# Dilated Filters for Edge-Detection Algorithms
## 3 Methodology
### Benchmarking the Edge Operators
\[PCM_{\eta}(f,g)=100\cdot(1-\frac{C(M_{opt}(f,g))}{(|f\cup g|)}) \tag{15}\]  
For each benchmark image, three different probability measures are computed: precision (\(P\)), recall (\(R\)) and F-measure (\(F1\)) defined in [70]. Precision (Equation (16)) is the probability that a resulting edge/boundary pixel was labeled as a true edge/boundary pixel. Recall (Equation (17)) is the probability that a true edge/boundary pixel was detected. F-measure (Equation (18)) is the accuracy measure computed as an average between precision and recall. Further, we need to specify that \(TP\) (true positive) represents the number of matched edge pixels, \(FP\) (false positive) the number of edge pixels that are incorrectly highlighted as edge pixel and \(FN\) (false negative) the number of pixels that have not been detected as an edge pixel but inthe dataset has been labeled as an edge pixel. The highest possible value of an \(F\)-score is 1.0, indicating perfect \(P\) and \(R\). The \(F1\) score is also known as the Sorensen-Dice coefficient or Dice similarity coefficient (DSC). \[P=\frac{TP}{TP+FP}. \tag{16}\]  
\[R=\frac{TP}{TP+FN}. \tag{17}\]  
\[F\text{-}measure=2*\frac{P*R}{P+R}=\frac{2*TP}{2*TP+FP+FN} \tag{18}\]  
The synthetic dataset chosen was used in our previous work [12] and was constructed using the principles presented in [68]. The gray-scale dataset contains various edge types (such as step edges, ramp and roof edge), widely varying angles, various junctions and brightness changes. To offer a suitable dataset, we introduced in certain images different levels and types of noise (such as uniform noise in Figure 6e or Gaussian noise in Figure 6h) and blurring (see Figure 6d). All these different aspects aim to provide a better evaluation of any certain edge detection algorithm. The benefit in using a synthetic dataset, rather than a natural dataset, is the fact that the subjective human-vision interpretation of the edge localization and nature is diminished. Figure 5: Example of images and corresponding ground truth from \(BSDS500\). Figure of merit (\(FoM\)) is a widely used and popular discrepancy measure for edge images. This distance measure has ranges from 0 to 1, where 1 corresponds to a perfect evaluation. In Equation (19), we used the following notation: \(f\) is the predicted image, \(g\) is the ground truth image, \(k\) is the constant \(\frac{1}{g}\) and \(d_{g}(p)\) represents the minimal Euclidean distance between pixel \(p\) from \(f\) and \(g\)[71]. \[FoM(f,g)=\frac{1}{max(|f|,|g|)}\cdot\sum_{p\in g}1+k\cdot d_{g}^{2}(p) \tag{19}\]  
Symmetric figure of merit (\(SFoM\)) is inspired by other evaluation measures, such as [72], as a way to avoid the computation of only the distance of \(FPs\) in \(FoM\) and to consider a combination of \(FoM(f,g)\) and \(FoM(g,f)\) as we can see in Equation (20). Similar to \(FoM\), \(SFoM\) is also normalized but takes into consideration both distance of \(FNs\) and \(FPs\) resulting in a better global evaluation of the predicted image [58]. \[SFoM(f,g)=\frac{1}{2}\cdot FoM(f,g)+\frac{1}{2}\cdot FoM(g,f) \tag{20}\]