# Efficient Filtering for Edge Extraction under Perspective Effect
## 1 Introduction  
Usually, edge extraction serves as an early stage for computer vision tasks, such as object detection [1], image classification [2], and damage localization [3]. As stated by [PERSON] and [PERSON] [4], the concept of an edge has a partly visual and partly physical meaning. For the visual aspect of the edge, edge is defined as a location of a mathematically precipitous variation in intensity [5]. The discontinuities of the image can be detected as points of high gradient magnitude. For the physical aspect of the edge, edge corresponds not only to the real contours of the interested target but also to differences in texture, changes in pigment, shadows, shadings, and even the real contours of the nearby objects which we do not care about. The scale of the edge also has both visual and physical meaning, and plays a vital role in edge extraction methods [6, 7]. Originated from the works of [PERSON] et al. [8] and [PERSON] et al. [9], scale is expressed as a parameter of image resolution in the scale space. Many works strive to determine the scale for edge extraction. For example, [PERSON] et al. [10, 11] presented an automatic scale selection approach based on locally maximal normalized derivatives over scales, but the \"scale\" in this work is just a measurement of the diffusion degree of the step edge. [PERSON] et al. [12] refined this problem by the 3D Harris detector under an anisotropic diffusion assumption, and got better results that are consistent with the subjective experience of human beings. [PERSON] et al. [13] presented the concept of minimum reliable scale, estimated from the directed second derivatives, and then used it as the assigned scale for every location to compute the edge gradients. The scale of the edge discussed in the above works is a resolution-related parameter, connected with an isotropic/anisotropic diffusion process to extract edge. It is more closely related to a bottom-up attribute of the edge in a local range. In contrast, we suggest estimating both the scale and orientation of the edge in a top-down manner for better edge response. In real applications, people may have certain prior knowledge about the scene and perspective effect from an imaging process [14; 15; 16]. As shown in Figure 1, it is easy for us to attain the imaging parameters (intrinsic and extrinsic parameters of the camera), some properties of the lane markings (e.g., the width of individual lane markings and the distance between adjacent lane markings), and the relative positions of the road and the camera. If we take lane marking as the concerned target, these kinds of top-down information are much more useful to estimate the scale (referred to as _localized target size_) and orientation (referred to as _localized target orientation_). Here, the _localized target size_ for any candidate pixel of a local feature is defined as the minimal distance between the real local features from the same target along the normal direction of the _localized target orientation_ nearest to that pixel. And, the _localized target orientation_ for any candidate pixel of a local feature is defined as the tangent direction of the real boundary of the interested target nearest to that pixel about the edge of the target. With the help of perspective effect, we know that, for the parts of the image nearer to the camera, a large scale should be chosen; for remote parts of the image, we should employ a small scale to account for the pop-out effect of the expected edges. This perspective effect is universal and does exist in both human vision system and mechanical imaging processes [17; 18]. To extract the edge of the lane marking under the perspective effect, we associate the scale of edge in the image with the target size in the real world and assign different scales and orientations for the local filtering kernels. Naively performing filters with these kernels from different scales and orientations is very time consuming.