Elsevier

Pattern Recognition

Volume 36, Issue 2, February 2003, Pages 483-503
Pattern Recognition

Geometric and algebraic properties of point-to-line mappings

https://doi.org/10.1016/S0031-3203(02)00079-1Get rights and content

Abstract

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.

Introduction

Linear point-to-line mappings (PTLMs) are studied in projective geometry, where they are called correlations. In 1962, Hough [1] used a correlation to detect collinear sets of points in an image, but it was pointed out in 1972 by Duda and Hart [2] that Hough's method was not practical, and the authors have recently shown [3] that the same objection must apply to any PTLM-based method of detecting collinear points. But correlations have other uses in the geometric analysis of images; for example, it can be shown [4] that three lines L,M,N in the plane are the images of three mutually perpendicular lines in space iff there exists a correlation that maps the vertices of triangle LMN into their opposite sides.

This paper discusses a variety of mathematical properties of PTLMs; some of them can be found in the geometry literature, but most of them are new. Section 2 reviews some basic facts about correlations, particularly about their point-line incidence properties, and Section 3 discusses canonical forms for the matrices of correlations, with emphasis on correlations defined on the real projective plane. Section 4 discusses arbitrary PTLMs, and proves that if a PTLM has an incidence-symmetry property, it must be linear. Section 5 and the appendix discuss PTLMs of polygons; in particular, we show how to construct polygons whose vertices are mapped into their sides by an incidence-symmetric PTLM. Section 6 shows how a PTLM can be used to define binary operations on points, and how algebraic properties of the operations can be derived from geometric properties of the PTLM. Finally, Section 7 discusses possible generalizations of our results.

Section snippets

Properties of correlations

Points in the projective plane are defined by triples x=[x0,x1,x2]T of homogeneous coordinates, where x and y define the same point if the y's are proportional to the x's. Lines in the projective plane are defined by triples u=[u0,u1,u2]T of coefficients; here again, proportional triples define the same line. The point x lies on the line u iff xTu=0, where T denotes the transpose.

Canonical forms for the matrices of correlations

In this section we discuss how the matrix of a correlation can be put into a “canonical form”—specifically, with respect to an equivalence relation on matrices called “congruence”.

Let f,f′ be correlations and let A,A be the matrices of f,f′, respectively. We call f and f′ (or A and A) congruent if there exists a nonsingular matrix D such that A′=DTAD. We can regard D as a matrix of a linear point-to-point mapping (a collineation) and DT as a matrix of a line-to-line collineation; the effect

General PTLMs

In this section we consider (not necessarily linear) PTLMs that have the incidence-symmetry and incidence-preserving properties described in Section 2.1, and we discuss conditions under which these properties imply linearity.

Let f be a PTLM defined on the (real) projective plane. We call f incidence-symmetric if Qf(P) implies Pf(Q). Similarly, if g is a line-to-point mapping (LTPM) we call g incidence-symmetric if g(L)∈M implies g(M)∈L. As we saw in the corollary to Proposition 4, if f is a

PTLMs of polygons

The polygons (more precisely: “complete polygons”) usually studied in projective geometry are figures defined by finite sets of points and all the lines determined by pairs of these points, or by finite sets of lines and all the points where pairs of these lines intersect. Thus an “n-angle” is a figure defined by n points and the n(n−1)/2 lines determined by pairs of the points, and an “n-lateral” is a figure defined by n lines and all the n(n−1)/2 points in which pairs of the lines intersect.

Hough algebra: PTLMs and point compositions

In this section we show that PTLMs and LTPMs defined on the (real or complex) projective plane can be used to define point (or line) compositions. We discuss how to derive algebraic properties of the compositions from geometric properties of the mappings and how to construct sets that are closed under the compositions.

Concluding remarks

This paper has made many basic observations about PTLMs. We summarize a few of these observations here:

  • (1)

    A one-to-one PTLM that preserves point-line incidence must be linear, and its matrix must be symmetric, so it must be a polarity. Because of their incidence-symmetry property, invertible polarities are an especially well-behaved class of point-to-line mappings. For example, they take nondegenerate polygons into nondegenerate polylines and take infinitely many nondegenerate triangles into

About the Author—PRABIR BHATTACHARYA is currently working as a Principal Scientist at the Panasonic Information and Networking Technological Laboratories, Princeton, New Jersey, USA. Till recently, he served as a tenured Full Professor at the University of Nebraska-Lincoln, Department of Computer Science and Engineering, USA. He also holds a Visiting Full Professorship at the Center for Automation Research, University of Maryland, College Park. He obtained a D. Phil. from the University of

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    About the Author—PRABIR BHATTACHARYA is currently working as a Principal Scientist at the Panasonic Information and Networking Technological Laboratories, Princeton, New Jersey, USA. Till recently, he served as a tenured Full Professor at the University of Nebraska-Lincoln, Department of Computer Science and Engineering, USA. He also holds a Visiting Full Professorship at the Center for Automation Research, University of Maryland, College Park. He obtained a D. Phil. from the University of Oxford, England in 1979, specializing in group theory. He has authored or co-authored 78 papers in refereed journals, over 50 published papers in conference proceedings, and co-edited a book on Vision Geometry. He is on the editorial boards of the Pattern Recognition, Pattern Recognition Letters, IEEE Transactions on Systems Man and Cybernetics, International Journal of Pattern Recognition and Artificial Intelligence, and Machine Graphics and Vision. During 1995–98, he was on the editorial board of the IEEE Computer Society Press. During 1995–98, he was a Distinguished Visitor of the IEEE Computer Society, and also a National Lecturer of the ACM. In the year 2002 he is on the program committees of six conferences in the areas of image understanding and multimedia. He is a Fellow of the IEEE, and a Fellow of the IMA, UK.

    About the Author—AZRIEL ROSENFELD is a Distinguished University Professor emeritus and the founding Director of the Center for Automation Research at the University of Maryland in College Park. He also held Affiliate Professorships in the Departments of Computer Science, Electrical Engineering, and Psychology. He holds a Ph.D. in Mathematics from Columbia University (1957) as well as several other earned and honorary doctoral degrees. He is a widely known researcher in the field of computer image analysis. He has published over 30 books and over 600 book chapters and journal articles, is an Associate Editor of over 25 journals, and has directed nearly 60 Ph.D. dissertations. He is a Fellow of several professional societies and has won numerous professional society awards.

    About the Author—ISAAC WEISS received his Ph.D. from the Department of Physics and Astronomy of the Tel-Aviv University. He was subsequently a Research Scientist at New York University's Courant Institute of Mathematics (one year) and at the Massachusetts Institute of Technology (two years). He is now a Senior Research Scientist at the Center for Automation Research of the University of Maryland. His current research interests are computer vision, object recognition, pattern recognition and robotics. Specific areas of focus are applications of geometric and physics-based invariance in object recognition, and methods of robust estimation.

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