Camera Extrinsic Denoising Process

• Camera extrinsic denoising is a method that iteratively refines the spatial alignment between camera and LiDAR by operating in the Lie algebra space of SE(3).
• It leverages calibration networks as surrogate denoisers to progressively correct initial pose estimates, leading to improved RMSE, robustness, and stability metrics.
• This process enhances sensor fusion accuracy in autonomous systems by reducing calibration errors and enabling precise multi-sensor alignment for applications such as 3D object detection and SLAM.

Camera extrinsic denoising is a process for iteratively refining the estimated spatial relationship (extrinsic parameters) between cameras and other sensors, primarily LiDAR, using a surrogate diffusion methodology. This procedure operates in the Lie algebra space representing SE(3) transformations, employing existing calibration networks as surrogate denoisers to progressively correct the pose estimate until it converges toward the ground truth. The approach enhances sensor fusion accuracy for perception tasks in autonomous systems, offering improved error metrics, robustness, and stability compared to prior single-step or simple iterative calibration methods.

1. Mathematical Foundations of Extrinsic Denoising

The camera extrinsic denoising process addresses the estimation of the rigid body transformation $T_{CL} \in SE(3)$ between camera and LiDAR. Let $T_{CL}^{(0)}$ be the initial extrinsic and $T_{CL}^{(gt)}$ the ground truth. The difference is represented in Lie algebra space as:

$$x_0 = \mathcal{G}^{-1}(T_{CL}^{(gt)} \cdot (T_{CL}^{(0)})^{-1})$$

where $\mathcal{G}$ and its inverse map between SE(3) and $\mathfrak{se}(3)$.

A forward diffusion process generates noisy states via linear interpolation:

$$x_t = \sqrt{\bar{\alpha}_t} \, x_0 + \sqrt{1-\bar{\alpha}_t} \, \varepsilon$$

with $\varepsilon=0$ so $x_T=0$ and thus $\mathcal{G}(x_T)T_{CL}^{(0)} = T_{CL}^{(0)}$. The reverse (“denoising”) process seeks to recover $x_0$ from $x_T$ through iterative application of a surrogate denoiser using learned calibration networks.

2. Surrogate Diffusion Framework

The surrogate diffusion framework is agnostic to the choice of calibration model. At each reverse step, the surrogate denoiser receives the current noisy extrinsic $\mathcal{G}(x_t)T_{CL}^{(0)}$ and associated sensor data $C = [I, P, K]$, where $I$ is the image, $P$ the point cloud, and $K$ the camera intrinsic matrix. This calibration method, $D_\theta$, is repurposed as a denoiser:

$$\hat{x}_0 = \mathcal{G}^{-1} \left( \mathcal{G}(D_\theta(C, \mathcal{G}(x_t)T_{CL}^{(0)})) \cdot \mathcal{G}(x_t) \right)$$

The updated extrinsic correction is computed as:

$$\hat{T}_{CL}^{(gt)} = \mathcal{G}(\hat{x}_0)T_{CL}^{(0)}$$

The denoising step follows a deterministic process analogous to diffusion models, with the update:

$$x_{t-1} = \mu_\theta(x_t, \hat{x}_0, t) + \Sigma(t) \cdot \varepsilon$$

Given $\varepsilon=0$, the updates proceed via a linear combination in $\mathfrak{se}(3)$.

3. Comparative Evaluation Methodology

The efficacy of surrogate diffusion for extrinsic denoising is evaluated using state-of-the-art calibration networks: CalibNet, RGGNet, LCCNet, and LCCRAFT. These models, when embedded in the linear surrogate diffusion (LSD) framework, are benchmarked against two iterative baselines—NaIter (naive iteration) and NLSD (nonlinear surrogate diffusion) adapted from point cloud registration literature.

Key SE(3)-domain metrics employed are:

Metric	Description	Thresholds/Formula
RMSE	Root mean squared error for rotation/translation	Euler/translation RMSE
Robustness	$\%$ samples below error thresholds	(#1{3}{3}, #1{5}{5}: 3°/3cm, 5°/5cm)
Stability	Monotonic error decrease across iterations	$\rho\%$, RMSE$_2 \ge$ RMSE$_5 \ge$ RMSE$_{10}$

The transformation error is:

$$\epsilon_{T} = \hat{T}_{CL}^{(gt)} \cdot (T_{CL}^{(gt)})^{-1}$$

4. Experimental Results and Findings

Evaluation on the KITTI Odometry dataset demonstrates that LSD yields lower median and variance for rotation and translation RMSE compared to both single-step prediction and baseline iterative approaches. Robustness metrics also increase: the proportion of samples achieving error under 3°/3cm and 5°/5cm thresholds is highest for LSD across all denoisers (per Table I of the source). Stability, assessed via the monotonic error decrease metric $\rho\%$, is likewise superior in LSD, with consistent improvement over successive steps, as reflected in error curves and box plots (cf. Fig. 4 and 5 in the source).

The process optimizes the diffused Lie algebra error with the loss:

$$\mathcal{L}_{LSD}(\hat{x}_0, x_0) = ||\hat{x}_0 - x_0||_1$$

This deterministic denoising procedure leads to more stable and accurate calibration convergence.

5. Functional Implications and Applications

The camera extrinsic denoising process has direct implications for systems requiring precision multi-sensor fusion, notably autonomous vehicles. It improves calibration accuracy, directly benefiting perception tasks such as 3D object detection, SLAM, and scene flow estimation. The iterative surrogate diffusion reduces error and increases robustness, contributing to safer navigation and better environmental understanding. Further, the model-agnostic nature of the denoising process suggests potential application in robot navigation, UAV sensor alignment, and other multimodal systems demanding robust cross-sensor calibration.

A plausible implication is that surrogate diffusion methods may be generalized to other sensor pairs and modalities by operating within the appropriate transformation Lie algebra, potentially extending beyond rigid registration to deformable or time-varying extrinsics.

6. Limitations and Prospects

No empirical evidence in the primary source addresses real-time performance constraints or resource requirements for LSD in production environments. The approach is shown to enhance calibration models without architectural changes but is evaluated under the deterministic scenario $\varepsilon = 0$; stochastic variants are unexamined. Possible future directions include adaptive diffusion scheduling, automatic selection of surrogate denoisers, and extensions to dynamic or time-varying sensor configurations. The reported findings establish surrogate diffusion as an effective paradigm for camera extrinsic denoising, but full scalability and deployment in safety-critical or highly dynamic contexts remain open for investigation.