# From a few Accurate 2D Correspondences to 3D Point Clouds

arXiv: https://arxiv.org/abs/2206.08749 · HTML: https://mlsanity.com/paper/2206.08749/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 56 references that are in the map.
- First posted on arXiv 13 Jun 2022.

## Builds on (most cited first, 6 of 6)

- [Past, Present, and Future of Simultaneous Localization And Mapping: Towards the Robust-Perception Age](https://mlsanity.com/paper/1606.05830/) (2016)
- [PointTriNet: Learned Triangulation of 3D Point Sets](https://mlsanity.com/paper/2005.02138/) (2020)
- [A General Multi-Graph Matching Approach via Graduated Consistency-regularized Boosting](https://mlsanity.com/paper/1502.05840/) (2015)
- [Five-point Fundamental Matrix Estimation for Uncalibrated Cameras](https://mlsanity.com/paper/1803.00260/) (2018)
- [Free-boundary conformal parameterization of point clouds](https://mlsanity.com/paper/2010.15399/) (2020)
- [Multi-view Geometry: Correspondences Refinement Based on Algebraic Properties](https://mlsanity.com/paper/2205.01634/) (2022)

## Abstract (arXiv)

Key points, correspondences, projection matrices, point clouds and dense clouds are the skeletons in image-based 3D reconstruction, of which point clouds have the important role in generating a realistic and natural model for a 3D reconstructed object. To achieve a good 3D reconstruction, the point clouds must be almost everywhere in the surface of the object. In this article, with a main purpose to build the point clouds covering the entire surface of the object, we propose a new feature named a geodesic feature or geo-feature. Based on the new geo-feature, if there are several (given) initial world points on the object's surface along with all accurately estimated projection matrices, some new world points on the geodesics connecting any two of these given world points will be reconstructed. Then the regions on the surface bordering by these initial world points will be covered by the point clouds. Thus, if the initial world points are around the surface, the point clouds will cover the entire surface. This article proposes a new method to estimate the world points and projection matrices from their correspondences. This method derives the closed-form and iterative solutions for the world points and projection matrices and proves that when the number of world points is less than seven and the number of images is at least five, the proposed solutions are global optimal. We propose an algorithm named World points from their Correspondences (WPfC) to estimate the world points and projection matrices from their correspondences, and another algorithm named Creating Point Clouds (CrPC) to create the point clouds from the world points and projection matrices given by the first algorithm.