Q17Digital Image Processing

These are first-order derivative (gradient) operators that estimate the gradient $\nabla f=[G_x,G_y]$ at each pixel by convolving the image with a pair of masks, then compute edge magnitude as $|\nabla f|\approx |G_x|+|G_y|$ and edge direction as $\theta ={\tan }^{-1}(G_y/G_x)$.

Roberts Cross Operator

Uses two $2\times 2$ masks that compute diagonal differences: $G_x=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$, $G_y=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$. It is computationally the cheapest but highly sensitive to noise because it operates on the smallest possible neighborhood.

Prewitt Operator

Uses $3\times 3$ masks with equal weighting: $G_x=\left[\begin{array}{ccc}-1& 0& 1\\ -1& 0& 1\\ -1& 0& 1\end{array}\right]$, $G_y=\left[\begin{array}{ccc}-1& -1& -1\\ 0& 0& 0\\ 1& 1& 1\end{array}\right]$. The larger neighborhood gives some averaging/noise suppression compared to Roberts.

Sobel Operator

Uses $3\times 3$ masks similar to Prewitt but with a weight of 2 on the center row/column: $G_x=\left[\begin{array}{ccc}-1& 0& 1\\ -2& 0& 2\\ -1& 0& 1\end{array}\right]$, $G_y=\left[\begin{array}{ccc}-1& -2& -1\\ 0& 0& 0\\ 1& 2& 1\end{array}\right]$. This extra weighting gives the center row/column more influence, providing better noise suppression via a smoothing effect while still detecting edges, which is why Sobel is the most widely used of the three in practice.