## 1 Introduction
Each method employs different kernels: [PERSON] kernel is intended for edges at 45\({}^{\circ}\), whereas [PERSON] and [PERSON] respond better to horizontal and vertical edges (Equation (2)). \[[PERSON]\left\{ \begin{array}{c}G_{x}=\left(\begin{array}{cc}1&0\\ 0&-1\end{array}\right)\\ G_{y}=\left(\begin{array}{cc}0&1\\ -1&0\end{array}\right)\end{array} \right.\begin{array}{c}Prewitt\\ \left\{ \begin{array}{cc}G_{x}=\left(\begin{array}{cc}1&0&-1\\ 1&0&-1\\ 1&0&-1\end{array}\right)\\ G_{y}=\left(\begin{array}{cc}1&1&1\\ 0&0&0\\ -1&-1&-1\end{array}\right)\end{array} \right. \begin{array}{c}G_{x}=\left( \begin{array}{cc}-1&0&1\\ -2&0&2\\ -1&0&1\end{array}\right)\\ G_{y}=\left(\begin{array}{cc}1&1&1\\ 0&0&0\\ -1&-1&-1\end{array}\right)\end{array} \right. \begin{array}{c}G_{y}=\left( \begin{array}{cc}1&2&1\\ 0&0&0\\ -1&-2&-1\end{array}\right)\end{array}  \tag{2}\]  
Once the gradient for each direction is calculated (vertical and horizontal direction), a general magnitude (Equation (3)) and direction (Equation (4)) can be calculated:  
\[|G|=\sqrt{{G_{x}}^{2}+{G_{y}}^{2}} \tag{3}\]  
\[\theta=\arctan\frac{G_{x}}{G_{y}} \tag{4}\]  
A new grey scale image is obtained, with each grey value representing the gradient magnitude \(|G|\) calculated for the corresponding pixel on the original image. In these methods, any pixel of the new image will be considered a boundary if the value of its gradient \(|G|\) is higher than a threshold T. It should be noted that the Prewitt operator is expected to provide noisier results, because it uses a simpler approximation matrix than the Sobel technique [27]. Second derivative detectors, on the other hand, are based on the fact that a peak or valley on the first derivative would appear as a zero-crossing in the second derivative. One of these methods is based on the Laplacian calculation of the original greyscaleimage. An approximation of the second derivative can be calculated by means of a discrete convolution kernel, that can include diagonals or not (Equation (5)). \[K=\left( \begin{array}{ccc}0&1&0\\ 1&-4&1\\ 0&1&0\end{array} \right)\textit{or}\;K=\left( \begin{array}{ccc}1&1&1\\ 1&-8&1\\ 1&1&1\end{array} \right) \tag{5}\]  
Using one of these kernels, the Laplacian can be calculated using standard convolution methods. These kernels are approximating a second derivative measurement on the image; therefore, they are very sensitive to noise. Due to this approach also being isotropic, no information about edge orientation is obtained. Another possible second derivative method uses the Laplacian of Gaussian (LoG) kernel, that can be calculated in advance, so that only one convolution needs to be performed at run-time on the image. The 2-D LoG function centred on zero, with Gaussian standard deviation \(\sigma\), can be expressed as:  
\[LoG(x,y)=-\frac{1}{\pi\sigma^{4}}\left[1-\frac{x^{2}+y^{2}}{2\sigma^{2}} \right]e^{-\frac{x^{2}+y^{2}}{2\sigma^{2}}} \tag{6}\]  
Note that as the Gaussian is made increasingly narrow, the LoG kernel becomes the same as the simple Laplacian kernels. The smoothing level is controlled by means of the standard deviation \(\sigma\). In a subsequent step, zero-crossing must be found for each row and column. The Canny method has also been considered in this study because, according to [PERSON] [21], it is by far the most used differential operator. Canny is actually a multi-step algorithm that computes the gradient using Sobel or a derivative Gaussian filter. The steps that compose this method are reflected in Figure 1.  
The first step involves applying a Gaussian filter to smooth the image to remove the noise. The next step is used to calculate the intensity gradients. The original formulation of this method used the Sobel kernel, whereas alternative formulations employ a derivative Gaussian filter. The result of this stage is a new image with wider edges; thus, a thinning stage by means of a non-maximum suppression technique is applied.