# Perspective-1-Ellipsoid: Formulation, Analysis and Solutions of the Camera Pose Estimation Problem from One Ellipse-Ellipsoid Correspondence

arXiv: https://arxiv.org/abs/2208.12513 · HTML: https://mlsanity.com/paper/2208.12513/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 48 references that are in the map.
- First posted on arXiv 26 Aug 2022.

## Builds on (most cited first, 16 of 16)

- [You Only Look Once: Unified, Real-Time Object Detection](https://mlsanity.com/paper/1506.02640/) (2015)
- [SSD: Single Shot MultiBox Detector](https://mlsanity.com/paper/1512.02325/) (2015)
- [Fast R-CNN](https://mlsanity.com/paper/1504.08083/) (2015)
- [YOLOv3: An Incremental Improvement](https://mlsanity.com/paper/1804.02767/) (2018)
- [Rich feature hierarchies for accurate object detection and semantic segmentation](https://mlsanity.com/paper/1311.2524/) (2013)
- [YOLO9000: Better, Faster, Stronger](https://mlsanity.com/paper/1612.08242/) (2016)
- [QuadricSLAM: Dual Quadrics from Object Detections as Landmarks in Object-oriented SLAM](https://mlsanity.com/paper/1804.04011/) (2018)
- [Satellite Pose Estimation Challenge: Dataset, Competition Design and Results](https://mlsanity.com/paper/1911.02050/) (2019)
- [Semantics for Robotic Mapping, Perception and Interaction: A Survey](https://mlsanity.com/paper/2101.00443/) (2021)
- [Ellipse R-CNN: Learning to Infer Elliptical Object from Clustering and Occlusion](https://mlsanity.com/paper/2001.11584/) (2020)
- [Ellipse Detection and Localization with Applications to Knots in Sawn Lumber Images](https://mlsanity.com/paper/2011.04844/) (2020)
- [Detecting Lesion Bounding Ellipses With Gaussian Proposal Networks](https://mlsanity.com/paper/1902.09658/) (2019)
- [3D-Aware Ellipse Prediction for Object-Based Camera Pose Estimation](https://mlsanity.com/paper/2105.11494/) (2021)
- [Object-oriented SLAM using Quadrics and Symmetry Properties for Indoor Environments](https://mlsanity.com/paper/2004.05303/) (2020)
- [Ellipse Regression with Predicted Uncertainties for Accurate Multi-View 3D Object Estimation](https://mlsanity.com/paper/2101.05212/) (2020)
- [Accurate and Robust Object-oriented SLAM with 3D Quadric Landmark Construction in Outdoor Environment](https://mlsanity.com/paper/2110.08977/) (2021)

## Built on by (most cited first, 5 of 5)

- [3D-Aware Object Localization using Gaussian Implicit Occupancy Function](https://mlsanity.com/paper/2303.02058/) (2023)
- [Projecting Gaussian Ellipsoids While Avoiding Affine Projection Approximation](https://mlsanity.com/paper/2411.07579/) (2024)
- [AI-Enabled Crater-Based Navigation for Lunar Mapping](https://mlsanity.com/paper/2509.20748/) (2025)
- [Globally Optimal Pose from Orthographic Silhouettes](https://mlsanity.com/paper/2604.09199/) (2026)
- [Relative State Estimation using Event-Based Propeller Sensing](https://mlsanity.com/paper/2604.18289/) (2026)

## Abstract (arXiv)

In computer vision, camera pose estimation from correspondences between 3D geometric entities and their projections into the image has been a widely investigated problem. Although most state-of-the-art methods exploit low-level primitives such as points or lines, the emergence of very effective CNN-based object detectors in the recent years has paved the way to the use of higher-level features carrying semantically meaningful information. Pioneering works in that direction have shown that modelling 3D objects by ellipsoids and 2D detections by ellipses offers a convenient manner to link 2D and 3D data. However, the mathematical formalism most often used in the related litterature does not enable to easily distinguish ellipsoids and ellipses from other quadrics and conics, leading to a loss of specificity potentially detrimental in some developments. Moreover, the linearization process of the projection equation creates an over-representation of the camera parameters, also possibly causing an efficiency loss. In this paper, we therefore introduce an ellipsoid-specific theoretical framework and demonstrate its beneficial properties in the context of pose estimation. More precisely, we first show that the proposed formalism enables to reduce the pose estimation problem to a position or orientation-only estimation problem in which the remaining unknowns can be derived in closed-form. Then, we demonstrate that it can be further reduced to a 1 Degree-of-Freedom (1DoF) problem and provide the analytical derivations of the pose as a function of that unique scalar unknown. We illustrate our theoretical considerations by visual examples and include a discussion on the practical aspects. Finally, we release this paper along with the corresponding source code in order to contribute towards more efficient resolutions of ellipsoid-related pose estimation problems.