Accelerating Outlier-robust Rotation Estimation by Stereographic Projection

TL;DR

The paper addresses robust rotation estimation in the presence of heavy outliers and noise by decoupling rotation into an axis and an angle. It employs stereographic projection to map 3D circle constraints on the rotation axis to a 2D plane, enabling efficient axis estimation via voting, followed by histogram voting to determine the rotation angle and Rodrigues’ formula to recover the rotation. The approach supports single and multiple rotation estimation, delivering high accuracy and speed, including impressive results on large-scale data (e.g., $10^6$ points) with up to 90% outliers. Extensive experiments on synthetic and real-world datasets (3DMatch, KITTI) show competitive or superior performance against state-of-the-art methods, with notable gains in robustness and computational efficiency, especially under multi-rotation scenarios. The method’s practical impact lies in enabling fast, reliable pose estimation for applications like autonomous driving and 3D reconstruction where outliers are prevalent and real-time processing is essential.

Abstract

Rotation estimation plays a fundamental role in many computer vision and robot tasks. However, efficiently estimating rotation in large inputs containing numerous outliers (i.e., mismatches) and noise is a recognized challenge. Many robust rotation estimation methods have been designed to address this challenge. Unfortunately, existing methods are often inapplicable due to their long computation time and the risk of local optima. In this paper, we propose an efficient and robust rotation estimation method. Specifically, our method first investigates geometric constraints involving only the rotation axis. Then, it uses stereographic projection and spatial voting techniques to identify the rotation axis and angle. Furthermore, our method efficiently obtains the optimal rotation estimation and can estimate multiple rotations simultaneously. To verify the feasibility of our method, we conduct comparative experiments using both synthetic and real-world data. The results show that, with GPU assistance, our method can solve large-scale ($10^6$ points) and severely corrupted (90\% outlier rate) rotation estimation problems within 0.07 seconds, with an angular error of only 0.01 degrees, which is superior to existing methods in terms of accuracy and efficiency.

Figures (8)

• Figure 1: (a) Given the $i$-th $\bm{\mathit{z}}_{i}$, all possible rotation axes are in a circle (a green dashed circle). (b) Given $\bm{\mathit{z}}_{1}$ and $\bm{\mathit{z}}_{2}$, all possible rotation axes should be in the intersections of two corresponding circles. (c) The stereographic projection is circle-preserving, and a 3D circle on the sphere is into a 2D circle when projected onto the plane. (d) The intersections of 3D circles are projected to the intersections of 2D circles in the plane by stereographic projection.
• Figure 2: Diagram of rotation motion, illustrating that any rotation can be represented by the rotation axis and rotation angle.
• Figure 3: (a) With the perfect inputs, many project circles are across the same point, corresponding to the rotation axis. (b) With incorrect inputs, there are multiple wrong intersections, which will seriously affect the results of the pose estimation if not handled properly. (c) The number in each cell represents the number of circles that pass through the cell. (d) There are two rotation axes in this setting. (e) The 3D visualization histogram, with the highest peak being the position of the desired optimal rotation axis. (f) The two-dimensional projections of the two rotational axes are obtained from the histogram.
• Figure 4: Comparative experiments. The left calculates the rotation errors using inputs with different outlier rates when the total number is $10^{5}$. The right calculates the rotation errors using different amounts of inputs when the outlier rate is 90%.
• Figure 5: Comparative experiments. The left calculates the running time using inputs with different outlier rates when the total number is $10^{5}$. The right calculates the running time using different amounts of inputs when the outlier rate is 90%.

Theorems & Definitions (1)

• Definition 1: stereographic projection wilkins2017mobius