MonoSE(3)-Diffusion: Robust Camera-Robot Pose Estimation

• The paper introduces a novel framework that models pose estimation as a conditional denoising diffusion process, overcoming limitations of direct regression.
• It employs a two-stage architecture—visibility-constrained diffusion for pose augmentation and timestep-aware reverse diffusion for fine pose refinement.
• Empirical results on benchmarks, including a 32.3% AUC improvement on AzureKinect-Franka, demonstrate enhanced robustness and generalization.

MonoSE(3)-Diffusion is a monocular SE(3) diffusion framework designed for robust, markerless camera-to-robot pose estimation. Unlike conventional direct regression or keypoint-based methods, MonoSE(3)-Diffusion formulates pose estimation as a conditional denoising diffusion process, wherein the network iteratively refines noisy transformations originating from ground-truth poses. The two-stage architecture—visibility-constrained diffusion for pose augmentation and timestep-aware reverse diffusion for pose refinement—integrates geometric visibility constraints to ensure all generated and predicted poses remain within the camera's field-of-view, yielding improvements in generalization and robustness across benchmark datasets.

1. Framework Structure and Motivation

MonoSE(3)-Diffusion consists of two principal stages:

• Visibility-Constrained Diffusion Process: During training, the framework perturbs ground-truth poses via a Gaussian diffusion process on SE(3), constrained to preserve visibility. The process augments training data with diverse but always in-view poses, improving generalization.
• Timestep-Aware Reverse Process: During inference, the pose denoising network, conditioned on the current timestep, utilizes a coarse-to-fine refinement schedule to reconstruct accurate poses from corrupted samples. The reverse process employs DDIM sampling, iteratively updating and refining pose estimates while maintaining visibility priors.

This probabilistic iterative methodology avoids the limitations of direct regression, particularly premature convergence and reduced training diversity, by leveraging conditional noising and stepwise refinement.

2. Diffusion Process and Pose Normalization

The visibility-constrained diffusion process augments poses by introducing noise to the normalized pose representation with constraints:

• Monocular-Normalized Formulation: Poses are normalized to decouple rotation and translation by redefining rotation at the robot centroid (preventing rotation-induced translation). Translation is decomposed into $(x, y)$ in-plane components and depth $z$, with in-plane normalization using camera focal length $f$ and image dimensions $(w, h)$:
  • $t_0^x \mapsto (f t_0^x / w)$
  • $t_0^y \mapsto (f t_0^y / h)$
• Gaussian Diffusion Model: At each training step,

$$H_t = \sqrt{\bar{\alpha}_t} \cdot H_0 + \sqrt{1 - \bar{\alpha}_t} \cdot \epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$

where $H_0$ is the normalized ground-truth pose and $\bar{\alpha}_t$ is the cumulative noise schedule.

Using the monocular-normalized pose (denoted as $\Delta H_0$) ensures that noise does not cause the robot to vanish from the field-of-view. Consequently, training samples exhibit realistic within-view disturbances, vital for downstream inference robustness.

3. Reverse Diffusion: Denoising and Refinement

Inference involves iterative “denoising” of input poses via the reverse diffusion process:

• Timestep Conditioning: Each reverse step $t$ is associated with transformation scale, embedded as a sinusoidal position encoding. Early timesteps effect coarser adjustments; later timesteps make fine refinements, mitigating premature convergence and enabling scheduled correction.
• DDIM-Based Reverse Sampling: The reverse update in normalized pose space is given by

$$H_{t-1} = \sqrt{\bar{\alpha}_{t-1}} \cdot \tilde{H}_0 + \sqrt{1 - \bar{\alpha}_{t-1} - \sigma_t^2} \cdot \varepsilon_t$$

where $\tilde{H}_0$ is the network’s current (denoised) prediction, and $\varepsilon_t$ is the predicted noise.

• Rendering-Based Pose Denoiser: At each step, the network renders the robot at the estimated pose and compares it to the observed image (cropped appropriately), outputting displacement, rotation correction (in 6D), and depth update. The integration formula (Equation 10 in the paper) updates the components:
  • $$t_0^{(xy)} = \frac{v_{xy} \cdot t_t^z / f + t_t^{(xy)}}{t_t^z}$$
  • $R_0 = \Delta R \cdot R_t$
  • $t_0^z = v_z \cdot t_t^z$

This multicomponent update ensures simultaneous refinement of spatial and rotational pose terms, adhering to visibility constraints.

4. Visibility Constraints and Pose Representation

Critical to MonoSE(3)-Diffusion is the explicit integration of visibility constraints at all stages:

• Rotation-Induced Translation Decoupling: Rotational perturbations are applied at the robot centroid to avoid undesirable translation effects that could push the robot projection out of view.
• Translation Normalization: By expressing translation in image-plane and depth-aligned terms, and normalizing via camera intrinsics, spatial changes in latent space correspond to in-view variations in the scene.
• Monocular-Normalized Formulation and Inverse: Normalization and denormalization equations (see Equations 5–7 in the paper) guarantee both diffusion samples and reverse-step refinements map to physically plausible within-frustum poses in the camera setting.

This ensures the forward and reverse processes consistently respect the image formation geometry, avoiding creation or selection of poses with missing or occluded robot regions.

5. Benchmarking and Empirical Results

Performance is validated on established camera-to-robot pose estimation benchmarks:

Benchmark	AUC Achieved	State-of-the-Art Gain
AzureKinect-Franka	66.75	32.3%
DREAM	Improved	(Refer to paper)
RealSense-Franka	Improved	(Refer to paper)

• On the AzureKinect-Franka dataset, the framework reports an AUC of 66.75, corresponding to a 32.3% improvement over prior state-of-the-art methods.
• Ablation studies indicate substantial benefits from both visibility-constrained diffusion (enabling diverse in-view training samples) and the timestep-aware reverse procedure (addressing premature convergence).
• The pose denoising network loss is defined as the distance between transformed 3D keypoints of robot models (ADD metric), directly guiding corrections in both rotation and translation.

These results suggest a robust and generalizable solution for monocular robot localization, particularly in challenging and variable scenarios.

6. Mathematical Formulations and Loss Function

MonoSE(3)-Diffusion adopts rigorous mathematical structures:

• Pose Representation: Utilizes $H_0 = [R_0\ t_0;\ 0^\top\ 1]$ as a $4 \times 4$ homogeneous transformation. The normalized form:

$$\tilde{H}_0 = (r_0^1, r_0^2, \ldots, \frac{f \cdot t_0^x}{w}, \frac{f \cdot t_0^y}{h}, t_0^z - c_z)$$

with $c_z$ a depth normalization constant.

• Diffusion Mechanics: Euclidean diffusion is applied to normalized pose vectors, maintaining isotropic noise distribution in latent space while conforming to the visibility priors post-denormalization.
• DDIM Reverse Sampling: Reverse updates depend on network predictions and stepwise schedule, delivering both computational efficiency and systematic refinement.
• Loss Computation: The principal loss operates on 3D model keypoints, comparing their locations under ground-truth and predicted transforms, using the ADD metric:

$$\text{ADD} = \frac{1}{N} \sum_{i=1}^N \left\| H_{gt} p_i - H_{pred} p_i \right\|$$

This guides the network towards physically correct pose estimation.

7. Implications and Further Directions

MonoSE(3)-Diffusion demonstrates that embedding visibility priors in a denoising diffusion framework for SE(3) significantly benefits monocular pose estimation, particularly in scenarios demanding diverse, robust predictions under geometric constraints. The two-stage structure—diffusion for in-view augmentation and reverse process for iterative refinement—provides a paradigm for future work exploring conditional geometric priors in stochastic generative models for spatial tasks. A plausible implication is the suitability of visibility-constrained diffusion in other computer vision applications where scene geometry and object visibility are critical.