# _M\({}^{2}\)_Gf: Multi-Scale and Multi-Directional Gabor Filters for Image Edge Detection
## 1 Introduction  
Edge detection [1] is a foundational task in image processing and computer vision. Edges refer to positions where the information of pixels change suddenly in gray or color images [2; 3; 4; 5; 6; 7]. The information of edges is used in many image processing and analysis techniques, such as image segmentation [8; 9; 10], image retrieval [11; 12; 13], corner detection [14; 15; 16; 17; 18], etc. The edge detection results of images usually play a key role in certain applications [19; 20]. In addition, humans perceive color images more abundantly than colorless images [21; 22]. In conclusion, for color images, an efficient and convenient edge detection algorithm is indispensable for many downstream tasks of computer vision. In the past 40 years, there has been some progress in the research of edge detection approaches, which can be roughly divided into three branches [23]: fusion-based methods, gradient-based methods and vector-based methods. The former obtains the final edge result by fusing individual detected edge of channel components [24]. The core idea of the multi-dimensional gradient-based method aims to calculate the direction and intensity of each edge pixel [25]. The way that the vector properties of image pixels are not changed in the whole process of detection is described as vector-based methods;the edge algorithm based on the color morphological gradient [26], for example, avoids the processing of pixel value fusion. This paper mainly explores the multi-dimensional gradient-based method. The main idea of the multi-dimensional gradient method is to find the direction and intensity of the edge and fuse them effectively before the hysteresis threshold. In 1977, [PERSON], one of the first researchers to study edge detectors for color images, proposed a method that presents image data through brightness and two chrominance channels, and obtains edge maps in each individual component independently using Huckel edge detectors [22]. [PERSON] [27] also proposed a method for color edge detection in thesame year. The 24 directional derivatives were computed and the one with the largest magnitude was chosen as the gradient. A color edge detection algorithm [28] was designed by [PERSON] that can suppress pseudo-edge pixels, which are the most important and common problem in edge detection. In order to solve the problem of the missing direction of edge pixels, [PERSON] and [PERSON] [29] proposed a method of edge detection using local direction operators. It is precisely because the directional information of edge pixels is essential and relevant in various image analysis tasks [30; 31; 32; 33; 34]. Furthermore, a color edge detection method based on edge rendering, which combines gradient and intelligent routing, was proposed [35]. The Canny edge detector [36], as a milestone, has been in development for nearly four decades, and many improved algorithms have been proposed on this basis. [PERSON] believed that the magnitude and direction of the edge extracted by the Canny detector is the key to obtain high-quality edge detection [37], so it was introduced into color edge detection. It has been illustrated that Gabor filters resemble the perception of the human visual system [38; 39; 40]. [PERSON] and [PERSON] pointed out the merits of Gabor filters in extracting local features of images [41]. [PERSON] used a set of multi-scale and multi-directional Gabor filters to smooth edges and detect corners using the amplitude response and angle response [16]. In this paper, the CIE \(L^{a}\)*\(b\) space is adopted because it is sensitive to human visual perception. Then, the edge intensity map is obtained by smoothing the image with multi-scale and multi-direction Gabor filters. The Gabor filters with smaller scale are sensitive to noise but have high edge resolution and edge localization ability. In contrast, Gabor filters with large scale have better noise robustness, but poor accuracy.