Old and new straight-line detectors: Description and comparison

Abstract

Straight-line detection is important in several fields such as robotics, remote sensing, and imagery. The objective of this paper is to present several methods, old and new, used for straight-line detection. We begin by reviewing the standard Hough transform (SHT), then three new methods are suggested: the revisited Hough transform (RHT), the parallel-axis transform (PAT), and the circle transform (CT). These transforms utilize a point-line duality to detect straight lines in an image. The RHT and the PAT should be faster than the SHT and the CT because they use line segments whereas the SHT uses sinusoids and CT uses circles. Moreover, the PAT, RHT, and CT use additions and multiplications whereas the SHT uses trigonometric functions (sine and cosine) for calculation. To compare the methods we analyze the distribution of the frequencies in the accumulators and observe the effect on the detection of false local maxima. We also compare the robustness to noise of the four transforms. Finally, an example with a real image is given.

Introduction

The goal of this paper is to propose and analyze old and new methods for detecting straight lines in images, using a point-line duality approach. After a review of the standard Hough transform (SHT), three new methods are proposed. These new methods are: the revisited Hough transform (RHT), which uses the Cartesian equation of a line; the parallel-axis transform (PAT), which is obtained from the parallel-axis coordinate system; and the circle transform (CT), which uses the normal equation of a line.

Point-line duality was introduced by Hough [1] in 1962 to detect straight lines in an image. It was extended by Rosenfeld [2] in 1969 and by Duda and Hart [3] in 1972. They developed the (standard) Hough transform, which has since been a traditional method for straight-line detection in images. It has been used in a number of applications: for road detection in satellite images by Geman and Jedynak [4], for robot localization by Hoppenot et al. [5], for local constraint analysis by Kramer et al. [6], for robust bar-code reading by Muniz et al. [7], and for the detection of boat wakes by Rey et al. [8] and Magli et al. [9]. Since this method use sinusoids, Tuytelaars et al. [10] have recently suggested an extension of the SHT, called the cascade Hough transform (CHT), which uses only straight lines. We also present an extension of the SHT which uses only straight lines and which is simpler than both the CHT and SHT. This method has recently been introduced simultaneously and independently in Refs. [11], [12].

A point-line duality also appears in the parallel-axis coordinates system. A point in ${\mathbb{R}}^n$ is represented by a poly-line in the parallel-axis coordinate system and conversely, a poly-line represents a point in ${\mathbb{R}}^n$. The parallel-axis coordinate system is used for the visualization of multidimensional information. Sets of data in several dimensions are visualized by drawing a poly-line for each point through regularly spaced parallel axes. This technique introduced by Inselberg [13] and Inselberg and Dimsdale [14], [15] has been used for several applications: in robotics by Cohan and Yang [16], in aerospace by Helly [17], in management by Desai and Walers [18], and in aerial navigation by Inselberg [19]. We use the parallel-axis coordinate system to develop a new method of straight-line detection as we did for the RHT [11].

In Section 2, we present and review how the SHT is used to detect the points aligned on a straight line in an image. We give the definition and properties of the SHT and describe its implementation. Section 3 is devoted to the RHT. This transform uses the Cartesian equation of a line directly. We present its definition and properties and also describe its implementation. In Section 4, we present a new method called PAT.

First we introduce the parallel-axis frame of reference proposed by Inselberg [13]. Then we give the definition of the PAT, which uses a parallel-axis representation of the plane and the point-line duality to define a new algorithm for straight-line detection in an image. We present the properties of this transform and describe its implementation. The CT is presented in Section 5. We give the definition of the CT, which uses a point-line duality to define a new algorithm for straight-line detection in an image. It is in fact the Hough transform but with a different parameter space [20]. We also present the properties of this new transform and discuss its implementation. In Section 6 we compare the four methods under four different aspects. We consider the transformation of the white image and the corresponding distributions of the votes in the accumulators. Then we illustrate the robustness to the occurrence of artifacts by detecting parallel lines. The analysis of the robustness of the methods to noise follows. Finally, we apply the methods on a real image. Conclusions are given in Section 7.