader(
    FileName=["67P-Churyumov-Gerasimenko.vtp"]
)
tTKDelaunayRipsPersistenceGenerators1 = TTKDelaunayRipsPersistenceGenerators(
    Input=comet67PChuryumovGerasimenkovtp
)
threshold1 = Threshold(Input=OutputPort(tTKDelaunayRipsPersistenceGenerators1, 2))
threshold1.Set(
    Scalars=["CELLS", "ClassPersistence"], LowerThreshold=1, UpperThreshold=2
)
SaveData("67PChuryumovGerasimenko_generator.obj", threshold1)
SaveData(
    "67PChuryumovGerasimenko_diagram.csv",
    OutputPort(tTKDelaunayRipsPersistenceGenerators1, 0),
)

k4csv = CSVReader(FileName=["K4.csv"])
tableToPoints2 = TableToPoints(Input=k4csv)
tableToPoints2.Set(XColumn="x", YColumn="y", ZColumn="z")
tTKDelaunayRipsPersistenceGenerators2 = TTKDelaunayRipsPersistenceGenerators(
    Input=tableToPoints2
)
SaveData("K4_diagram.csv", OutputPort(tTKDelaunayRipsPersistenceGenerators2, 0))

hypersphere5Dcsv = CSVReader(FileName=["hypersphere5D.csv"])
hypersphere5Dcsv.HaveHeaders = 0
tTKDelaunayRipsPersistenceDiagram3 = TTKDelaunayRipsPersistenceDiagram(
    Input=hypersphere5Dcsv
)
tTKDelaunayRipsPersistenceDiagram3.InputColumns = [
    "Field 0",
    "Field 1",
    "Field 2",
    "Field 3",
    "Field 4",
]
SaveData("hypersphere5D_diagram.csv", tTKDelaunayRipsPersistenceDiagram3)

To run the above Python script, go to your ttk-data directory and enter the following command: 

pvpython python/delaunayRipsPersistence.py

Inputs

Outputs

  • 67PChuryumovGerasimenko_generator.csv: a generator associated with the most persistent 2-dimensional persistent pair of the 67P-Churyumov-Gerasimenko.vtp dataset in OBJ format.
  • 67PChuryumovGerasimenko_diagram.csv: the persistence diagrams of the 67P-Churyumov-Gerasimenko.vtp dataset in CSV ASCII format.
  • K4_diagram.csv: the persistence diagrams of the K4.csv dataset in CSV ASCII format.
  • hypersphere5D_diagram.csv: the persistence diagrams of the hypersphere5.csv dataset in CSV ASCII format.

C++/Python API

DelaunayRipsPersistenceDiagram

DelaunayRipsPersistenceGenerators