# The Alignment of the Spheres: Globally-Optimal Spherical Mixture Alignment for Camera Pose Estimation

arXiv: https://arxiv.org/abs/1812.01232 · HTML: https://mlsanity.com/paper/1812.01232/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2018.

## Built on by (most cited first, 20 of 20)

- [Robust Point Cloud Registration Framework Based on Deep Graph Matching](https://mlsanity.com/paper/2103.04256/) (2021)
- [Wide-Baseline Relative Camera Pose Estimation with Directional Learning](https://mlsanity.com/paper/2106.03336/) (2021)
- [Is Geometry Enough for Matching in Visual Localization?](https://mlsanity.com/paper/2203.12979/) (2022)
- [Globally-Optimal Contrast Maximisation for Event Cameras](https://mlsanity.com/paper/2206.05127/) (2022)
- [FreeReg: Image-to-Point Cloud Registration Leveraging Pretrained Diffusion Models and Monocular Depth Estimators](https://mlsanity.com/paper/2310.03420/) (2023)
- [Line-based 6-DoF Object Pose Estimation and Tracking With an Event Camera](https://mlsanity.com/paper/2408.03225/) (2024)
- [PICCOLO: Point Cloud-Centric Omnidirectional Localization](https://mlsanity.com/paper/2108.06545/) (2021)
- [Accelerating Globally Optimal Consensus Maximization in Geometric Vision](https://mlsanity.com/paper/2304.05156/) (2023)
- [Advances in Global Solvers for 3D Vision](https://mlsanity.com/paper/2602.14662/) (2026)
- [Fully Geometric Panoramic Localization](https://mlsanity.com/paper/2403.19904/) (2024)
- [Self-Supervised Point Cloud Registration with Deep Versatile Descriptors](https://mlsanity.com/paper/2201.10034/) (2022)
- [Coarse-to-Fine Point Cloud Registration with SE(3)-Equivariant Representations](https://mlsanity.com/paper/2210.02045/) (2022)
- [Full Transformer Framework for Robust Point Cloud Registration with Deep Information Interaction](https://mlsanity.com/paper/2112.09385/) (2021)
- [Efficient Globally-Optimal Correspondence-Less Visual Odometry for Planar Ground Vehicles](https://mlsanity.com/paper/2203.00291/) (2022)
- [Self-Supervised Domain Adaptation for Visual Navigation with Global Map Consistency](https://mlsanity.com/paper/2110.07184/) (2021)
- [CPO: Change Robust Panorama to Point Cloud Localization](https://mlsanity.com/paper/2207.05317/) (2022)
- [LDL: Line Distance Functions for Panoramic Localization](https://mlsanity.com/paper/2308.13989/) (2023)
- [Robust Point Cloud Registration Framework Based on Deep Graph Matching(TPAMI Version)](https://mlsanity.com/paper/2211.04696/) (2022)
- [Unsupervised Discovery of Continuous Skills on a Sphere](https://mlsanity.com/paper/2305.14377/) (2023)
- [Global Truncated Loss Minimization for Robust and Threshold-Resilient Geometric Estimation](https://mlsanity.com/paper/2603.14796/) (2026)

## Abstract (arXiv)

Determining the position and orientation of a calibrated camera from a single image with respect to a 3D model is an essential task for many applications. When 2D-3D correspondences can be obtained reliably, perspective-n-point solvers can be used to recover the camera pose. However, without the pose it is non-trivial to find cross-modality correspondences between 2D images and 3D models, particularly when the latter only contains geometric information. Consequently, the problem becomes one of estimating pose and correspondences jointly. Since outliers and local optima are so prevalent, robust objective functions and global search strategies are desirable. Hence, we cast the problem as a 2D-3D mixture model alignment task and propose the first globally-optimal solution to this formulation under the robust L2 distance between mixture distributions. We search the 6D camera pose space using branch-and-bound, which requires novel bounds, to obviate the need for a pose estimate and guarantee global optimality. To accelerate convergence, we integrate local optimization, implement GPU bound computations, and provide an intuitive way to incorporate side information such as semantic labels. The algorithm is evaluated on challenging synthetic and real datasets, outperforming existing approaches and reliably converging to the global optimum.