MobiusConv

The official PyTorch implementation of Möbius Convolution from the SIGGRAPH 2022 paper.

[Paper: Möbius Convolutions for Spherical CNNs]

[Talk: SIGGRAPH 2022]

Dependencies

• PyTorch >= 1.10
• CMake
• TS2Kit (Automatically cloned into the repo with the --recursive-submodules command)

Our implementation relies on PyTorch's support for complex numbers and other functionalities introduced in version 1.10. The majority of this code is not compatable with earlier PyTorch versions. We also use the python packages progressbar2 and mpmath which can be installed with pip.

Installation

Clone this repository and its submodules

$ git clone --recurse-submodules https://github.com/twmitchel/MobiusConv.git

The C++ executable convcoeff is called automatically during model initalization to precompute and store the coefficients in Equation (13) using a closed form solution. On Linux, the model can be built by running the following sequence of commands in main directory:

$ cd precomp
$ mkdir build
$ cd build
$ cmake ..
$ make

Using Möbius Convolutions

The principal layer is an MCResNetBlock -- two Möbius convolutions, each followed by filter response normalization and a thresholded nonlinearity, with a residual connection between the input and output features. The layer can be initalized as follows

from MobiusConv.nn import MCResNetBlock

# Spherical band-limit of input signals (corresponds to a 2B x 2B DH spherical grid - see TS2Kit doc for more information)
B = 64

# Number of input channels
CIn = 16

# Number of output channels
COut = 16

# Radial (longitudinal) band-limit of log-polar filters
D1 = 1

# Angular (latitudinal) band-limit of log-polar filters
D2 = 1

# Angular band-limit of representation
M = D2 + 1

# Number of quadrature samples in representation
Q = 30

# Whether or not to use checkpointing (trade speed for less memory overhead, useful in large networks or at high resolutions)
checkpoint = False;

# Initalize an MCResNetBlock
MCRN = MCResNetBlock(CIn, COut, B, D1, D2, M, Q, checkpoint=checkpoint)

MobiusConv and MCResNetBlock modules initalized with band-limit B, CIn input channels, and COut output channels, expect input features to be torch.float tensors of dimension b X CIn X 2B X 2B corresponding to CIn-channel features sampled on a 2B X 2B Driscoll-Healy spherical grid (see TS2Kit documentation) with b batch dimensions. Each module contains additional documentation in the form of inline comments.

Simple equivariance demo

An example of how to set up Möbius Convolutions and a simple equivariance demo can be found in the demo_mobius_conv.ipynb notebook.

UNet example: pooling + unpooling

An example of a UNet architecture with Möbius Convolutions can be found in the nn/mc_unet.py, including pooling and unpooling operations.

Authorship and citation information

Author: Thomas (Tommy) Mitchel (thomas.w.mitchel 'at' gmail 'dot' com)

Please cite our paper if this code or our method contributes to a publication:

@inproceedings{10.1145/3528233.3530724,
author = {Mitchel, Thomas W. and Aigerman, Noam and Kim, Vladimir G. and Kazhdan, Michael},
title = {M\"{o}bius Convolutions for Spherical CNNs},
year = {2022},
isbn = {9781450393379},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/3528233.3530724},
doi = {10.1145/3528233.3530724},
booktitle = {ACM SIGGRAPH 2022 Conference Proceedings},
articleno = {30},
numpages = {9},
location = {Vancouver, BC, Canada},
series = {SIGGRAPH '22}
}