ProBA: Probabilistic Bundle Adjustment with the Bhattacharyya Coefficient

TL;DR

ProBA addresses the challenge of performing bundle adjustment without reliable initialization or known camera intrinsics by modeling both image observations and 3D structure as probabilistic Gaussians. It introduces a probabilistic reprojection loss and a Bhattacharyya-coefficient-based overlap term to enforce geometric consistency, yielding a single, robust objective that broadens the convergence basin. The method outperforms classical BA and prior initialization-free approaches on diverse datasets (DTU, LLFF, Replica, NeRF-360v2), improving accuracy and uncertainty calibration. By removing the need for strong priors, ProBA enhances the practicality of SLAM systems in unstructured environments and provides a foundation for further probabilistic, uncertainty-aware refinements in 3D reconstruction.

Abstract

Classical Bundle Adjustment (BA) methods require accurate initial estimates for convergence and typically assume known camera intrinsics, which limits their applicability when such information is uncertain or unavailable. We propose a novel probabilistic formulation of BA (ProBA) that explicitly models and propagates uncertainty in both the 2D observations and the 3D scene structure, enabling optimization without any prior knowledge of camera poses or focal length. Our method uses 3D Gaussians instead of point-like landmarks and we introduce uncertainty-aware reprojection losses by projecting the 3D Gaussians onto the 2D image space, and enforce geometric consistency across multiple 3D Gaussians using the Bhattacharyya coefficient to encourage overlap between their corresponding Gaussian distributions. This probabilistic framework leads to more robust and reliable optimization, even in the presence of outliers in the correspondence set, reducing the likelihood of converging to poor local minima. Experimental results show that \textit{ProBA} outperforms traditional methods in challenging real-world conditions. By removing the need for strong initialization and known intrinsics, ProBA enhances the practicality of SLAM systems deployed in unstructured environments.

Figures (14)

• Figure 1: We introduce ProBA as a probabilistic formulation of bundle adjustment. The key idea is to model the distribution of reconstructed 3D points as isotropic Gaussian distributions and enforce geometrid consistency across frames by means of the Bhattacharyya coefficient. This reduces the nonlinearity of the problem and entails a larger basin of convergence, leading to a significant boost in accuracy -- see Fig. \ref{['fig:conver']}. ProBA can work with any number of camera frames -- the above examples show reconstructions from 100 frames (a) and 2 frames (b). Estimated camera poses are shown in color whereas ground truth poses are shown in gray.
• Figure 2: Visualization of the Bhattacharyya coefficient. Left: Two overlapping 2D Gaussians separated with a distance $d$. Right: The Bhattacharyya coefficient of the distance $d$ between Gaussian centers.
• Figure 3: The comparison of performance across different datasets. These charts show the performance of each method on individual datasets. The x-axis lists the dataset names, and the bars represent the final evaluation metrics, enabling a comparison of method robustness across varying scenarios. ProBA consistently performs the best among all baseline methods.
• Figure 4: The Comparison of performance across varying numbers of input frames. The charts show the performance of each method when using different numbers of frames (from 2 to 10). The x-axis indicates the number of input frames, illustrating how each method scales with more observations. ProBA can still obtain good pose estimation despite larger uncertainty in the focal length when using a smaller number of frames.
• Figure 5: Convergence plots. Each plot shows the mean performance of different methods over training iterations across various metrics, illustrating convergence speed and stability. ProBA-1 converges faster and reaches the lowest error.

Theorems & Definitions (2)

• Proposition 3.1
• proof