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Geometric and algebraic properties of point-to-line mappings

机译:点到线映射的几何和代数性质

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摘要

Point-to-line mappings (PTLMs) have several uses in image analysis and computer vision; a linear PTLM was used by Hough to detect sets of collinear points in an image. and it can be shown that three lines L,M,N in the plane are the images of three mutually perpendicular lines in space iff there exists a PTLM that maps the vertices of triangle into their opposite sides. This paper discusses a variety of mathematical properties of PTLMs. It begins by reviewing some facts about linear PTLMs, with emphasis on their point-line incidence properties, and discusses canonical forms for the matrices of such PTLMs. It then shows that any PTLM that has an incidence-symmetry property must be linear and must have a symmetric matrix. It also discusses PTLMs of polygons, and shows how to construct polygons whose vertices are mapped into their sides by a PTLM. Finally, it shows how a PTLM can be used to define binary operations on points. and discuss algebraic proper-ties of these operations. (C) 2002 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. [References: 44]
机译:点对线映射(PTLM)在图像分析和计算机视觉中有多种用途。 Hough使用线性PTLM来检测图像中的共线点集。可以看出,平面中的三条线L,M,N是空间中三条相互垂直的线的图像,如果存在一个PTLM,它将三角形的顶点映射到它们的相对侧。本文讨论了PTLM的各种数学特性。首先回顾一些有关线性PTLM的事实,重点是它们的点线入射特性,然后讨论此类PTLM矩阵的规范形式。然后,它表明具有入射对称性的任何PTLM必须是线性的,并且必须具有对称矩阵。它还讨论了多边形的PTLM,并说明了如何构造其顶点通过PTLM映射到其侧面的多边形。最后,它显示了如何使用PTLM定义点上的二进制操作。并讨论这些运算的代数性质。 (C)2002模式识别学会。由Elsevier Science Ltd.出版。保留所有权利。 [参考:44]

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