Roberts Cross - Formulation

Formulation

In order to perform edge detection with the Roberts operator we first convolve the original image, with the following two kernels:

$\begin{bmatrix} +1 & 0 \\ 0 & -1\\ \end{bmatrix} \quad \mbox{and} \quad \begin{bmatrix} 0 & +1 \\ -1 & 0 \\ \end{bmatrix}.$

Let be a point in the original image and be a point in an image formed by convolving with the first kernel and be a point in an image formed by convolving with the second kernel. The gradient can then be defined as:

$\nabla I(x,y) = G(x,y) = \sqrt{ G_x^2 + G_y^2 }.$

The direction of the gradient can also be defined as follows:

$\Theta(x,y) = \arctan{\left(\frac{G_y(x,y)}{G_x(x,y)}\right)}.$

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