# Dilated Filters for Edge-Detection Algorithms
## 4 Experimental Results
### Effect of Dilation
#### 4.1.2 Operators Based on Second-Order Discrete Kernels  
The Laplace edge operator is one of the most popular edge detectors based on second-order derivative discrete kernels, described in Section 3.5. We have chosen to vary the threshold from 15 to 245, the results can be observed in Figure 9a, where \(TG\) is the notation of the threshold. Figure 9: Finding the best results for classic, extended and dilated using different parameters for: (**a**) Laplace edge operator; (**b**) LoG edge operator; (**c**) Marr–Hildreth edge operator; (**d**) Shen–Castan edge operator. A natural extension of the Laplace operator is the Laplacian of Log, described in Section 3.6. We searched for the best combination of sigma for constructing the LoG kernel and the threshold. The results can be observed in Figure 9b. We have chosen to vary the threshold from 5 to 60, avoiding a bigger threshold. Bigger threshold values would not generate better results because of the small intensity resulted after the convolution. Regarding the sigma for the Gaussian blur kernel, we chose to vary between 0.2 and 2.0, while going higher would generate a lack of resulting edge points. For the images, we use the following notations: Gaussian sigma is \(S\) and threshold is \(TG\). Marr-Hildreth operator, presented in Section 3.7, is similar to the Log operator. We first searched the best combination of sigma for constructing the LoG kernel and the threshold of zero crossing. The results can be observed in Figure 9c and for a better visualization, we have chosen to show only the best results in \(F1\) order. We have tried the combination of \(sigma\) value from 0.2 to 3.0 with gradient threshold of 30 to 90 of the gradient image, considering that is the interval where these parameters bring good results to the output. For the images, we use the following notations: Gaussian sigma is \(S\) and threshold is \(TG\). The results and evaluation of Shen-Castan operator, described in Section 3.9, are presented in this section. We searched again for the best combination of threshold of Laplace, smoothing factor, window size, thinning factor and ratio factor of the algorithm. We chose a threshold value of 40 for the Laplace edge detector and vary the rest of the values as follows: smoothing factor of ISEF filter (\(SF\)) from 0.5 to 0.9; adaptive zero crossing window size (\(W\)) of 5, 7, 9; threshold for zero crossing (\(R\)) from 0.5 to 0.9; thinning factor (\(TN\)) of 0, 0.5 and 0.9. There are multiple hyperparameter value combinations that yield the same \(F1\) score. For example, the dilated \(7\times 7\) filter obtains an \(F1\) score equal to 0.576 for 4 different combinations of values for R and TN: \((TG=4,SF=0.9,W=11,R=0.5,TN=0)\), \((TG=4,SF=0.9,W=11,R=0.5,TN=0.5)\), \((TG=4,SF=0.9,W=11,R=0.9,TN=0)\), \((TG=4,SF=0.9,W=11,R=0.5,TN=0.5)\). Looking over the results from Figure 9a-d, we can observe that the best results were obtained using the \(7\times 7\) dilated version, followed by the \(5\times 5\) dilated version. We can also  
Figure 10: Finding the best results for classic, extended and dilated using different parameters for: (**a**) Laplace edge operator; (**b**) LoG edge operator; (**c**) Marr–Hildreth edge operator; (**d**) Shen–Castan edge operator. conclude that for this operator using the 5 \(\times\) 5 extended kernel deteriorates the resulted edge map. Similar to first-order-based algorithm, we chose to vary all the parameters in the same limits for these evaluation. These is important so we can correlate the results on the different datasets. First we look over the results we obtained when evaluating the Laplace operator, see Figure 9(a) and we see that dilating bring forward significant improvements. We see a peak value of 0.478 for 5 \(\times\) 5 dilated and 0.470 for 7 \(\times\) 7. If we look over the results from LoG operator, see Figure 9(b), the dilatation produces slightly better results, but an interesting fact is the compression that we can see in the evaluation results interval.