## 7 Conclusions  
In this paper, we proposed a new class of edge-oriented filters for grayscale images, determined the filter's parameters, conducted a series of experiments to assess both its  
Figure 6: Frequency of \(K_{p}\) values expressed as a percentage. (**a**) GCMSE, (**b**) ESSIM, (**c**) MSSSIM. Figure 7: Output of the proposed FCCF for binary images. (**a**) Input binary image. (**b**) Output binary image. These figures illustrate how the FCCF fails in finding the edge map of a binary images. performance as an edge filter and the influence of parameters on its overall behavior, and compared the proposed system to standard edge filters. We determined that parameter \(K_{p}\) influences the edge map of the filter outputs; experimentation revealed that values within the interval \(\frac{1}{10}\leq K_{p}\leq\frac{1}{2}\) are suitable for edge filtering according to the criteria of maximizing the MSSSIM and GCMSE metrics. On the other hand, we explored the possibility that the output edge map would be related to some statistics of the image; however, we did not ascertain a definitive relationship between the image's statistics and the value of \(K_{p}\) that would lead to a model for calculating an optimal \(K_{p}\) for maximizing the ESSIM metric. The investigation on this subject involved exploring whether \(K_{p}\) correlates with some image statistics such as the mean, variance, kurtosis, skewness, and entropy, which at first we hypothesized might relate to edge content in an image; however, our experimental results did not support this hypothesis, suggesting that these metrics and \(K_{p}\) are unrelated. Therefore, this parameter must be determined by the user. From our experiments, we concluded that a \(K_{p}\leq\frac{1}{2}\) value is adequate for most images. We recommend setting \(K_{p}\) at smaller values for images that contain textures such as grass or thick animal hair, such as the second illustrative example in Appendix C.  
The evidence indicates that by using this value of \(K_{p}\), the FCCF computes an edge map with higher quality than the well-established filters we considered in this study. The main drawback we identified is that the FCCF is unable to find edge maps in binary images, and thus, we do not recommend the FCCF for applications where binary images need to be processed; however, the FCCF preserved the edges well, thus it leads to a study focused on adapting the FCCF for filtering binary images; this research direction was not followed here since it requires a re-design of the filter, and we think this new design will be incompatible with the theory presented in this paper, and as a consequence, we believe such a study will lead to another class of filters. Future works include assessing the performance under challenging scenarios common in real-world applications, such as noise, variable light conditions, and real-time image processing. Another research direction is to explore the potential improvements in image processing algorithms resulting from replacing the corresponding edge filter with the FCCF approach, which envisions promising perspectives given the wide variety of topics that make use of edge filters to accomplish various computer vision tasks. Yet, another research direction is to assess the performance of the FCCF for filtering medical images, and finally, another work is tuning the filter for finding edges of binary images, which is currently a limitation as discussed above. Based on the experimental results, we believe that target applications might include machine inspection, object recognition, and pattern recognition, among others, in cases where designers prefer to avoid solutions based on machine learning models or deep learning models. Conceptualization, M.G.-L. and H.V.-L.; methodology, M.G.-L., H.V.-L. and M.N.-M.; formal analysis, J.R.G.-M.; software, L.J.M.-M. and E.G.P.-R.; investigation, J.R.G.-M.; validation, M.N.-[PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript. This research received no external funding.