Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Special Unitary Parameterized Estimators of Rotation

Akshay Chandrasekhar

TL;DR

This work introduces Special Unitary Parameterized Estimators of Rotation by recasting Wahba's problem in $SU(2)$, yielding linear quaternion constraints across three solution paradigms: stereographic projection, Möbius transformations, and direct 3D sphere mappings. It then derives two neural-network-friendly rotation representations, 2-vec and QuadMobius, each with differentiable mappings to $SU(2)$ and quaternions, enabling end-to-end learning of rotations. Through synthetic Wahba experiments and three learning benchmarks (ModelNet10-SO3, inverse kinematics, and Cambridge camera pose), the paper demonstrates that these SU(2)-based methods can match or surpass traditional solvers and existing neural representations while providing efficient or robust gradient behavior. The results highlight the practical impact of integrating $SU(2)$-driven constraints into rotation estimation tasks, with potential extensions to broader pose estimation problems and analytical camera models.

Abstract

This paper revisits the topic of rotation estimation through the lens of special unitary matrices. We begin by reformulating Wahba's problem using $SU(2)$ to derive multiple solutions that yield linear constraints on corresponding quaternion parameters. We then explore applications of these constraints by formulating efficient methods for related problems. Finally, from this theoretical foundation, we propose two novel continuous representations for learning rotations in neural networks. Extensive experiments validate the effectiveness of the proposed methods.

Special Unitary Parameterized Estimators of Rotation

TL;DR

This work introduces Special Unitary Parameterized Estimators of Rotation by recasting Wahba's problem in , yielding linear quaternion constraints across three solution paradigms: stereographic projection, Möbius transformations, and direct 3D sphere mappings. It then derives two neural-network-friendly rotation representations, 2-vec and QuadMobius, each with differentiable mappings to and quaternions, enabling end-to-end learning of rotations. Through synthetic Wahba experiments and three learning benchmarks (ModelNet10-SO3, inverse kinematics, and Cambridge camera pose), the paper demonstrates that these SU(2)-based methods can match or surpass traditional solvers and existing neural representations while providing efficient or robust gradient behavior. The results highlight the practical impact of integrating -driven constraints into rotation estimation tasks, with potential extensions to broader pose estimation problems and analytical camera models.

Abstract

This paper revisits the topic of rotation estimation through the lens of special unitary matrices. We begin by reformulating Wahba's problem using to derive multiple solutions that yield linear constraints on corresponding quaternion parameters. We then explore applications of these constraints by formulating efficient methods for related problems. Finally, from this theoretical foundation, we propose two novel continuous representations for learning rotations in neural networks. Extensive experiments validate the effectiveness of the proposed methods.

Paper Structure

This paper contains 63 sections, 97 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Wahba's Problem
  3. Representations for Learning Rotations
  4. Contributions
  5. Solutions to Wahba's Problem via SU(2)
  6. Stereographic Plane Solution
  7. Approximation via Möbius Transformations
  8. 3D Sphere Solution
  9. Optimization Methods From Linear Quaternion Constraints
  10. Residual Based Optimization
  11. Constrained Optimization
  12. Two-Point Case for Wahba's Problem
  13. Weighted
  14. Unweighted
  15. Representations for Learning Rotations
  16. ...and 48 more sections

Figures (7)

  • Figure 1: (a)-(b) Illustration of difference between Gram-Schmidt and 2-vec in 2D. $\mathbf{b_x}$, $\mathbf{b_y}$ are predicted axes directions from the model, and $\mathbf{R_x}$, $\mathbf{R_y}$ are the orthogonalized coordinate axes from each mapping. Gram-Schmidt favors $\mathbf{b_x}$, aligning $\mathbf{R_x}$ with it greedily while 2-vec uses $\mathbf{b_x}, \mathbf{b_y}$ in a balanced way. (c)-(d) Conceptual illustration of QCQP, SVD, and QuadMobius maps in context of Wahba's problem in 3D. QCQP/SVD can be interpreted as direct projection of target points (red) to an orthogonal frame. QuadMobius first maps those points to an intermediate representation—a Möbius transformation, defined by three points (blue)—before projecting to an $SU(2)$ rotation.
  • Figure 3: Density plot of loss gradient ratios for Gram-Schmidt and 2-vec. The x-axis represents the loss $\mathcal{L}$, and the y-axis shows the ratio of loss gradient magnitudes $\|\nabla_{\mathbf{b}_x}\mathcal{L}\| / \|\nabla_{\mathbf{b}_y}\mathcal{L}\|$ for the predicted rotation axes $\mathbf{b}_x$ and $\mathbf{b}_y$. See \ref{['sec:theory_investigation']} for details. 2-vec exhibits noticeably lower variance, suggesting more stable gradients during learning.
  • Figure 4: Plot of mean loss (Chordal L2) against dropout rate of map representations. $\Theta$ and $\mathbf{M}$ denote whether dropout was applied to map inputs or intermediate representation for QuadMobius.
  • Figure 5: Distribution plot of loss gradient magnitudes against loss $\mathcal{L}$ (Chordal L2). The left shows the gradient with respect to the map inputs $\Theta$, while the right shows the gradient with respect to the Möbius transformation $\mathbf{M}$ estimated from eigendecomposition in QuadMobius.
  • Figure :
  • ...and 2 more figures