# General techniques for approximate incidences and their application to the camera posing problem

## TL;DR

This paper introduces and analyzes three techniques for solving approximate incidences problems, applying them to camera pose estimation in computer vision, and demonstrates their effectiveness through experiments on real and synthetic data.

## Contribution

The paper presents a novel reduction of camera pose estimation to approximate incidences and introduces three techniques, including a generalized data structure, for solving this problem efficiently.

## Key findings

- Primal-dual method is the fastest for typical data sizes.
- The reduction to approximate incidences applies to various camera pose problems.
- Experimental results validate the efficiency of the proposed methods.

## Abstract

We consider the classical camera pose estimation problem that arises in many computer vision applications, in which we are given n 2D-3D correspondences between points in the scene and points in the camera image (some of which are incorrect associations), and where we aim to determine the camera pose (the position and orientation of the camera in the scene) from this data. We demonstrate that this posing problem can be reduced to the problem of computing {\epsilon}-approximate incidences between two-dimensional surfaces (derived from the input correspondences) and points (on a grid) in a four-dimensional pose space. Similar reductions can be applied to other camera pose problems, as well as to similar problems in related application areas. We describe and analyze three techniques for solving the resulting {\epsilon}-approximate incidences problem in the context of our camera posing application. The first is a straightforward assignment of surfaces to the cells of a grid (of side-length {\epsilon}) that they intersect. The second is a variant of a primal-dual technique, recently introduced by a subset of the authors [2] for different (and simpler) applications. The third is a non-trivial generalization of a data structure Fonseca and Mount [3], originally designed for the case of hyperplanes. We present and analyze this technique in full generality, and then apply it to the camera posing problem at hand. We compare our methods experimentally on real and synthetic data. Our experiments show that for the typical values of n and {\epsilon}, the primal-dual method is the fastest, also in practice.