Output sensitive algorithms for approximate incidences and their applications
Abstract
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.
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.
Research Areas
