# LieTorch: Tangent Space Backpropagation


## Introduction

The LieTorch library generalizes PyTorch to 3D transformation groups. Just as `torch.Tensor` is a multi-dimensional matrix of scalar elements, `lietorch.SE3` is a multi-dimensional matrix of SE3 elements. We support common tensor manipulations such as indexing, reshaping, and broadcasting. Group operations can be composed into computation graphs and backpropagation is automatically peformed in the tangent space of each element. For more details, please see our paper:

<center><img src="lietorch.png" width="480" style="center"></center>

[Tangent Space Backpropagation for 3D Transformation Groups](https://arxiv.org/pdf/2103.12032.pdf)  
Zachary Teed and Jia Deng, CVPR 2021

```
@inproceedings{teed2021tangent,
  title={Tangent Space Backpropagation for 3D Transformation Groups},
  author={Teed, Zachary and Deng, Jia},
  booktitle={Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)},
  year={2021},
}
```


## Installation


### Requirements: 
 * Cuda >= 10.1 (with nvcc compiler)
 * PyTorch >= 1.7

We recommend installing within a virtual enviornment. Make sure you clone using the `--recursive` flag. If you are using Anaconda, the following command can be used to install all dependencies
```
git clone --recursive https://github.com/princeton-vl/lietorch.git
cd lietorch

conda create -n lie_env
conda activate lie_env
conda install scipy pyyaml pytorch torchvision torchaudio cudatoolkit=10.2 -c pytorch
```

To run the examples, you will need OpenCV and Open3D. Depending on your operating system, OpenCV and Open3D can either be installed with pip or may need to be built from source
```
pip install opencv-python open3d
```

### Installing (from source)

Clone the repo using the `--recursive` flag and install using `setup.py` (may take up to 10 minutes)
```
git clone --recursive https://github.com/princeton-vl/lietorch.git
python setup.py install
./run_tests.sh
```

### Installing (pip)
You can install the library directly using pip
```bash
pip install git+https://github.com/princeton-vl/lietorch.git
```



## Overview

LieTorch currently supports the 3D transformation groups. 

| Group  | Dimension | Action |
| -------| --------- | ------------- |
| SO3    | 3  | rotation |
| RxSO3  | 4  | rotation + scaling |
| SE3    | 6  | rotation + translation |
| Sim3   | 7  | rotation + translation + scaling |

Each group supports the following differentiable operations:

| Operation | Map | Description |
| -------| --------| ------------- |
| exp    | g -> G | exponential map |
| log    | G -> g | logarithm map |
| inv    | G -> G | group inverse |
| mul   | G x G -> G | group multiplication |
| adj    | G x g -> g | adjoint |
| adjT   | G x g*-> g* | dual adjoint |
| act    | G x R^3 -> R^3 | action on point (set) |
| act4   | G x P^3 -> P^3 | action on homogeneous point (set) |
| matrix | G -> R^{4x4} | convert to 4x4 matrix
| vec    | G -> R^D | map to Euclidean embedding vector |
| InitFromVec | R^D -> G | initialize group from Euclidean embedding



&nbsp;
### Simple Example:
Compute the angles between all pairs of rotation matrices

```python
import torch
from lietorch import SO3

phi = torch.randn(8000, 3, device='cuda', requires_grad=True)
R = SO3.exp(phi)

# relative rotation matrix, SO3 ^ {8000 x 8000}
dR = R[:,None].inv() * R[None,:]

# 8000x8000 matrix of angles
ang = dR.log().norm(dim=-1)

# backpropogation in tangent space
loss = ang.sum()
loss.backward()
```


### Converting between Groups Elements and Euclidean Embeddings
We provide differentiable `FromVec` and `ToVec` functions which can be used to convert between LieGroup elements and their vector embeddings. Additional, the `.matrix` function returns a 4x4 transformation matrix.
```python

# random quaternion
q = torch.randn(1, 4, requires_grad=True)
q = q / q.norm(dim=-1, keepdim=True)

# create SO3 object from quaternion (differentiable w.r.t q)
R = SO3.InitFromVec(q)

# 4x4 transformation matrix (differentiable w.r.t R)
T = R.matrix()

# map back to quaterion (differentiable w.r.t R)
q = R.vec()

```


## Examples
We provide real use cases in the examples directory
1. Pose Graph Optimization
2. Deep SE3/Sim3 Registrtion
3. RGB-D SLAM / VO

### Acknowledgements
Many of the Lie Group implementations are adapted from [Sophus](https://github.com/strasdat/Sophus).