# Efficient Filtering for Edge Extraction under Perspective Effect
## 2 Optimal Scale and Orientation Assignment
### Local Edge Extractor Selection and Performance Analysis  
We compared the FDOG and DoB as the local feature detector. Here, we choose to use DoB as the local edge extractor. The reasons for this are as follows. Inspired by the integration map wildly used in the computer vision society, we want to use DoB to produce a fast algorithm. With the integration map, we can compute the response of DoB at any scale using only four addition and subtraction operations. By rotating the original image to multiple orientations before computing the integration map, we can tune the orientation using density orientations. The DoB also has the shape parameters of the aspect ratio, which can be used to determine a balance between angular selectivity and angular invariance. These two factors are important for the tuning of local features to any scale  
Figure 4: Angular response of FDOG for different aspect ratio \(\lambda\). and a density orientation for fast realization in real-world applications. Similar reason can account for why the gPb edge detector was proposed by [PERSON] et al. [7]. These points highlight the advantages of the proposed method with respect to the other existing approaches:  
1. For the system to distinguish the real contour of an object from the interference, such as false edges caused by noise or irrelevant events, the maximum acceptable scale is more important than the localization criteria. So, when we have some pre-knowledge about the localized target size, we prefer to deduce the maximum acceptable scale using only the best detection criteria while ignoring the minimal localization error and minimal multiple response criteria. 2. Considering the discrete expression of Canny's criteria of localization under 2D conditions, the localization error will always be acceptable with different scale parameters for the DoB filter [29]. 3. As an FIR filter, the DoB filter is limited with respect to filter quality, but its 2D shape allows it much more freedom than steerable filters. For example, by adjusting the aspect ratio properly, we can attain a sharp angular response profile to meet the balance between angular selectivity and angular invariance. For a local feature detector to be tuned by the scale and orientation, the filter should show certain invariance to the photometrical and geometrical deformation while keeping certain selectivity to the specified scale and orientation. If the responses of the local feature detectors is too robust with respect to scale and orientation, specifying the local scale and orientation becomes useless. Here, we investigate the performance of the optimal local filter over variations in scale and orientation. We use the orientation and scale-tuned DoB as the local feature detector, and will focus on the performance evaluation of DoB. The signal is still a strip, but the width of it varies from \(0.1d\) to \(2d\), and the orientation varies from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). The optimal filter has the optimal scale parameter with \(a=d\) and the optimal aspect ratio parameter with \(\lambda=2.98\). The contour plot of the response of the DoB, having optimal parameters with the pre-known localized target orientation, varying from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), and the pre-known localized target size varying from \(0.1d\) to \(2d\), is shown in Figure 6.  
As shown in Figure 6, the optimal DoB has a rather good angular performance when the localized target orientation varies around the specified orientation. Somehow, the scale performance is not so good. When the real localized target size is less than the pre-known localized target size, the selectivity of DoB is rather good. However when the real localized target size is greater than the pre-known localized target size, the response of DoB remains constant, which means it has no selectivity at all. This is an important issue that should be considered in future works.