# SNR Analysis for Quantitative Comparison of Line Detection Methods
## 1 Introduction  
In the field of image processing, edges and lines are important features used to detect object shapes in a scene. In the literature, lines are classified as \"ridge\" or \"valley\" depending on their relative intensity compared to their neighbor intensities [1; 2; 3; 4; 5; 6]. Lines are the primary features observed in various types of images and are used to detect and recognize the appearance of objects in an image. Recently, by developing various applications, the demand for efficient line extraction methods is rapidly increasing [7; 8; 9; 10; 11; 12; 13]. Frequently, edge detection (ED) is used to indirectly extract line features [7; 8; 9; 10; 11; 12], and different operators have been proposed for ED [14; 15; 16; 17; 18]. However, ED-based methods are often inefficient for detecting lines because they require the detection of edge pixels on both sides of the lines. Quality line extraction requires line detection with high signal-to-noise (SNR) values and high quality subpixel line localization and linking processes [5]. Moreover, although the criteria for establishing a method for detecting linear features in noisy images are always necessary, not many studies have compared the performances of line detection methods to help establish such criteria. Therefore, in this study, a quantitative comparison of the effectiveness of line detection methods, and their merits and demerits in detecting line features under varying conditions, has been described based on SNR analysis. In this study, the influence of image smoothing on both the signal and noise strengths was first investigated. For the comparative study of line feature detectors, a second-order derivative (SD)-based method [19] was first selected and compared with an ED-based indirect method. Moreover, the SD-based line detection method was shown to result in dislocalized line pixels under relatively large line widths [6]. A line detection method based on the addition of gradient angle differences (SGAD) was proposed to overcome the problems [6]. Evaluating line detectors based on experiments with a limited number of test imagesdoes not comprehensively demonstrate the performance of line detectors under varying conditions. Thus, the analytical quantification of the SNR of line detectors is required to determine their efficiency. A SNR analysis to indentify an optimal edge detector was used in [18]. However, a few studies have considered evaluating line detectors based on analytical quantification of their SNR. This primarily motivated the quantitative derivation of the SNRs of line detectors in this study. A set of line detectors were compared based on their experimental characteristic curves in [20]. Moreover, multiple line detection methods were evaluated using multiple images in [21]. However, evaluating line detectors with a set of images does not comprehensively demonstrate the performance of them under varying conditions. In this study, quantitative derivation of the SNRs of line detectors overcomes the limitations of the existing evaluation methods. The contributions of this study are listed below:  
* Streamlined derivation of the SNRs of line detectors including the derivation of correlations among noises after smoothing; derivations of signal and noise strengths of elementary operators; combination of SNRs for line detectors; and application of a penalty function for considering the influence of blurring, smoothing, and line width on line detectors. * Analytical quantification of the SNRs of line detectors using error propagation and visualization of the quantity of the derived values including signal strengths, noise strengths, and SNRs. * Verification of the validity of the SNRs derived for the SD- and SGAD-based line detectors by investigating the relationship of the SNRs with completeness and correctness of their line detection results. The rest of this study is organized as follows. Section 2 describes the related studies for line detection.