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 $\epsilon >0$, and our goal is to report all pairs $(p,s)\in P\times S$ 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 $\alpha >1$. 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.

当点和某个几何对象（线，圆，平面，球体）之间的距离最大为ε时，就会发生ε近似入射。给定一组点和一组对象，计算它们之间的近似发生率是计算机视觉和图形中许多数据库和基于Web的应用程序中的主要步骤，其中包括健壮的模型拟合，近似点模式匹配和估计基本矩阵在对极（立体）几何中。

在这种典型的近似入射问题中，我们在二维或三维中获得了一组P个m个点，一组S个n个对象（线，圆，平面，球体）和一个误差参数$\epsilon >0$，我们的目标是报告所有对 $（p，s）\in P\times \mathrm{小号}$彼此之间的距离最大为ε。我们针对许多情况提供了有效的输出敏感近似算法，包括平面中的点和线或圆，以及三维中的点和平面，球体，线或圆。在上面提到的应用程序中会出现其中几种情况。我们的算法报告距离≤ε的所有对，但也可能报告另外的对，对于某些与问题相关的常数，所有对都保证在最大距离αε处$\alpha >\mathrm{1个}$。我们的算法基于简单的原始和对偶网格分解，并且易于实现。我们注意到（a）对偶性的使用，导致算法开销成本的显着改善，对于这种问题似乎是新颖的；（b）在这些问题中，对偶性的正确选择相当复杂，需要谨慎对待；（c）算法的正确性和性能分析（尤其是在更高级的版本中）相当重要。我们分析了算法，并证明了算法运行时间和“失真”参数α的有保证的上限。