Authors: Takeki Sudo, Kazushi Ahara (Meiji University), Shizuo Kaji (Kyushu University / Kyoto University)

CubicalRipser is an adaptation of Ripser by Ulrich Bauer, specialised in fast computation of persistent homology for cubical complexes.

• High performance (among the fastest for cubical complexes up to 4D)
• Flexible filtrations: supports both V- and T-constructions (see: V and T Constructions)
• Binary coefficients (field F2)
• Python module and standalone CLI
• Birth/death locations with creator/destroyer cells

For details, see: Cubical Ripser: Software for Computing Persistent Homology of Image and Volume Data (https://arxiv.org/abs/2005.12692)

Distributed under the GNU Lesser General Public License v3.0 or later.

• Google Colab Demo: CubicalRipser in Action
• Topological Data Analysis (TDA) Tutorial: Hands-On Guide
• Applications in Deep Learning:
  • Example 1: Homology-enhanced CNNs
  • Example 2: Pretraining CNNs without Data

Install the Python module directly:

pip install -U cripser

If you encounter architecture compatibility issues, try:

pip uninstall cripser
pip install --no-binary cripser cripser

Requires a C++11-compatible compiler (e.g., GCC, Clang, MSVC).

1. Clone the repository:

   git clone https://github.com/shizuo-kaji/CubicalRipser_3dim.git
   cd CubicalRipser_3dim

2. Build the command-line executable:

   cd build
   cmake ..
   make

   The executable cubicalripser will be created.

3. Alternatively, without cmake:

   cd src
   make all

   Modify the Makefile if needed.

4. Install the Python module:

   pip install .

CubicalRipser works on 1D / 2D / 3D / 4D NumPy arrays (dtype convertible to float64).

Basic example (V-construction, default):

import numpy as np, cripser

arr = np.load("input.npy")
ph = cripser.compute_ph(arr, maxdim=3, filtration="V")   # alias: computePH(...)

Result:

• For 1D–3D input: ph has shape (n, 9) Columns: dim, birth, death, x1, y1, z1, x2, y2, z2
• For 4D input: shape (n, 11) Columns: dim, birth, death, x1, y1, z1, w1, x2, y2, z2, w2
• death is DBL_MAX for essential features

Creator (x1,...) gives birth; destroyer (x2,...) kills the class (see Creator and Destroyer cells).

T-construction (8-neighborhood in 2D, etc.):

ph_T = cripser.compute_ph(arr, maxdim=3, filtration="T")

Convert to GUDHI-style structures (see section below):

dgms = cripser.to_gudhi_diagrams(ph)
pers = cripser.to_gudhi_persistence(ph)

For convenience, a small utility package is included to convert the raw output into GUDHI-compatible formats.

import numpy as np
import cripser

arr = np.load("input.npy")
ph = cripser.compute_ph(arr)

# List of diagrams per dimension, each an array of shape (k, 2)
dgms = cripser.to_gudhi_diagrams(ph)

# Or, GUDHI SimplexTree-like list of (dim, (birth, death))
persistence = cripser.to_gudhi_persistence(ph)

# Example: plot using GUDHI
import gudhi as gd
gd.plot_persistence_diagram(diagrams=dgms)

Infinite deaths encoded internally as DBL_MAX are automatically converted to np.inf.

A convenience wrapper around the core library for quick experiments without writing code.

Typical capabilities:

• Accepts a single file (e.g. .npy, image, DICOM) or a directory of sequential image / DICOM slices
• Builds a 1D–4D NumPy array
• Chooses V- or T-construction
• Computes persistent homology up to a chosen max dimension
• Writes raw PH pairs (CSV) or serialized NumPy results
• Optional sorting of input slice filenames (useful for DICOM)

Basic help:

python demo/cr.py -h

Examples

1. Single NumPy array (default: V-construction, maxdim=2):

python demo/cr.py input.npy -o ph.csv

2. Increase max dimension and use T-construction:

python demo/cr.py volume.npy -o ph.csv --maxdim 3 --filtration T

3. Directory of DICOM files (sorted), output CSV:

python demo/cr.py dicom/ --sort -it dcm -o ph.csv

4. Directory of PNG slices -> PH (auto-detect by extension):

python demo/cr.py slices/ -o ph.csv

5. Save raw PH as NumPy (to reuse in Python):

python demo/cr.py volume.npy -o ph.npy --format npy

6. Invert intensities (example flag; use only if present in -h):

python demo/cr.py volume.npy -o ph.csv --invert

Typical options (exact list: see -h):

• --maxdim k maximum homology dimension
• --filtration V|T choose construction
• --sort lexicographically sort input filenames
• -it EXT explicit input slice extension (e.g. dcm, png)
• -o FILE output file (.csv or .npy)
• --format csv|npy override format if extension is ambiguous
• --embedded Alexander dual interpretation (matches CLI flag)
• --invert intensity inversion (if implemented)

Basic example (text / Perseus-style input):

./cubicalripser --print --maxdim 2 --output out.csv demo/3dimsample.txt

Output file (CSV): each row

dim, birth, death, x1, y1, z1, x2, y2, z2

Meaning:

• dim: homology dimension
• birth, death: filtration values (death = DBL_MAX if essential)
• (x1,y1,z1): creator cell coordinates
• (x2,y2,z2): destroyer cell coordinates (omitted / meaningless if death is infinite)

Numpy array input (1D–4D):

./cubicalripser --output result.csv input.npy

Common options:

• --maxdim k compute up to dimension k (default: 2)
• --print also print pairs to stdout
• --embedded use embedded (Alexander dual) interpretation
• --filtration V|T choose construction (default: V); T alternative executable: tcubicalripser
• --output FILE write CSV (omit to print only)

Example (T-construction on a 3D volume):

./tcubicalripser --maxdim 3 --output volume_ph.csv volume.npy

Note: For 4D input, two extra coordinates (w1, w2) are appended:

dim, birth, death, x1, y1, z1, w1, x2, y2, z2, w2

• NUMPY (.npy): Native format for both Python and CLI.
• Perseus Text (.txt): Specification.
• CSV (.csv): Simplified input for 2D images.
• DIPHA (.complex): Specification.

A small utility is included that converts images in various formats into NUMPY arrays.

• Convert images to .npy:

  python demo/img2npy.py input.jpg output.npy

  A series of image files such as JPEG and PNG files (as long as the Pillow library can handle them) can also be made into a volume in a similar way:

    python demo/img2npy.py input*.jpg volume.npy

  Note that here we rely on the shell's path expansion. If your shell does not support it, you can manually specify file names as in the following:

    python demo/img2npy.py input00.dcm input01.dcm input02.dcm volume.npy

• Handle DICOM volumes: Given a series of DICOM files named input00.dcm, input01.dcm, input02.dcm... under the directory dicom, we can convert the DICOM files to a single 3D Numpy array volume.npy that is compatible with Cubical Ripser by

  python demo/img2npy.py dicom/*.dcm output.npy

  Or, we can compute persistent homology directly by

    python demo/cr.py dicom  --sort -it dcm -o output.csv

  by reading .dcm files from the directry dicom in a sorted order.

The filename should end with ".complex". Look at DIPHA binary format for specification.

We can convert input and output files between Cubical Ripser and DIPHA.

• to convert an Numpy array img.npy into DIPHA's format img.complex

    python dipha2npy.py img.npy img.complex

• the other way around

    python dipha2npy.py img.complex img.npy

• convert DIPHA's output result.output into an Numpy array result.npy

    python dipha2npy.py result.output result.npy

A scalar time-series can be considered as a 1D image, so Cubical Ripser can compute its persistent homology. Note that other software would be more efficient for this purpose.

An example of regressing the frequency of noisy sine curves is demonstrated here.

• V-Construction: Pixels represent 0-cells (4-neighbor connectivity in 2D).
• T-Construction: Pixels represent top-cells (8-neighbor connectivity in 2D).

Use the appropriate executable for your needs:

• V-construction: cubicalripser (Python module: cripser).
• T-construction: tcubicalripser (Python module: tcripser).

By the Alexander duality, the following two give essentially the same results:

./cubicalripser input.npy
./tcubicalripser --embedded input.npy

The difference is in the sign of the filtration and the permanent cycle. Here, (--embedded) converts the input I to -I^\infty described in the paper below.

For more details, see Duality in Persistent Homology of Images by Adélie Garin et al.

The creator of a cycle is the cell which gives birth to the cycle. For example, the voxel in a connected component with the lowest filtration value creates a 0-dimensional cycle, and the voxel which connects two separate connected components destroys the component with a higher birth time. The creator and the destroyer cells are not uniquely determined, but they provide useful information to localise the cycle. Cubical Ripser adopts the following convention on the location of these cells: when the lifetime of a cycle is finte,

arr[x2,y2,z2] - arr[x1,y1,z1] = death - birth = lifetime

where arr is the image, (x1,y1,z1) is the location of the creator cell, and (x2,y2,z2) is the location of the destroyer cell. Note that when computed with the (--embedded) option, the roles of creator and destroyer are switched:

arr[x1,y1,z1] - arr[x2,y2,z2] = death - birth = lifetime

The authors thank Nicholas Byrne for suggesting the convention and providing a test code.

• Lifetime Enhanced Image: Adds topological features as additional channels for CNNs.

  ./cubicalripser --output result.npy input.npy
  python demo/stackPH.py result.npy -o lifetime_image.npy -i input.npy

  In lifetime_image.npy, persistent homology is encoded as the extra channels so that it can be used as input for CNNs.

  Please look at the example section of our paper.

• Persistent Histogram Image: Similarly, the persistent histogram image can be obtained by

  python demo/stackPH.py result.npy -o hist_image.npy -t hist -i input.npy

  For practical examples, see HomologyCNN.

We give a referece to various software for persistent homology of images. The comments are based on our limited understanding and tests, and hence, could be wrong.

• Cubicle by Hubert Wagner

It computes for the V-construction of the image. Its parallelised algorithm offers faster computation on multi-core machines. Also, it reads the input image in small chunks so that it requires much less memory footprint.

• HomcCube By Ippei Obayashi.

It computes for the V-construction of the image. It is integrated into Homcloud developed by the same author.

• DIPHA by Ulrich Bauer and Michael Kerber

It computes for the V-construction of the image. It is parallelised with MPI so it works on a cluster. The software has been used in various projects. The memory footprint is relatively large.

• GUDHI developed at INRIA

It computes for the V- and T-construction of an array of any dimension. It is well-documented and offers a well-organised and easy to use interface. It focuses more on usability than performance.

• diamorse developed at The Australian National University.

It computes for the V-construction of the image.

• Perseus by Vidit Nanda

It computes for the V-construction of the image.

• (v0.0.19) added support for 4D cubical complexes
• (v0.0.8) fixed memory leak in Python bindings (pointed out by Nicholas Byrne)
• (v0.0.7) slight speed up
• (v0.0.6) changes in the definition of birth/death location (suggested by Nicholas Byrne)
• (up to v0.0.5, difference from the original version
  • optimised codes (much less memory footprint, much faster for certain data; sometimes more than 100 times.)
  • Python friendly: see the Jupyter Notebook example found under the demo directory.
  • virtually infinite input size (compared to 510x510x510)
  • cache control
  • option to use the Alexander duality for the highest degree persistent homology
  • V and T construction for building cubical complexes from an image
  • output birth/death location