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.
Highlights
With the goal of obtaining algorithms that are really simple to implement, and that run in time linear in the input and output sizes, we adopt the approach of using approximation schemes, in which we still report all the pairs (s1, s2) that satisfy dist(s1, s2) ≤ ε, but are willing to report additional pairs, provided that all pairs that we report satisfy dist(s1, s2) ≤ αε, for some constant problem-dependent parameter α > 1
In this paper we address this issue, and develop a series of “primal-dual” grid-based algorithms for several approximate incidence reporting problems, that are faster than this naive scheme for suitable ranges of the parameters m, n, and ε
In a more recent work, Aiger and Kedem [2] describe another algorithm for computing approximate incidences of points and circles, following a similar approach by Fonseca and Mount [7] for points and lines, which is better than the one of [10] for n = Ω(1/ε3/2), and use this for approximate point pattern matching
Summary
In a more recent work, Aiger and Kedem [2] describe another algorithm for computing approximate incidences of points and circles, following a similar approach by Fonseca and Mount [7] for points and lines, which is better than the one of [10] for n = Ω(1/ε3/2), and use this for approximate point pattern matching This algorithm has been used in Mellado et al [14], to reduce the running time of the 4PCS algorithm in [3] to be asymptotically linear in n and in the output size. Due to lack of space, this version contains full details of only the first algorithm (for points and lines in the plane), and of the last one (finding all nearly congruent triangles in R3), and describes, briefly and informally, the main features of the rest
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