Point-to-Line Mappings and Hough Transforms

Nearly 40 years ago Hough showed how a point-to-line mapping that takes collinear points into concurrent lines can be used to detect collinear sets of points, since such points give rise to peaks where the corresponding lines intersect. Over the past 30 years many variations and generalizations of Hough's idea have been proposed, Hough's mapping was linear, but most or all of the mappings studied since then have been nonlinear, and take collinear points into concurrent curves rather than concurrent lines; little or no work has appeared in the pattern recognition literature on mappings that take points into lines.This paper deals with point-to-line mappings in the real projective plane. (We work in the projective plane to avoid the need to deal with special cases in which collinear points are mapped into parallel, rather than concurrent, lines.) We review basic properties of linear point-to-point mappings (collineations) and point-to-line mappings (correlations), and show that any one-to-one point-to-line mapping that takes collinear points into concurrent lines must in fact be linear. We describe ways in which the matrices of such mappings can be put into canonical form, and show that Hough's mapping is only one of a large class of inequivalent mappings. We show that any one-to-one point-to-line mapping that has an incidence-symmetry property must be linear and must have a symmetric matrix which has a diagonal canonical form. We establish useful geometric properties of such mappings, especially in cases where their matrices define nonempty conics.