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Differential filters next up previous contents index Russian
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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

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


next up previous contents index Russian
Next: Subtraction of surfaces of Up: Data Processing Functions (Mathematics Previous: Morphological Filters   Contents   Index
Filonov 2005-02-04