Abstract
An epsilonapproximate 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 webbased 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 outputsensitive 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. Our algorithms report all pairs at distance <= epsilon, but may also report additional pairs, all of which are guaranteed to be at distance at most alphaepsilon, for some constant alpha>1. Our algorithms are based on simple primal and dual grid decompositions and are easy to implement. We note though that (a) the use of duality, which leads to significant improvements in the overhead cost of the algorithms, appears to be novel for this kind of problems; (b) the correct choice of duality in some of these problems is fairly intricate and requires some care; and (c) the correctness and performance analysis of the algorithms (especially in the more advanced versions) is fairly nontrivial.
We analyze our algorithms and prove guaranteed upper bounds on their running time and on the "distortion" parameter alpha. We also briefly describe the motivating applications, and show how they can effectively exploit our solutions. The superior theoretical bounds on the performance of our algorithms, and their simplicity, make them indeed ideal tools for these applications. In a series of preliminary experimentations (not included in this abstract), we substantiate this feeling, and show that our algorithms lead in practice to significant improved performance of the aforementioned applications.
BibTeX  Entry
@InProceedings{aiger_et_al:LIPIcs:2017:7822,
author = {Dror Aiger and Haim Kaplan and Micha Sharir},
title = {{Output Sensitive Algorithms for Approximate Incidences and Their Applications}},
booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)},
pages = {5:15:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770491},
ISSN = {18688969},
year = {2017},
volume = {87},
editor = {Kirk Pruhs and Christian Sohler},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7822},
URN = {urn:nbn:de:0030drops78224},
doi = {10.4230/LIPIcs.ESA.2017.5},
annote = {Keywords: Approximate incidences, nearneighbor reporting, duality, gridbased approximation}
}
Keywords: 

Approximate incidences, nearneighbor reporting, duality, gridbased approximation 
Collection: 

25th Annual European Symposium on Algorithms (ESA 2017) 
Issue Date: 

2017 
Date of publication: 

01.09.2017 