Sobel filter

Sobel filters are used to compute the magnitude and direction of the local gradient in 2-D.

Kernels

The horizontal gradient (G_x) is approximated by the convolution of image 'I' with kernel F_x:

$F_x = \begin{bmatrix} -1 & 0 & 1\\ -2& 0& 2\\ -1 & 0& 1 \end{bmatrix}$

The vertical gradient (G_y is approximated by the convolution of image 'I' with kernel F_y:

$F_y = \begin{bmatrix} 1 & 2 & 1\\ 0 & 0 & 0 \\ -1 & -2 & -1 \end{bmatrix}$

The magnitude of the local gradient is computed as:

$|G| = \sqrt{{G_x}^2 + {G_y}^2}$

And its direction is given by:

$\Theta = \tan^{-1}(G_y/G_x)$

Sobel, I. (1990). An isotropic 3× 3 image gradient operator, Machine vision for three-dimensional scenes, 376-379.