# Dilated Filters for Edge-Detection Algorithms
## 3 Methodology
### Benchmarking the Edge Operators  
Edge-detection evaluation methods are an important aspect and multiple solutions are presented in the literature, but it still is a challenging and not totally solved problem. Edge-detection evaluation methods can be categorized in several ways. First, they can be classified as subjective or objective methods. The subjective method uses human observation to decide the edge detection evaluation. In objective methods, quantitative measures are defined based solely on images and the edge-detection results. Secondly, edge-detection evaluation methods can be categorized based on test images: synthetic-image-based methods and real-image-based methods [55; 56; 57]. In edge detection evaluation, the measurement process can be classified as: unsupervised or supervised evaluation criteria. The first method concerns only the input data image and generates a score of coherence that qualifies the algorithm result. The second computes a similarity/dissimilarity measure between the resulted image and a ground truth obtained from synthetic data or an expert judgment [58]. Multiple evaluation schemes are found in the literature for measuring supervised evaluation criteria, as in [3; 58; 59; 60]. For the evaluation of edge detection, the confusion matrix remains a cornerstone in boundary detection evaluation methods. If we consider the ground truth image as \(I_{gt}\) and the predicted image as \(I_{p}\) we have the following possible set of points:  
* True positive points (TPs), common points of \(I_{gt}\) and \(I_{p}\). * False positive points (FPs), false detected edges of \(I_{p}\). * False negative points (FNs), missing edge points of \(I_{p}\). * True negative points (TNs), common non-edge points. To evaluate our proposed algorithms, we used two different supervised evaluation criteria on two different datasets: one synthetic and one constructed on expert judgment. In order to select the correct dataset, we analyzed multiple datasets [61; 62; 63; 64; 65; 66; 12; 67; 68]. For the expert judgment, we used the Berkeley Segmentation Data Set and Benchmarks 500 (\(BSDS500\)) in correlation with the pixel corresponding metric (_PCM_) [69]. Regarding the synthetic dataset, we used the dataset from [12] in correlation with the symmetric figure of merit [58] (\(SFoM\)) metric. Due to the different nature of the datasets and their different evaluation criteria, we performed an objective evaluation. \(BSDS500\) is a widely used dataset in the field of computer vision for benchmarking edge-detection algorithms as a measures of boundary detection. It contains natural images that have been manually segmented and is considered the ground truth in many boundary detection comparisons. The benchmark is used to evaluate the result images generated by a specific algorithm with the segmented images in the dataset. For edge detection evaluation we used the Pixel Correspondence Metric (_PCM_) algorithm, defined in [69]. The \(BSDS500\) benchmark offers 500 images for testing, presented in few different sets. The images arenatural images marked for the boundaries and edges of the objects and structures that they represent or contain, as we can see in Figure 5.  
For evaluating the obtained edge maps we used the \(PCM\) algorithm [69]. This metric is reliable for correlating similarities because it searches for the optimal matching of the pixels between the edge images and then estimates the error produced by this matching. In Equation (15), we present the definition of \(PCM\) between two images \(f\) and \(g\), where \(C(M_{opt}(f,g))\) is the cost of an optimal matching, \(\eta\) is the maximum localization error and \(|f\cup g|\) is the total number of pixels that are not zero in both images. The optimal matching algorithm used is an approximation of the weighted matching algorithm (more details in [69]), with a depth of 5 and localization error of 5 pixels.