Geometric and algebraic properties of point-to-line mappings

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 LMN 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 properties of these operations.