Differential filters

The derivative along $x$ axis ($y$ axis) for a discrete function is presented as $h_x \otimes S$ ($h_y \otimes S$), where $\otimes$ means convolution of two functions, $h_x$ and $h_y$ -- some discrete functions (matrices), $S$ -- function to be derivated. Knowing the derivatives along two axis it is easy to find the module of gradient as a square root of the sum of two squares. Three variants for the calculation of gradients are available in FemtoScan program: Gradient, Prewitt and Sobel. The difference between them lies in the used matrices $h_x$ and $h_y$. They are shown in the Table 3.4.

Table 3.4: Matrices used in the calculations of the first derivatives

	$h_x$	$h_y$
Gradient	$\left( \begin{array}{ccc} 0 & 0 & 0\\ 1 & 0 & -1\\ 0 & 0 & 0 \end{array}\right)$	$\left( \begin{array}{ccc} 0 & 1 & 0\\ 0 & 0 & 0\\ 0 & -1 & 0 \end{array}\right)$
Prewitt	$\frac{1}{3} \left( \begin{array}{ccc} 1 & 0 & -1\\ 1 & 0 & -1\\ 1 & 0 & -1 \end{array}\right)$	$\frac{1}{3} \left( \begin{array}{ccc} 1 & 1 & 1\\ 0 & 0 & 0\\ -1 & -1 & -1 \end{array}\right)$
Sobel	$\frac{1}{4} \left( \begin{array}{ccc} 1 & 0 & -1\\ 2 & 0 & -2\\ 1 & 0 & -1 \end{array}\right)$	$\frac{1}{4} \left( \begin{array}{ccc} 1 & 2 & 1\\ \par 0 & 0 & 0\\ -1 & -2 & -1 \end{array}\right)$

Laplace Filter calculates the laplacian in the following way: $Laplace(S)=h \otimes S$, where

$$h=\left(\begin{array}{ccc} 0 & 1 & 0 1 & -4 & 1 0 & 1 & 0 \end{array}\right)$$