Geometric and algebraic properties of point-to-line mappings

✍ Scribed by Prabir Bhattacharya; Azriel Rosenfeld; Isaac Weiss

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 266 KB
• Volume: 36
• Category: Article
• ISSN: 0031-3203
• DOI: 10.1016/s0031-3203(02)00079-1

✦ Synopsis

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 i 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 deÿne binary operations on points, and discuss algebraic properties of these operations.