# Fractional Order Processing of Satellite Images
## 2 Definitions and Formulations
### Edge Detection Operators  
All the edge detection operators used in the algorithm above make use of the Grunwald-Letnikoff definition of fractional derivative:  
\[{}_{c}D_{t}^{a}f(t)=\lim_{h\to 0}h^{-\alpha}\sum_{k=0}^{N}(-1)^{k}\Big{(} \begin{array}{c}\alpha\\ k\end{array} \Big{)}f(t-kh),\quad N=\bigg{\lfloor}\frac{t-c}{h}\bigg{\rfloor}, \quad t>c \tag{1}\]  
where combinations of \(a\) things, \(b\) at a time, are given by [21]  
\[\Big{(} \begin{array}{c}a\\ b\end{array} \Big{)}=\left\{ \begin{array}{l}\frac{\Gamma(a+1)}{\Gamma(b+1) \Gamma(a-b+1)},\quad\text{ if \ \ a,b,\text{a-b}\
ot\in\mathbb{Z}^{-}}\\ \frac{(-1)^{b}\Gamma(b-a)}{\Gamma(b+1)\Gamma(-a)},\quad\text{ if \ \ a\in\mathbb{Z}^{-} \wedge b\in\mathbb{Z}_{0}^{+}}\\ 0,\text{ if }\big{[}\big{(}b\in\mathbb{Z}^{-}\lor b-a\in\mathbb{N}\big{)} \wedge a\
ot\in\mathbb{Z}^{-}\big{]}\vee\\ (a,b\in\mathbb{Z}^{-}\wedge|a|>|b|)\end{array} \right. \tag{2}\]  
The operators employed were the following. They are further detailed in [5]. * **Canny Edge Detector. ** In this popular edge detection algorithm [22], the image is first convolved with a Gaussian Filter, and then a derivative operator is applied to the smoothed image to compute gradients. This is done determining the first partial derivatives of the smoothed image in both the \(x\) and \(y\) directions for each color channel, defining the following Jacobian [3]: \[\mathbf{J}=\left( \begin{array}{cc}R_{x}&R_{y}\\ G_{x}&G_{y}\\ B_{x}&B_{y}\end{array} \right)=\left(C_{x},C_{y}\right)\] (3)  
Figure 7: Totally Processed Image. The direction in the image along which the largest variation in the chromatic image function occurs is represented by the eigenvector of \(J^{T}J\) corresponding to the largest eigenvalue. \[\mathbf{J^{T}J} = \left( \begin{array}{cc}J_{x}&J_{xy}\\ J_{yx}&J_{y}\end{array} \right)\] (4) \[\lambda = \frac{J_{y}+J_{x}\pm\sqrt{(J_{y}+J_{x})^{2}-4(J_{x}J_{y}-J_{xy}^{2} )}}{2}\] (5)
* **Sobel Edge Detector. ** The integer Sobel operator performs a derivative operation on an image and so it highlights regions where there are sudden increases of pixel intensity which correspond to edges, and consists of the two masks [23] \[ \begin{bmatrix}-1&0&1\\ -2&0&2\\ -1&0&1\end{bmatrix} \quad \begin{bmatrix}1&2&1\\ 0&0&0\\ -1&-2&-1\end{bmatrix} \] (6) The fractional Sobel operator given in [13] has the mask \[ \begin{bmatrix}\frac{\alpha(-a+1)(-a+2)}{2}&\frac{\alpha(-a+1)(-a+2)}{6}& \frac{\alpha(-a+1)(-a+2)}{12}\\ \frac{\alpha(-a+1)}{4}&\frac{\alpha(-a+1)}{2}&\frac{\alpha(-a+1)}{4}\\ \frac{\alpha}{2}-\frac{\alpha(-a+1)(-a+2)}{12}&\alpha-\frac{\alpha(-a+1)(-a+2) }{6}&\frac{\alpha}{2}-\frac{\alpha(-a+1)(-a+2)}{12}\\ -\frac{1}{2}-\frac{\alpha(-a+1)}{4}&-1-\frac{\alpha(-a+1)}{2}&-\frac{1}{2}- \frac{\alpha(-a+1)}{4}\\ -\frac{\alpha}{2}&-\alpha&\frac{\alpha}{2}\\ \frac{1}{2}&1&\frac{1}{2}\end{bmatrix} \] (7) A novel color-based fractional Sobel was implemented by applying the same algorithm of the Canny detertor, with the mask in (7). * **Roberts Edge Detector. ** The integer Roberts operator [24] consists of a pair of \(2\times 2\) masks [23]: \[ \begin{bmatrix}1&0\\ 0&-1\end{bmatrix} \quad \begin{bmatrix}0&1\\ -1&0\end{bmatrix} \] (8) The fractional Roberts detector presented in [11] and based on the truncated coefficients of the GL definition applies derivatives of arbitrary order \(\alpha\): \[ \begin{bmatrix}\frac{\alpha^{2}-a+2}{2}&\frac{\alpha^{2}-a+2}{2}&\frac{\alpha ^{2}-a+2}{2}\\ \frac{\alpha^{2}-a+2}{2}&-8\alpha&\frac{\alpha^{2}-a+2}{2}\\ \frac{\alpha^{2}-2+2}{2}&\frac{\alpha^{2}-a+2}{2}&\frac{\alpha^{2}-a+2}{2} \end{bmatrix} \] (9) This mask is applied after the convolution of an image with the conventional integer Sobel. A color-based version of this detector was developed using the same approach as the Sobel algorithm. Since the fractional operator requires two convolutions, the color-based computations were applied only to the first one with the integer mask. After this, the result is convolved with (9). * **Laplacian of Gaussian (LoG) Detector. ** Three possible Laplacian operators are \[