Output sensitive algorithms for approximate incidences and their applications

TL;DR

This paper introduces efficient output-sensitive algorithms for approximate incidences between points and geometric objects like lines, circles, planes, and spheres, which are crucial in computer vision and graphics applications.

Contribution

It presents novel output-sensitive approximation algorithms for computing approximate incidences between points and various geometric objects in 2D and 3D.

Findings

Algorithms are efficient and output-sensitive.

Applicable to points and lines, circles, planes, spheres.

Useful in computer vision and graphics applications.

Abstract

An $\epsilon$-approximate incidence between a point and some geometric object (line, circle, plane, sphere) occurs when the point and the object lie at distance at most $\epsilon$ from each other. Given a set of points and a set of objects, computing the approximate incidences between them is a major step in many database and web-based applications in computer vision and graphics, including robust model fitting, approximate point pattern matching, and estimating the fundamental matrix in epipolar (stereo) geometry. In a typical approximate incidence problem of this sort, we are given a set $P$ of $m$ points in two or three dimensions, a set $S$ of $n$ objects (lines, circles, planes, spheres), and an error parameter $\epsilon>0$, and our goal is to report all pairs $(p,s)\in P\times S$ that lie at distance at most $\epsilon$ from one another. We present efficient output-sensitive approximation algorithms for quite a few cases, including points and lines or circles in the plane, and points and planes, spheres, lines, or circles in three dimensions. Several of these cases arise in the applications mentioned above.