Probabilistic Rotation Representation With an Efficiently Computable Bingham Loss Function and Its Application to Pose Estimation

TL;DR

This work proposes a fast-computable and easy-to-implement loss function for Bingham distribution and shows not only to examine the parametrization of Bingham Distribution but also an application based on the loss function.

Abstract

In recent years, a deep learning framework has been widely used for object pose estimation. While quaternion is a common choice for rotation representation of 6D pose, it cannot represent an uncertainty of the observation. In order to handle the uncertainty, Bingham distribution is one promising solution because this has suitable features, such as a smooth representation over SO(3), in addition to the ambiguity representation. However, it requires the complex computation of the normalizing constants. This is the bottleneck of loss computation in training neural networks based on Bingham representation. As such, we propose a fast-computable and easy-to-implement loss function for Bingham distribution. We also show not only to examine the parametrization of Bingham distribution but also an application based on our loss function.

Figures (5)

• Figure 1: Inference sample of our Bingham model. The upper figure shows that the handle is visible and the corresponding distribution is low-variance, while the lower figure shows that the mug's handle is occluded and the distribution is widely spread.
• Figure 2: Overview of our network implementation. We changed the loss function and the dimension of final output from 4 to $d$. As we will describe in Section \ref{['section:experimentsresults']}, we decided $d=10$. Our method can also obtain the pose ambiguity.
• Figure 3: An example of the inference result of our network. The upper figure shows the relation between $\operatorname{E}[\Delta Q]$ and $\operatorname{tr}(A_\text{shifted})$. The red and the green point are the minimum and the maximum trace, respectively. The lower figures are the distribution of $\Delta Q$ at the red and the green point shown in the upper figure.
• Figure 4: Results on ours and PoseCNN xiang2018posecnn: Top row represents Average distance threshold curves. Middle row represents Translation threshold curves. Bottom row represents the histogram of rotation angle error. These metrics are described in xiang2018posecnn.
• Figure 5: Some toy example for an explanation of the mechanism how the network learns the Bingham parameter. Groundtruths are shown in rotation angles around the $z$-axis instead of quaternions. The first two and the last entries of inferred 10D parameter are shown in the bottom row.