A statistical method for line segment detection

Highlights

• Voting in each column around an initial peak is considered to be a random variable.

• The optimal θ is determined by fitting and minimizing a 2nd-order curve.

• The optimal ρ is determined by fitting and interpolating a sine curve.

• We calculate voting boundaries instead of searching for non-zero voting cells.

• The endpoint coordinates are determined by fitting instead of by solving equations.

Abstract

Line segment detection is a fundamental procedure in computer vision, pattern recognition, or image analysis applications. This paper proposes a statistical method based on the Hough transform for line segment detection by considering quantization error, image noise, pixel disturbance, and peak spreading, also taking the choice of the coordinate origin into account.

A random variable is defined in each column in a peak region. Statistical means and statistical variances are calculated; the statistical non-zero cells are analyzed and computed. The normal angle is determined by minimizing the function which fits the statistical variances; the normal distance is calculated by interpolating the function which fits the statistical means. Endpoint coordinates of a detected line segment are determined by fitting a sine curve (rather than searching for the first and last non-zero voting cells, and solving equations containing coordinates of such cells).

Experimental results on simulated data and real world images validate the performance of the proposed method for line segment detection.

Introduction

The Hough transform (HT) [1], [2], [3] is widely used for extracting geometric features in images. The standard HT is inefficient with respect to run time and storage requirements; it also suffers from the peak-spreading problem in the Hough space [4], [5].

In order to improve the computation and storage efficiency, many methods have been proposed, such as the fast HT [6], adaptive HT [7], special architectures HT [8], probabilistic HT [9], randomized HT [10], or generalized HT [11].

In order to locate accurately the peak, and to derive accurate normal parameters, various extensions to the standard HT have been proposed. Numerous methods put the emphasis on generating more distinct peaks by modifying the voting scheme of a Hough transform. Edge information [12], [13], [14] or image preprocessing techniques [15], [16], [17] are used to enhance the peak in the accumulator array. In addition, a continuous statistical kernel [18] is proposed for modeling the Hough votes instead of using a discrete accumulator array. This approach is computationally more expensive because it considers and models all pixels in the given image.

After finding a peak, there are three different ways for supporting the computation of accurate peak parameters:

• A peak cell (θ, ρ) is simply selected for specifying the normal parameters of the detected line.

• A weighted averaging [19], [20] is used by considering neighboring cells (θ_i, ρ_i) around a peak cell.

• A specially designed peak localization technique is employed. For example, two accurate peak localization methods are presented in [21], [22] using smoothing windows and weighted averaging, a two-stage method [23] is described by finding a median position in narrow strips passing through an initial peak, a high-accuracy HT, proposed in [24], is based on the theoretical symmetry of the butterfly pattern around a peak, and a subcell-accuracy method [25] locates the peak by using fitting and interpolation techniques while analyzing the butterflies.

The HT can also be used for extracting the endpoints of a detected line segment [26] although the standard HT only provides normal angle θ and normal distance ρ.

We classify HT-based methods for detecting line segments into two categories. One class of methods is based on projection [27], [28], [29]. After applying the standard HT, the detected line is cut into pairwise disjoint segments; the endpoints of one of those line segments are determined by analyzing the projection of the feature points on either the x- or y-axis in image space. Two thresholds are needed to control the length of a line segment and the width of a gap between line segments. But, how to determine these two thresholds defines then a new problem.

The second class of methods is motivated by the butterfly shape [30], [31] of a peak region in Hough space [32], [33], [34]. Butterfly features are used to extract parameters of a line segment. The first and last cell with non-zero voting values are identified and used to compute the endpoints of the line segment by solving two sets of simultaneous equations [35], [36], [37], [38]. However, how to obtain the first and the last cell containing a non-zero voting value is a difficult problem in the presence of image noise, pixel disturbance, and a wide line-segment.

In a recent publication [39], we described a method for detecting line segments based on minimum-entropy analysis. Direction and length of a line segment are simultaneously determined by analyzing the voting entropies. The midpoint is detected by fitting voting means. The endpoint coordinates are finally calculated based on the detected direction, length and midpoint.

This paper proposes a novel statistical method based on HT for extracting line segments. The vote in each column (around a peak) is considered to be a random variable. The normal angle and distance of a line segment are computed by fitting and interpolating statistical variances and means. Endpoint coordinates of a line segment are computed by fitting a sine curve to the statistical non-zero voting cells, computed from statistical means and variances. Thus, there is no need for searching non-zero cells and solving two equations generated from just two cells.

In distinction to [39], this paper does not apply any minimum-entropy analysis. It extracts the normal angle, the normal distance, and endpoint coordinates using statistical characteristics. This paper also proposes a solution to calculate the statistical non-zero cells instead of searching for the first and the last non-zero voting cells in each column of a butterfly.

The rest of the paper is organized as follows. Section 2 introduces a way for peak distribution analysis in Hough space. Section 3 defines the random variable and statistical characteristics. Section 4 describes our normal angle and distance detection method based on statistical characteristics. Section 5 outlines endpoint detection by fitting sine curves. Section 6 compares experimental results with simulate data and real images. Section 7 concludes.