Duality and Geometry Straightness, Characterization and Envelope

Duality applied to geometrical problems is widely used in many applications in computer vision or computational geometry. A classical example is the Hough Transform to detect linear structures in images. In this paper, we focus on two kinds of duality/polarity applied to geometrical problems: digital straightness detection and envelope computation. Introduction In domain of geometry, notion of duality is often used to represent the same structure in different domains like spatial domain or parametric one. The objective is to facilitate transformations like characterization, detection, recognition or classical ones such as intersection or union. A first example is illustrated with Voronoi partition in which polygonal regions are not homogeneous in terms of number of vertices. Nevertheless, the corresponding dual mesh, called Delaunay mesh, is composed of triangles. According to applications the choice of the alternative representations can be used on optimality criteria (computational cost, database structure, ...). Following this first example, we focus in this paper on dual transformations illustrated by problems of digital straightness and envelope. 1 Example of the Hough Transform The Hough transform (HT for short) is a very classical tool in image analysis to detect geometric features in images. These features may be line segments, circles, ellipses or any other parameterized curve. The HT, introduced in 1962 by Hough [1], is a dual transformation that enables to find a set of global structures, without any a priori knowledge on the number of structures to be found. Note also that this method is robust to noise and disconnected features. 1.1 Definition of Hough transform The general idea of this transform is that every point of the image contributes to the definition of the solution set for a given parameterized structure. Consider for instance a point p0 of coordinates (x0, y0) and the parameterization of lines y = αx + β. Then the set of lines going through p0 are the ones of parameters (α, β) fulfilling the equality y0 = αx0 + β. This equality may be rewritten as β = −αx0 + y0, and if a new geometrical space (αβ), called dual space, or parameter space, is defined, this equation defines a line : in this dual space, each point of this line represents a line of the (xy) space going through the point p0. An illustration of three points and the three corresponding lines in the dual space (αβ) are represented in Figure 1 (a)-(b): note that the three lines in (αβ) space are concurrent in one point, the coordinates of which defines a line going through the three points in (xy) space. However, as noticed by Duda in [2], the linear parameterization of lines defined by y = αx + β is not the handiest one since the two parameters α and β are unbounded. Thus, another transform consists in using the polar parameterization of straight lines ρ = x cos θ+ y sin θ. Any point in the (xy) space defines a sinusoidal curve in the (θρ) space, where only the parameter ρ has unbounded values (see Figure 1(c) for an illustration). General properties fulfilled by these two representations, and suitable for straight line detection in images were expressed by Duda [2]: Property 1. • A point in the (xy) space matches up with one curve in the dual space; • A point in the dual space matches up with a straight line in the (xy) space; • Points lying on a same line in the (xy) space match up with concurrent curves in the dual space; • Points on a same curve in the dual space match up with concurrent straight lines in the (xy) space.