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A Smooth Representation of Belief over SO(3) for Deep Rotation Learning with Uncertainty

Valentin Peretroukhin, Matthew Giamou, David M. Rosen, W. Nicholas Greene, Nicholas Roy, Jonathan Kelly

arXiv:2006.01031v4cs.CVcs.LG

TL;DR

Rotation regression needs representations that support continuous learning, while common point representations do not quantify network uncertainty. This paper uses a symmetric matrix to represent rotations and a Bingham belief, then reports lower synthetic-data errors and OOD rejection results on vehicle imagery.

  • Problem

    Standard unit quaternions lack the continuity property needed for learning arbitrary rotation targets, while point representations do not quantify network uncertainty.

    Source: §I · p. 1

  • Method

    A symmetric-matrix representation provides a smooth global section, encodes a Bingham belief over unit quaternions, and supports OOD detection without stochastic sampling or an auxiliary classifier.

    Source: §I · p. 1

  • Results

    For arbitrary synthetic rotation targets, the symmetric representation achieves the lowest training and testing errors; on all three MAV test sets, A + DT performs on par with 6D + AE.

    Source: A. Wahba Problem with Synthetic Data · p. 6, D. MAV Indoor-Outdoor Dataset · p. 8

  • Takeaways & Limitations

    The representation provides an eigenvalue-based OOD rejection method, and the authors identify improving learned perception reliability and robustness in real-world settings as a future direction.

    Source: §VIII · p. 9

    The differentiable layer adds training cost, especially for compact models, and excludes matrices whose minimum eigenvalue is not simple.

    Source: §VII · p. 8

Abstract

from arXiv · show

Accurate rotation estimation is at the heart of robot perception tasks such as visual odometry and object pose estimation. Deep neural networks have provided a new way to perform these tasks, and the choice of rotation representation is an important part of network design. In this work, we present a novel symmetric matrix representation of the 3D rotation group, SO(3), with two important properties that make it particularly suitable for learned models: (1) it satisfies a smoothness property that improves convergence and generalization when regressing large rotation targets, and (2) it encodes a symmetric Bingham belief over the space of unit quaternions, permitting the training of uncertainty-aware models. We empirically validate the benefits of our formulation by training deep neural rotation regressors on two data modalities. First, we use synthetic point-cloud data to show that our representation leads to superior predictive accuracy over existing representations for arbitrary rotation targets. Second, we use image data collected onboard ground and aerial vehicles to demonstrate that our representation is amenable to an effective out-of-distribution (OOD) rejection technique that significantly improves the robustness of rotation estimates to unseen environmental effects and corrupted input images, without requiring the use of an explicit likelihood loss, stochastic sampling, or an auxiliary classifier. This capability is key for safety-critical applications where detecting novel inputs can prevent catastrophic failure of learned models.

I. INTRODUCTION

The paper introduces a symmetric-matrix representation intended to combine smooth rotation regression with uncertainty modeling. It situates this approach among rotation parameterizations and learning-based estimation methods.

I. INTRODUCTION

  • Standard unit quaternions are numerically efficient and nonsingular, but their parameterization lacks the continuity property needed for learning arbitrary rotation targets.Prior work proposed continuous alternatives that lead to better network performance.Source: §I · p. 1
  • The symmetric-matrix representation admits a smooth global section, defines a Bingham distribution over unit quaternions, and supports OOD detection without stochastic sampling or an auxiliary classifier.The paper presents these as properties for learned rotation models.Source: §I · p. 1

II. RELATED WORK

  • Rotation estimation has a long history in computer vision and robotics, and prior work surveys rotation averaging and methods estimating rotations from raw sensor data.The review emphasizes learning-based estimation and differentiable optimization or convex-relaxation approaches.Source: §II · p. 2

A. Rotation Parameterization

  • Rotation parameterizations vary by task: robotics often uses SO(3), Euler angles suit small steady-state aircraft perturbations, and unit quaternions are popular for spacecraft maneuvers.SO(3) supports Gauss-Newton optimization and local uncertainty quantification; spacecraft control often involves large-angle maneuvers.Source: A. Rotation Parameterization · p. 2

B. Learning-based Rotation Estimation

  • Learning-based rotation estimation has been used for outlier classification, RANSAC guidance, robust-loss tuning, and improved egomotion accuracy and robustness.Prior work also identifies a continuous right-inverse as important for learning rotation representations.Source: B. Learning-based Rotation Estimation · p. 2

C. Learning Rotations with Uncertainty

The paper places its representation in the context of learned rotation uncertainty and formulates rotation estimation as a quadratic optimization over quaternions. A learned symmetric matrix supplies the objective data for this optimization.

C. Learning Rotations with Uncertainty

  • Prior uncertainty approaches include dropout, ensembles, multiple network heads, and quaternion averaging with local uncertainty; OOD detection has also used autoencoders.The paper contrasts its global Bingham belief with these lines of work.Source: C. Learning Rotations with Uncertainty · p. 2
  • The representation uses one symmetric matrix with 10 parameters to encode both the mode and uncertainty of a Bingham belief, and offers analytic gradients without likelihood training.Compared with, it avoids computing the Bingham normalization constant and proves a smooth right-inverse.Source: C. Learning Rotations with Uncertainty · p. 2

III. SYMMETRIC MATRICES AND SO(3)

  • The representation uses real symmetric 4 × 4 matrices with a simple minimum eigenvalue, each mapped to a unique rotation through a differentiable QCQP.The section relates the matrix to a Bingham distribution and rotation averaging, and presents an analytic derivative.Source: §III · pp. 2–3

A. Rotations as Solutions to QCQPs

  • In the QCQP formulation, a unit quaternion parameterizes the rotation, while matrix A encodes error primitives, data association, and uncertainty.For learned estimation, the network predicts A from input data that lacks vector correspondences.Source: A. Rotations as Solutions to QCQPs · p. 3

1) Solving Problem 3:

The rotation is recovered from the minimum eigenspace of the learned symmetric matrix, and its derivative can be computed analytically when the minimum eigenvalue is simple. The construction also provides a smooth global section from rotations to matrices.

1) Solving Problem 3:

  • Problem 3 returns antipodal unit quaternions from A’s one-dimensional minimum eigenspace, representing a single rotation.The representation excludes matrices whose minimum eigenvalue is not simple; that excluded set has measure zero.Source: 1) Solving Problem 3 · p. 3
  • When the minimum eigenvalue is simple, the implicit function theorem gives the derivative of the minimizing quaternion with respect to A.The derivative uses a Moore-Penrose pseudoinverse and is implemented in PyTorch’s symeig function.Source: 1) Solving Problem 3 · p. 3
  • A smooth global section maps each rotation to a symmetric matrix whose minimum eigenspace represents that rotation.The construction uses the orthogonal projection I − qq^T and is well-defined for either antipodal quaternion representative.Source: B. A Smooth Global Section of SO(3) · p. 4

C. A and the Bingham Distribution

The representation matrix A defines a Bingham distribution over unit quaternions, encoding a belief over rotations. Its eigenvalue gaps relate to distribution dispersion, and its eigenstructure sets the distribution’s mode.

C. A and the Bingham Distribution

  • A defines a Bingham distribution over unit quaternions, so it encodes a belief over rotations.The Bingham distribution is antipodally symmetric and is derived from a zero-mean Gaussian conditioned to lie on the unit sphere.Source: C. A and the Bingham Distribution · p. 4
  • For unit quaternions, the Bingham density uses x ∈ S3, an orthogonal matrix D, dispersion coefficients Λ, and normalization constant N(Λ).D is formed from four mutually orthogonal unit vectors.Source: C. A and the Bingham Distribution · p. 4
  • Setting DΛD^T = −A makes the solution of Problem 3 the mode of the Bingham density.Because adding cI to Λ leaves the Bingham distribution unchanged, the equivalent density can use shifted dispersion coefficients.Source: C. A and the Bingham Distribution · p. 4
  • A’s eigenvalue gaps establish a relation to the dispersion of the Bingham density.Each dispersion coefficient controls spread along its corresponding direction; the mode is given by d4.Source: C. A and the Bingham Distribution · p. 4

IV. USING A WITH DEEP LEARNING

The paper frames learning with its formulation as fitting deep rotation-regressor parameters to minimize training loss while generalizing to unseen data.

IV. USING A WITH DEEP LEARNING

  • The formulation fits parameters π for a deep rotation regressor R = NN(x; π) by minimizing training loss L while generalizing to unseen data.Source: §IV · p. 4

A. Self-Supervised Learning

The representation can be transformed differentiably into a rotation matrix through a pair of antipodal unit quaternions. This smooth global section avoids the back-propagation discontinuity identified in prior work.

A. Self-Supervised Learning

  • The layer solves for antipodal unit quaternions ±q* and maps them differentiably to a rotation matrix, avoiding the discontinuity identified in during back-propagation.The section limits its focus to supervised rotation regression to compare directly with the continuous representation of.Source: A. Self-Supervised Learning · pp. 4–5

B. Supervised Learning: Rotation Loss Functions

The paper uses chordal loss for its experiments, although the representation can also support a Bingham likelihood loss. Computing the latter’s normalization constant is non-trivial, so comparisons with other belief representations are left to future work.

B. Supervised Learning: Rotation Loss Functions

V. A AND ROTATION AVERAGING

The section interprets the symmetric representation as learned chordal averaging and develops dispersion thresholding as an empirical, likelihood-free epistemic cue.

V. A AND ROTATION AVERAGING

  • Because the base rotations must span training data, their distance can represent epistemic uncertainty without an explicit likelihood loss.The authors relate this interpretation to neural pose regression as interpolation over base poses.Source: §V · p. 5
  • Regression with the symmetric representation is analogous to a weighted chordal average over learned base orientations, whose bases and weights are tuned during training.The network’s final layer defines the bases, while its input-dependent coefficients provide the weights.Source: §V · p. 5

A. Dispersion Thresholding (DT) as Epistemic Uncertainty

  • After canonicalization, the authors interpret Λ as a negative inertia matrix and use its trace as an empirical uncertainty measure without penalizing Λ during training.The text notes that this Λ is not itself the Bingham dispersion matrix, though the two can be related.Source: A. Dispersion Thresholding (DT) as… · p. 5
  • Dispersion thresholding rejects test inputs above a tr(Λ) threshold chosen from training-data quantiles, retaining the lowest qth quantile.The authors report that the measure works empirically without an explicit uncertainty penalty, while leaving the reason for this signal to future work.Source: A. Dispersion Thresholding (DT) as… · pp. 5–6
  • The authors conjecture that learned basis rotations must spread across the training distribution, so OOD inputs tend to average them out.They propose that in-distribution inputs are more likely to remain near learned bases and produce larger |tr(Λ)|.Source: A. Dispersion Thresholding (DT) as… · p. 6

VI. EXPERIMENTS

The experiments compare quaternion, 6D, and symmetric-matrix rotation regressors, including on synthetic vector-correspondence data. On arbitrary rotation targets, the continuous representations outperform quaternions, with the symmetric representation achieving the lowest errors.

VI. EXPERIMENTS

  • The experiments compare unit quaternions, the 6D representation, and the symmetric-matrix representation using rotational errors in degrees.The synthetic Wahba setup samples corresponding unit-vector pairs and trains all models with the chordal loss.Source: §VI · p. 6, A. Wahba Problem with Synthetic Data · p. 6

A. Wahba Problem with Synthetic Data

B. Wahba Problem with SHAPENET

The SHAPENET experiment applies random rotations to airplane point clouds and reports results similar to the synthetic-data comparison.

B. Wahba Problem with SHAPENET

  • Training uses 2,290 airplane point clouds and holds out 400, with 10 random rotations per training iteration and 100 per held-out cloud at test time.The test comparison includes the symmetric representation, unit quaternions, and the 6D representation.Source: B. Wahba Problem with SHAPENET · p. 7
  • On SHAPENET point clouds, the representation comparison produces results similar to the synthetic-data experiment.Figure 5b compares the symmetric representation against unit quaternions and the 6D representation.Source: B. Wahba Problem with SHAPENET · p. 7

C. Visual Rotation-Only Egomotion Estimation: KITTI

On KITTI, dispersion thresholding (DT) rejects corrupted and likely high-error inputs, and the proposed representation with DT outperforms a 6D representation with autoencoder-based rejection. On MAV, it performs on par with that comparator under a random split that places test images near training images.

C. Visual Rotation-Only Egomotion Estimation: KITTI

  • KITTI rotation targets are typically around one degree, and performance changed little across SO(3) representations.The evaluation tested three sequences and introduced random rectangular black regions only at test time.Source: C. Visual Rotation-Only Egomotion… · p. 7

1) Auto-Encoder (AE) OOD Method:

D. MAV Indoor-Outdoor Dataset

  • On MAV, A + DT performed on par with 6D + AE across all three test sets without a separate classifier, despite test images being near training images.The model was trained on outdoor images using rotation targets from onboard visual-inertial odometry.Source: D. MAV Indoor-Outdoor Dataset · pp. 7–8

VII. DISCUSSION AND LIMITATIONS

  • The differentiable layer may slow training versus basic-operation representations, especially for compact models, though its test-time cost is negligible.The representation excludes matrices with non-simple minimum eigenvalues; the constraint is assumed rather than explicitly enforced.Source: §VII · p. 8

VIII. CONCLUSIONS AND FUTURE WORK

  • The symmetric-matrix representation admits a Bingham-belief and weighted-rotation-average interpretation, and its eigenvalues support OOD rejection.Combining the representation with a Bingham likelihood loss and further investigating its uncertainty connection are future work.Source: §VIII · pp. 8–9
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