Linearly Solving Robust Rotation Estimation

Abstract: Rotation estimation plays a fundamental role in computer vision and robot tasks, and extremely robust rotation estimation is significantly useful for safety-critical applications. Typically, estimating a rotation is considered a non-linear and non-convex optimization problem that requires careful design. However, in this paper, we provide some new perspectives that solving a rotation estimation problem can be reformulated as solving a linear model fitting problem without dropping any constraints and without introducing any singularities. In addition, we explore the dual structure of a rotation motion, revealing that it can be represented as a great circle on a quaternion sphere surface. Accordingly, we propose an easily understandable voting-based method to solve rotation estimation. The proposed method exhibits exceptional robustness to noise and outliers and can be computed in parallel with graphics processing units (GPUs) effortlessly. Particularly, leveraging the power of GPUs, the proposed method can obtain a satisfactory rotation solution for large-scale($10^6$) and severely corrupted (99$\%$ outlier ratio) rotation estimation problems under 0.5 seconds. Furthermore, to validate our theoretical framework and demonstrate the superiority of our proposed method, we conduct controlled experiments and real-world dataset experiments. These experiments provide compelling evidence supporting the effectiveness and robustness of our approach in solving rotation estimation problems.

• The paper presents a novel linear formulation that reformulates rotation estimation as a quaternion circle fitting problem.
• It employs dual linear systems to achieve robust performance even with up to 99% outliers, surpassing traditional methods like RANSAC.
• The approach simplifies computation for real-time robotics and aerospace applications, paving the way for extensions to complex pose estimation tasks.

Overview of "Linearly Solving Robust Rotation Estimation"

The paper “Linearly Solving Robust Rotation Estimation” presents a novel approach to rotation estimation, a crucial process in computer vision and robotic tasks. Recognizing the complexity traditionally associated with rotation estimation tasks—characterized by non-linear and non-convex optimization challenges—the authors propose a fundamentally new perspective. They introduce a method to reformulate rotation estimation as a linear model fitting problem, specifically utilizing the dual structure of rotation motion to represent it as a great circle on a quaternion sphere.

Theoretical Insights

Rotation estimation typically involves solving for a rotation matrix $\mathbf{R} \in \mathbb{SO}(3)$ that aligns sets of observations. This paper approaches the problem by exploring quaternion circles—a concept where rotations satisfying certain conditions lie on great circles on the unit quaternion sphere $\mathbb{S}^3$. This geometric perspective allows the problem to be framed as solving a linear system without introducing singularities. The method leverages the quaternion representation of rotations to avoid the pitfalls of non-linear parameterizations, offering a linearly solvable model.

The authors provide insights into how a rotation's motion constraint can be represented with two linear equation systems derived from quaternion circle theory. This formulation takes advantage of the symmetry and orthogonality properties inherent in unit quaternions. One of the key theoretical assertions is that robust rotation estimation can be transformatively approached by exhaustively examining intersections on the quaternion sphere, analogous to broad concepts within line detection via the Hough Transform.

Numerical Results

Robustness and efficiency are validated through controlled experiments and comparisons with state-of-the-art methods. Particularly striking is the algorithm’s capacity to handle high-scale rotation estimation with 99% outlier ratios in less than half a second when accelerated by GPUs. Across various datasets and quantitative setups, the proposed method demonstrates competitive performance, matching or surpassing traditional methods like RANSAC and certifiably optimal solutions in efficiency and accuracy.

Implications and Future Directions

Practically, this research introduces a methodology potentially transformative for applications in robotics and aerospace where high reliability under substantial data corruption is crucial. The linear reformulation and voting scheme simplify the estimation processes, enabling implementation in settings that demand real-time performance or operate under stringent computational constraints.

Theoretically, the paper invites further exploration into the geometric and algebraic properties of quaternion circles that could be extended to broader classes of motion estimation problems. This research paves the way for investigating analogous linear formulations in $\mathbb{SE}(3)$, potentially simplifying pose estimation tasks that include both rotational and translational components.

Conclusion

Overall, the paper contributes significant advancements in rotation estimation by providing a robust, efficient solution characterized by reframing the problem into a linear context. Future developments might focus on extending these concepts to more complex scenarios involving multiple simultaneous rotations or integrating translation components, while continuing to address the intrinsic challenges in computational geometry.