# Line-Detection Based on the Sum of Gradient Angle Differences
## 2 Related Work  
As one of the earliest approaches to detecting linear features, [PERSON] [19] proposed a method to detect edge and line pixels. This method was based on the analysis of the coefficients resulting from the fitting of Legendre polynomials of degree two or lower to local intensities, in which a set of combinations of the coefficients was used to classify the pixels into edge, ridge, valley, and other geometric features. Although this approach had the ability to classify a given intensity surface into multiple types, it was majorly limited in terms of line detection because it needed to execute a sequence of steps to detect line pixels and a set of well-tuned threshold values, generating which is an arduous task. [PERSON] [1] proposed a method to detect ridges and valleys by finding the pixels of zero crossings of the first directional derivatives in the direction that extremizes the second directional derivative function. The method employed 10 coefficients generated by fitting a bivariate cubic polynomial to intensity surfaces within a local window. A drawback of this method was the possibility of the generation of non-trivial ringing artifacts that cannot be removed. In addition, this approach required a large window size to calculate the 10 coefficients of the bivariate cubic polynomial. Moreover, this approach encountered problems when localizing the ridge and valley pixels because the intensity surface tended to deviate from the bivariate cubic polynomial surface model, resulting in a relatively large number of false positives and ringing artifacts. [PERSON] and [PERSON] [20] proposed another parameter-based edge- and line-detection method that employed Zernike moments to calculate the parameters of edges and lines. [PERSON] and [PERSON] [2] and [PERSON] et al. [3] proposed methods to find ridge and valley pixels by using differential geometry, which finds zero crossings of the gradients of the curvature of the level curve. However, these approaches were limited by their high computational cost for calculating the first-, second-, and third-order derivatives of intensity surfaces. [PERSON] [18] and [PERSON] [17] proposed line-detection approaches that identify line pixels based on large second directional derivatives in a direction determined by the eigenvector corresponding to the maximum absolute eigenvalue of the Hessian matrix. In similar studies, [PERSON] et al. [21] and [PERSON] et al. [22] employed [PERSON]'s method to detect roads in aerial images and synthetic-aperture radar images, respectively. However, as will be demonstrated in the following sections, the SD-based line-detection methods used in the abovementioned studies suffer from problems such as non-uniqueresponses to a single line signal and dis-localization because of a relatively large line width. [PERSON] [17] proposed a multi-scale-based line-detection approach to find line pixels that respond most strongly to multi-scale filter operations. This approach may, however, be limited when used to detect line features located near other non-homogeneous features, which are observed frequently in many types of images. These limitations occur because, at a larger scale, the features in the neighborhood intersperse, causing line signals to mingle and become dis-localized. One solution to this problem is the use of line detection or edge detection when the line width is relatively small or relatively large, respectively. A simple rapid line-detection method that detects ridge and valley pixels by using EC, which is derived from relational operations, to compare the intensity and other properties of a center pixel with those of neighboring pixels has been proposed in [4]. Although the computational cost of this method is low, its performance is limited, especially when images are contaminated with noise, as will be demonstrated in the following sections. A line-detection method that uses multi-scale anisotropic Gaussian kernels for characterizing the second-order partial derivative of an intensity image was proposed in [23].