A Smooth Representation of Belief over SO(3) for Deep Rotation Learning with Uncertainty (2006.01031v4)

Abstract: 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.

• The paper introduces a symmetric matrix representation that achieves smoother convergence and improved uncertainty quantification with Bingham distributions.
• It demonstrates superior performance on synthetic and real-world datasets, outperforming traditional unit quaternion and 6D vector methods.
• It features a dispersion thresholding mechanism that robustly detects out-of-distribution inputs, enhancing model reliability under challenging conditions.

Overview of "A Smooth Representation of Belief over $\LieGroupSO{3}$ for Deep Rotation Learning with Uncertainty"

The paper "A Smooth Representation of Belief over $\LieGroupSO{3}$ for Deep Rotation Learning with Uncertainty," authored by Valentin Peretroukhin et al., addresses the challenge of accurately estimating rotations—an essential task in robotic perception used in various applications such as visual odometry and object pose estimation. The paper introduces a novel representation based on symmetric matrices for the 3D rotation group, $\LieGroupSO{3}$, that aids in improving deep learning models used for rotation tasks. The authors emphasize the representation's suitability due to two critical properties: smoothness which enhances convergence and model generalization, and its capacity to encode uncertainty through a symmetric Bingham belief over unit quaternions.

Key Contributions

1. Symmetric Matrix Representation: The paper proposes a symmetric matrix representation contributing to smoother solutions and improved learning performance in large rotation target regressions. This representation is pivotal as it adds flexibility by encoding not only a point estimate in $\LieGroupSO{3}$ but also a Bingham distribution, which is advantageous for uncertainty quantification.
2. Continuity with $\LieGroupSO{3}$: An important realization in the paper is the representation's support for a smooth global section of $\LieGroupSO{3}$, thus overcoming challenges related to discontinuity as encountered with traditional unit quaternion parametrizations at certain rotational angles.
3. Out-of-Distribution Detection: The authors deliver a mechanism to enhance the robustness of regression models to outliers or unseen distributions, not requiring explicit stochastic methods or auxiliary classifiers. This mechanism, named dispersion thresholding (DT), relates the dispersion in the learned representation directly to uncertainty, allowing effective rejection of inputs likely to cause errors.

Empirical Evaluation

The paper features comprehensive experiments testing the suggested formulation on synthetic and real-world datasets, with notable benchmarks against state-of-the-art rotation representations (including unit quaternions and 6D vector representations). Experiments show that symmetric matrices yield superior predictive coverage and robustness, particularly highlighting DT's efficacy under test-time distribution shifts and environmental perturbations.

1. Synthetic Data Experimentation: Through a controlled setup, the representation outperformed others when learning arbitrary rotational transformations, with a significant error reduction in scenarios involving large rotation angles.
2. Benchmarking with ShapeNet: Tests on the ShapeNet dataset demonstrated enhanced learning outcomes using the symmetric matrix representation, reinforcing its competitive advantage over existing thought-to-be optimal continuous representations.
3. Application on Real-world Datasets: Utilizing the KITTI odometry dataset and a novel multicopter dataset, the representation illustrated improved resilience against uncertain inputs, with DT effectively filtering corrupted and anomalous data, thus maintaining accurate estimations.

Theoretical and Practical Implications

The theoretical underpinning provided in terms of existence and smoothness of a global section provides confidence in deploying this method in practical systems where rotation learning is critical. Importantly, given it can be seamlessly adopted into conventional deep learning pipelines, the idea beckons further exploration and enhancement, particularly in scaling to more complex state estimations or other manifold-valued data.

Future Directions

The paper opens avenues for further research, such as integrating a likelihood-based training objective using Bingham distributions to potentially improve calibration of the uncertainty estimates. Moreover, deeper exploration into epistemic uncertainty tied to learned representations amidst neural architectures is envisioned, as well as system-level integration for autonomous robotics demanding high-reliability perception.

In summary, this work significantly advances the landscape of rotation learning by proposing a versatile and efficient representation that both meets demand for precision and guards against failures in critical applications. This approach sets a promising direction for future work in incorporating uncertainty awareness and self-supervision into robotic systems.