# 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


### Installing (from source):

Requires torch >= 2 and CUDA >= 11. Tested up to torch==2.7 and CUDA 12. Make sure PyTorch and CUDA major versions match. 

```bash
git clone --recursive https://github.com/princeton-vl/lietorch.git
cd lietorch

python3 -m venv .venv
source .venv/bin/activate

# install requirements
pip install torch torchvision torchaudio wheel

# optional: specify GPU architectures
export TORCH_CUDA_ARCH_LIST="7.5;8.6;8.9;9.0"

# install lietorch
pip install --no-build-isolation .
```

### Installing (with pip)
```bash
# optional: specify GPU architectures
export TORCH_CUDA_ARCH_LIST="7.5;8.6;8.9;9.0"

pip install git+https://github.com/princeton-vl/lietorch.git --no-build-isolation
```


To run the examples, you will need these additional libraries
```bash
pip install opencv-python open3d scipy pyyaml
```

### Running Tests

After building, you can run the tests
```bash
./run_tests.sh
```



## 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). 
