Tangent Space Backpropagation for 3D Transformation Groups

TL;DR

This paper introduces a novel backpropagation method for 3D transformation groups that operates in tangent spaces, improving numerical stability and ease of implementation for 3D vision and robotics applications.

Contribution

A new library that performs backpropagation on 3D transformation manifolds, leveraging their group structure for better stability and simplicity.

Findings

Numerically more stable than standard methods

Easier to implement in existing frameworks

Beneficial across diverse 3D tasks

Abstract

We address the problem of performing backpropagation for computation graphs involving 3D transformation groups SO(3), SE(3), and Sim(3). 3D transformation groups are widely used in 3D vision and robotics, but they do not form vector spaces and instead lie on smooth manifolds. The standard backpropagation approach, which embeds 3D transformations in Euclidean spaces, suffers from numerical difficulties. We introduce a new library, which exploits the group structure of 3D transformations and performs backpropagation in the tangent spaces of manifolds. We show that our approach is numerically more stable, easier to implement, and beneficial to a diverse set of tasks. Our plug-and-play PyTorch library is available at https://github.com/princeton-vl/lietorch.