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 approach, 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.

翻译：我们处理3D变换组3D变换组3SO(3)美元、3SE(3)美元和3SIM(3)美元的计算图进行反向调整的问题。 3D变换组在3D视觉和机器人中广泛使用,但它们并不形成矢量空间,而是在光滑的方块上。标准反向调整法将3D变换嵌入Euclidean空间,在数字上存在困难。我们引入了一种新的方法,利用3D变换组结构,在相近的多元体空间进行反向调整。我们表明我们的方法在数字上更加稳定,更容易执行,并且有益于一系列不同的任务。我们的插插和编PyTorch图书馆可在https://github.com/princenton-vl/lietorch查阅。