# Quadratic Curve Fitting-Based Image Edge Line Segment Detection: A Novel Methodology
## 1 Introduction  
Detecting line segments within images constitutes a significant challenge within the field of computer vision [1], and it has extensive ramifications across diverse applications such as camera calibration [2], pose estimation [3], image matching [4], image stitching [5], 3D object detection [6], and visual simultaneous localization and mapping (SLAM) [7]. Contemporary scholarly endeavors in the domain of edge line segment detection predominantly revolve around the identification of straight line segments within undistorted (or marginally distorted) images. Broadly, these techniques can be classified into the following categories: Hough transform [8], edge linking [9], and methodologies that are premised on gradient direction and magnitude [10]. The Hough transform technique translates image coordinates into a parameter space and identifies straight line segments through a process of accumulation voting, thereby delivering impressive accuracy and reliability. However, the technique necessitates parameter computation for each edge pixel within the image, leading to considerable computational load and compromising real-time efficiency. Furthermore, the Hough transform method relies solely on parametersto represent a line, rendering the precise localization of line segment endpoints challenging. In response to these shortcomings, [PERSON] et al. [11] implemented a perception grouping methodology based on the Hough transform technique to ascertain line segment endpoints and achieved commendable results. Conversely, edge linking methodologies extract initial line segments by pixel traversal, and are based on the substantial grayscale variations at image edges. The subsequent extension and merging of these initial segments are carried out using parameter fitting to yield reasonable and complete line segments. In comparison to the Hough transform method, edge linking techniques display superior noise resistance and computational efficiency. However, the quality of results obtained from edge linking methodologies heavily hinges on the reliability of the edge operator, potentially leading to a loss of some original image information. To counteract this issue, [PERSON] et al. [12] introduced indicators of edge main direction and image gradient direction during the edge detection process, enhancing the line segment detection's resistance to interference in edge linking methodologies. [PERSON] et al. [13] proposed an adaptive parameter Canny operator to extract image edges, thereby enhancing the reliability of line segment detection in edge linking methodologies. Additionally, the gradient direction-based technique [14], unlike its counterparts, computes the gradient direction of pixels directly and groups pixels with similar gradients to establish regions for line segment detection. Yet, this method demands further refinement as it overlooks gradient magnitude information. Building upon the work of [PERSON] et al. [14], [PERSON] et al. [10] proposed the classic line segment detector (LSD) method. This method considers both the gradient direction and the gradient magnitude of the image. By selecting pixels with extremum values of a gradient magnitude as seed points, it employs a region-growing approach to generate candidate line segment regions. Additionally, by combining hypothesis verification, it extracts the final line segments. The parameters of the LSD algorithm are adaptive, resulting in high accuracy and fast detection speed. Consequently, it has become a benchmark for modern line segment detection algorithms [15]. Extending the LSD algorithm, [PERSON] et al. [16] proposed a multiscale line segment detection algorithm, and [PERSON] et al. [17] developed a line segment detection algorithm for high-resolution color images, delivering satisfactory results. Based on the literature review presented above, it is evident that existing edge line segment detection methodologies are primarily designed for undistorted images.