Compared with the Sobel Operator which use 8 points, the Roberts Operator use only 4 points to compute the gradient magnitude.  The first order derivative represents the directional derivative:

\[
G_x =
\begin{bmatrix}
-1 & 0\\
0 & 1
\end{bmatrix} * A
~~~~~~~~~~~~
G_y =
\begin{bmatrix}
0 & -1\\
1 & 0
\end{bmatrix} * A
\]

We can compute the gradient magnitude using:

\[
G = \sqrt{G_x^2 + G_y^2}
\]

For example: if A is the input image

\[
A =
\begin{bmatrix}
P_1 & P_2 \\
P_3 & P_4
\end{bmatrix}
\]

For \(P_1\), the gradient vector components are computed using:

\[
G_x = P_4 - P_1  \\
G_y = P_3 - P_2
\]

 

2016-03-09 Comments

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