# Fractional Order Processing of Satellite Images
## 4 Discussion and Conclusions  
In this paper, an automatic coasts edge detection tool using five state-of-the-art edge detection versions and seven novel detector versions was presented. The conventional integer versions of the algorithms were also tested and compared to fractional solutions. Forty-three high definition (10,980p \(\times\) 10,980p) satellite images were tested using the different versions of the operators in a wide range of parameters. The grey-scale Fractional Derivatives operator was the detector that provided the best average scores regarding all figures, with varying parameters. The mean Jaccard coefficient for this operator was 0.9623, i.e. the algorithm identified correctly in average 96.23% of the pixels in the input images. In remote sensing, it is important not only that the algorithm can reach high efficiency but also that the parameters required are possible to implement for an automatic solution. Despite the exceptional performance achieved with varying parameters, this Jaccard coefficient is only possible using a great spread of parameters impossible to reach in automatic solutions. Thus, the search for a solution with fixed parameters was conducted. In this case, the color-based Fractional Derivatives operator was the method that provided the best average scores regarding all figures in the data set, with a Jaccard metric of 0.9436. The result obtained sets a fixed-parameter solution that allows automatic detection of coasts with a decrease in performance compared to the aforementioned varying-parameters of less than 2%. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Detector** & _Threshold_ & \(\mathbf{k}\) & \(\sigma\) & \(\mathbf{\alpha}\) & \(\overline{J}\) \\ \hline
**Color** Fractional Derivative Op. & 0.9 & - & - & 0.8 & 0.9436 \\ Fractional Derivative Op. & 0.7 & - & - & 0.8 & 0.9428 \\
**Color** CRONE & - & 5 & - & 0.9 & 0.9328 \\ CRONE & - & 5 & - & 1.1 & 0.9261 \\
**Color** Canny & - & - & 0.7 & 1.7 & 0.9193 \\
**Color** [PERSON] & 0.1 & - & - & 1.4 & 0.9174 \\
**Color** Sobel & 0.3 & - & - & \(-\)0.2 & 0.9153 \\ Sobel & 0.9 & - & - & 0.2 & 0.9111 \\
**Color** Laplacian of Gaussian & 0.1 & - & - & \(-\)0.9 & 0.9108 \\ [PERSON] & 0.1 & - & - & \(-\)1.3 & 0.9076 \\ Laplacian of Gaussian & 0.1 & - & - & \(-\)1.4 & 0.9028 \\ [PERSON] & - & - & 0.6 & 0 & 0.9011 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Full volume analysis of best results with fixed parameters for all detectors. Figure 26: Bar chart with best results average for all images (Fixed parameters). Further analyzing overall performance Tables 6 and 7, as well as the performance data in [20], one may draw a few more conclusions. The usage of fractional derivatives in edge detection for this application matched or improved, in most cases, the performance of the conventional integer methods, which is not surprising since more information is available to be used. The color based algorithms allowed to equal or improve the performance of grey-scale methods in most cases. However, often the increase in performance is low in percentage. Nevertheless, and since we are dealing with images that are composed of more than 120 million pixels, a percentage of 1% increase corresponds to more than 1 million pixels correctly identified. To summarize, in this study a successful automatic tool was developed that identifies and segments coasts in satellite images using both grey-scale and novel color-based fractional edge detectors. Subsequent parameters that allow the automation of this tool were also found. M.H. : methodology, software, investigation, data curation, writing - original draft preparation, visualization; D.V. : conceptualization, methodology, validation, investigation, writing - review and editing, supervision; R.M. : conceptualization, methodology, validation, investigation, data curation, writing - review and editing, supervision. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. See [20].