## 6 Experimental Results and Discussion  
As we previously mentioned, through the first experiment we designed, we examined the relationship between \(K_{p}\) and the resultant edge map. To achieve this objective, we observed the output of the filter over the entire BSD500 dataset for values of \(K_{p}\) ranging from 0.1 to 1. For clarity, we display here only selected results for the standard test image \"Lena\" in Figure 3. Additional edge maps for this example are detailed in Appendix B, and additional examples of images drawn from the BSD500 are included in Appendix C, which provides a more comprehensive insight into the relationship between \(K_{p}\) and the resultant edge maps. Figure 3 reveals a relationship between the number of edges in the filter outputs and the parameter \(K_{p}\); although this is a subjective assessment, we confirmed this relationship with the second experiment we proposed using metrics oriented towards edge preservation.  
With the second experiment, we aimed to determine a \(K_{p}\) value that would result in the highest edge preservation. We calculated the metrics discussed in the last paragraph of Section 5 using a grid of 10 equally spaced values within the interval \(\frac{1}{10}\leq K_{p}\leq\frac{1}{2}\); subsequently, we computed the edge-oriented metrics and presented the distribution of this data in the box plot shown in Figure 4, which depicts the distribution of \(K_{p}\) values that maximize each metric. This figure indicates that the median \(K_{p}\) value for maximizing the metrics MSSSIM and GCMSE is \(\frac{1}{2}\) for nearly all images in the BSDS500; in contrast, the median for ESSIM is 0.4. Moreover, these results imply that finding a fixed \(K_{p}\) value that ensures high edge preservation for the ESSIM metric is challenging in most cases. Using this premise, we evaluated the performance of the proposed filter, letting \(K_{p}=\frac{1}{2}\), and then we compared the resultant metrics with those computed using the [PERSON] and Roberts filters.  
Figure 5a-c displays the evaluation metrics from the output edge map computed using the FCCF, the Sobel filter, and the Roberts filter. As observed in Figure 5a, the proposed filter achieves higher performance compared to the [PERSON] and Roberts filters, as evidenced by attaining the lowest GCMSE. Additionally, Figure 5b illustrates the superior performance of the filter when assessed using ESSIM, indicated by a higher median value of ESSIM, and consequently, the FCCF exhibits a distribution skewed towards higher values in comparison to the Sobel and Roberts methods. Furthermore, the evaluation from the MSSSIM perspective also substantiates the enhanced performance of the FCCF, as Figure 5c clearly demonstrates.  
We observed that for this test, the metrics for the [PERSON] and Roberts filters are nearly identical; thus, we focused our comparison on the FCCF versus the Sobel filter. To facilitate this, we constructed a violin plot to depict the distributions of the edge-preservation metrics and positioned them side by side for an easier comparison in Figure 6. This comparative analysis reveals that the proposed filter surpasses the performance across all metrics considered in this study.  
Figure 4: The search of a value for maximizing each metric led us to letting \(K_{p}=\frac{1}{8}\).  
Figure 3: The effects of varying the value of the parameter \(K_{p}\) on the edge map.  
Figure 5: Frequency of \(K_{p}\) values expressed as a percentage. (**a**) GCMSE, (**b**) ESSIM, (**c**) MSSSIM.  
Figure 6: _Cont._