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A Probabilistic Rotation Representation for Symmetric Shapes With an Efficiently Computable Bingham Loss Function

Hiroya Sato, Takuya Ikeda, Koichi Nishiwaki

TL;DR

A fast-computable and easy-to-implement NLL loss function for Bingham distribution is introduced and the inference network is created and it is shown that the loss function can capture the symmetric property of target objects from their point clouds.

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, it cannot represent the ambiguity of the observation. In order to handle the ambiguity, the Bingham distribution is one promising solution. However, it requires complicated calculation when yielding the negative log-likelihood (NLL) loss. An alternative easy-to-implement loss function has been proposed to avoid complex computations but has difficulty expressing symmetric distribution. In this paper, we introduce a fast-computable and easy-to-implement NLL loss function for Bingham distribution. We also create the inference network and show that our loss function can capture the symmetric property of target objects from their point clouds.

A Probabilistic Rotation Representation for Symmetric Shapes With an Efficiently Computable Bingham Loss Function

TL;DR

A fast-computable and easy-to-implement NLL loss function for Bingham distribution is introduced and the inference network is created and it is shown that the loss function can capture the symmetric property of target objects from their point clouds.

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, it cannot represent the ambiguity of the observation. In order to handle the ambiguity, the Bingham distribution is one promising solution. However, it requires complicated calculation when yielding the negative log-likelihood (NLL) loss. An alternative easy-to-implement loss function has been proposed to avoid complex computations but has difficulty expressing symmetric distribution. In this paper, we introduce a fast-computable and easy-to-implement NLL loss function for Bingham distribution. We also create the inference network and show that our loss function can capture the symmetric property of target objects from their point clouds.
Paper Structure (28 sections, 33 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 28 sections, 33 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORKS
  3. Continuous Rotation Representations
  4. Expressions of Rotational Ambiguity
  5. Loss Functions for Bingham Representation
  6. ROTATION REPRESENTATIONS
  7. Quaternion and Spatial Rotation
  8. Quaternion
  9. Product of Quaternions
  10. Conjugate, Norm, and Unit Quaternion
  11. Unit Quaternion and Spatial rotation
  12. Distance Function of Quaternions
  13. Definition of Bingham Distribution and Its Properties
  14. EFFICIENT COMPUTATION OF BNLL LOSS
  15. Definition of BNLL Loss Function
  16. ...and 13 more sections

Figures (5)

  • Figure 1: Inference sample of our network. A translucent frame represents the mode of ground truth and an opaque frame the estimated mode. One can see that the inference result for point cloud of unambiguous shape (airplane) is unimodal and peaky, while that of axis-symmetric shape (wine bottle) becomes zonally spread around the axis.
  • Figure 2: The transition of Kullback-Leibler divergence (KLD) during optimization. "GT" stands for "ground truth". Each visualization corresponds to the distribution at the tip of the arrow. The left-top figure is the initial distribution. $N_\text{sample} = 100$ is applied here. The ground truth is $\mathfrak{B}(A_\text{true})$ shown in Table \ref{['tab:values_of_As']}. The eigenvalues of true parameter used for unimodal case is $[0, -1209.9, -2217.9, -2342.4]$. $A_\text{init}$ is common.
  • Figure 3: Variation of Kullback-Leibler divergence (KLD) with respect to $N_\text{sample}$ and $s$. At $s=10.0$ in (b), the resulting KLD of BNLL gets worse, but it decreases to 0.119 if keeping on computing until $N_\text{iter}=100000$. Note that the result of QCQP is already converged at $N_\text{iter}=20000$.
  • Figure 4: Examples of inference results. $N_\text{sample}$ stands for the number of points in the given point cloud.
  • Figure 5: Resulting KLDs for some $N_\text{sample}$ after 20000 iterations. 100 times of tests were held. The upper and lower whiskers show the maximum and minimum values of all data (including outliers), respectively. In the result of axis-symmetric with $N_\text{sample} = 5$, the maximum KLD is 204.244428.