Probabilistic 6D Representations

• Probabilistic 6D Representations are a framework that encodes full probability distributions over an object’s pose in SE(3), including both rotation and translation.
• They employ diverse mathematical parameterizations and probabilistic modeling strategies such as Bayesian filtering, particle methods, and deep neural flows for handling multi-modality and uncertainty.
• This approach improves uncertainty quantification, robust object tracking, and multi-view fusion in robotics while balancing trade-offs between analytical tractability and computational efficiency.

A probabilistic 6D representation encodes a full probability distribution over the rigid pose of objects in three-dimensional space, encompassing both orientation and translation (SE(3)), rather than returning a single optimum. This paradigm provides structured uncertainty quantification, multi-modal scene reasoning, and robust object tracking in the presence of symmetries, occlusions, and sensor noise. The approach is foundational in modern robotics, 3D scene understanding, and manipulation under perceptual uncertainty. Recent advances extend representations to nonparametric distributions, deep generative models, and analytic mixtures, enabling high-fidelity estimation, sampling, and probabilistic fusion of pose.

1. Mathematical Parameterizations of 6D Pose

The rigid 6D pose of an object is naturally modeled in SE(3), combining a rotation $R\in$ SO(3) (parameterized via Euler angles, angle–axis, quaternions, or rotation matrices) and a translation $t\in\mathbb{R}^3$. Multiple parameterizations are adopted depending on computational and representational requirements:

• SE(3) coordinate charts: Axis–angle with translation; locally minimal but with chart boundaries (Wüthrich et al., 2015).
• Quaternions and dual quaternions: Handle orientation (quaternion $q\in S^3$) and full pose (dual quaternion $Q=q+\varepsilon q_d$) with algebraic closure under composition (Feiten et al., 2017, Lang, 2017).
• Tangent-space linearization: Pose Gaussians defined over the tangent space at a reference point (e.g., via exponential map $\exp_{q_0}(\cdot)$ on $S^3$) supporting analytic distribution and fusion operations (Feiten et al., 2017, Lang, 2017).
• Samples and particles: Direct representation of $p(R,t)$ as a weighted set of samples (particles), universal but requiring many samples and non-analytic operations (Wüthrich et al., 2015, Zhou et al., 2023).
• Deep energy and neural fields: High-capacity networks define implicit densities or energy over SO(3) or SE(3) (Periyasamy et al., 2022, Jin et al., 3 Nov 2025).

This diversity affords trade-offs between analytic tractability (closure under composition/fusion), expressive power (multi-modality, heavy tails), and computational efficiency.

2. Probabilistic Modeling: Posterior, Fusion, and Filtering

Probabilistic 6D representations are employed to encode the posterior $p(R,t \mid \text{obs})$ given sensory inputs (RGB-D, range images, etc.). Core modeling strategies include:

• Bayesian Network Filtering: For dynamic tracking, a Markovian process over pose ($r_t$) and occlusion states ($o_t$) is coupled with observations (e.g., depth images $z_t$), marginalized over possible occlusions. Bayes-filter recursion tracks the distribution through time (Wüthrich et al., 2015):

$$p(x_t\mid z_{1:t},u_{1:t}) = \eta\,p(z_t\mid x_t) \int p(x_t\mid x_{t-1},u_t)\,p(x_{t-1}\mid z_{1:t-1},u_{1:t-1})\, dx_{t-1}$$

• Analytic Projected Gaussians: A local Gaussian in $T_{(q_0,t_0)}(S^3\times\mathbb{R}^3)$ pushed forward to SE(3) via manifold exponential projects to a non-parameteric density. Closed-form fusion is available for tangent-space aligned distributions (Feiten et al., 2017, Lang, 2017):

$$\Sigma_f = (\Sigma_A^{-1}+\Sigma_B^{-1})^{-1},\quad \mu_f = \Sigma_f(\Sigma_A^{-1}\mu_A + \Sigma_B^{-1}\mu_B)$$

• Particle and Sample-based Filtering: The full posterior is approximated as a weighted sum of samples, supporting multi-modality and strong nonlinearities but with increased computational cost (Wüthrich et al., 2015, Zhou et al., 2023).
• Neural or Flow-based Densities: Distribution is given by the pushforward of a base density along a learned flow on SE(3) to estimate $p(R,t\mid x)$ for ambiguous observations (Jin et al., 3 Nov 2025):

$\frac{dx}{dt} = v_\theta (t, x)$

Mixture models (MoPGs) enable universal approximation of arbitrarily complex pose pdfs, supporting analytic fusion for each component and numerically efficient reduction via merging (Lang, 2017). Particle filtering and RBPF techniques allow explicit marginalization of unobserved factors (e.g., occlusion, dynamics) (Wüthrich et al., 2015, Zhou et al., 2023).

3. Deep Probabilistic Pose Estimation Methodologies

Recent approaches employ deep networks to encode complex, data-driven 6D pose distributions:

• ImplicitPDF (nonparametric SO(3) orientation): For image-based orientation estimation, a network learns a log-likelihood field $\mathcal{F}_\theta(R, \mathcal{I})$ over SO(3). Symmetries are handled by forming the likelihood $$p_\theta(R \mid \mathcal{I})\propto \sum_{g\in G} \exp[\mathcal{F}_\theta(Rg,\mathcal{I})]$$ over group elements $g$ (Periyasamy et al., 2022).
• EPRO-GDR (Posterior over SE(3)): A geometry-guided detection network outputs dense 2D–3D correspondences and confidence weights, yielding an unnormalized posterior via a weighted reprojection error:

$$p(R, t \mid x) \propto \exp\left(-\frac{1}{2}\sum_{i=1}^N \|w^{2D}_i \odot (\pi(R x^{3D}_i + t) - x^{2D}_i)\|_2^2 \right)$$

Posterior is sampled or locally approximated by Laplace; network is trained with a combined detection, geometry, and negative log-likelihood (KL) loss (Pöllabauer et al., 2024).

• SE(3)-PoseFlow (Sample-based flow-matching): A deep network is trained to pushforward a base distribution to the data-dependent posterior on SE(3) by matching learned velocity fields to closed-form reference flows between source/target pose pairs, leveraging ODE integration for sampling (Jin et al., 3 Nov 2025).
• 3DNEL (Generative Likelihood with Embeddings): The likelihood of observed RGB-D is expressed as a mixture over rendered surface points, with per-pixel embedding comparison and depth noise models. Posterior tracking across sequences is performed with particle filtering (Zhou et al., 2023).

These neural methods natively encode symmetry, occlusion, and ambiguity, and admit multi-modal, non-Gaussian posteriors.

4. Application: Uncertainty Quantification, Fusion, and Robotics

Probabilistic 6D representations are leveraged for explicit uncertainty quantification, informed fusion, and improved downstream task performance:

• Uncertainty Metrics: Covariance on translation ($\Sigma_p$) and orientation ($\Sigma_R$, Karcher mean), entropy over pose samples, and explicit uncertainty intervals derived from the pose distribution (Jin et al., 3 Nov 2025).
• Fusion and Multi-view Integration: Analytic fusion (for MoPGs and tangent-space Gaussians) using closed-form updates; fusion of particle sets or multi-hypothesis integration for joint state tracking (Lang, 2017, Zhou et al., 2023).
• Downstream Use-Cases: Uncertainty-aware grasp planning (integrating over the pose posterior), active perception (selection of viewpoint with minimal expected entropy), and robust closed-loop manipulation under uncertainty (Jin et al., 3 Nov 2025, Wüthrich et al., 2015).
• Occlusion Handling: Explicit state tracking of occlusion variables, per-pixel likelihood marginalization, and particle spreading under occlusion with reconvergence when visible (Wüthrich et al., 2015, Zhou et al., 2023).

A key implication is that probabilistic pose representations enable principled reasoning in the presence of ambiguous or incomplete data, improving manipulation, tracking, and scene understanding robustness.

5. Comparisons: Mixture Models, Particle, and Neural Methods

Different families of probabilistic 6D pose models furnish trade-offs in expressiveness, analytic tractability, and computational efficiency, summarized in the following table:

Class	Key Property	Limitations
Projected Gaussian (PG)	Analytic fusion in tangent space; unimodal	Cannot represent strong multi-modality
Mixture of PGs (MoPG)	Universal approximation; analytic merging/fusion	Mixture size combinatorics
Particle (Sample-based)	Arbitrary pdfs; works for all Riemannian manifolds	No closed-form fusion; expensive
ImplicitPDF/Neural Flow	Data-driven, nonparametric; learns arbitrary densities	Requires large data/compute

Mixture representations and sample-based (particle) methods are preferred for highly non-Gaussian, multi-modal distributions. Analytic approaches are efficient for unimodal, low-uncertainty regimes and amenable to fusion and filtering; neural densities admit expressive, scalable, and multi-modal posteriors directly from high-dimensional observation data (Lang, 2017, Xu et al., 2024, Periyasamy et al., 2022).

6. Empirical Performance and Limitations

State-of-the-art probabilistic 6D pose estimators demonstrate superior recall and accuracy, especially in ambiguous, symmetric, or occluded scenes. Key empirical findings include:

• EPRO-GDR achieves AR$_{\text{BOP}}$ improvements of 0.786 (LM-O), 0.844 (YCB-V), and 0.412 (ITODD), outperforming deterministic baselines; the approach provides plausible pose samples per detection and robust scene fusion (Pöllabauer et al., 2024).
• SE(3)-PoseFlow reports 45.4%/$68.2\%$ recall on YCB-V for $[5^\circ,5 \text{cm}]$/$[10^\circ,5 \text{cm}]$ thresholds, and enables calibrated multi-hypothesis inference (Jin et al., 3 Nov 2025).
• Neural and analytic particle filters (RBPF, SIR) allow explicit tracking of multi-modal posteriors, maintaining robustness to occlusion and uncertainty spikes (Wüthrich et al., 2015, Zhou et al., 2023).
• Key limitations include computational cost (especially for particle methods), susceptibility to embedding network failures, and the need for careful specification of noise and proposal models (Zhou et al., 2023).

This suggests probabilistic 6D representations are essential for robust perception and manipulation under real-world uncertainty, with method choice driven by the statistics and operational constraints of the underlying task domain.