TY - JOUR

T1 - Output sensitive algorithms for approximate incidences and their applications

AU - Aiger, Dror

AU - Kaplan, Haim

AU - Sharir, Micha

N1 - Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2020/12

Y1 - 2020/12

N2 - An ε-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 ε 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 ε>0, and our goal is to report all pairs (p,s)∈P×S that lie at distance at most ε 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. Our algorithms report all pairs at distance ≤ε, but may also report additional pairs, all of which are guaranteed to be at distance at most αε, for some problem-dependent constant α>1. Our algorithms are based on simple primal and dual grid decompositions and are easy to implement. We note 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 non-trivial. We analyze our algorithms and prove guaranteed upper bounds on their running time and on the “distortion” parameter α.

AB - An ε-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 ε 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 ε>0, and our goal is to report all pairs (p,s)∈P×S that lie at distance at most ε 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. Our algorithms report all pairs at distance ≤ε, but may also report additional pairs, all of which are guaranteed to be at distance at most αε, for some problem-dependent constant α>1. Our algorithms are based on simple primal and dual grid decompositions and are easy to implement. We note 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 non-trivial. We analyze our algorithms and prove guaranteed upper bounds on their running time and on the “distortion” parameter α.

KW - Approximate incidences

KW - Data structures

KW - Model fitting

KW - Pattern matching

KW - Primal-dual grids

UR - http://www.scopus.com/inward/record.url?scp=85085274029&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2020.101666

DO - 10.1016/j.comgeo.2020.101666

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AN - SCOPUS:85085274029

SN - 0925-7721

VL - 91

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

M1 - 101666

ER -