Sobel Operator - Technical Details

Technical Details

As a consequence of its definition, the Sobel operator can be implemented by simple means in both hardware and software: only eight image points around a point are needed to compute the corresponding result and only integer arithmetic is needed to compute the gradient vector approximation. Furthermore, the two discrete filters described above are both separable:

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

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

and the two derivatives G_x and G_y can therefore be computed as

$\mathbf{G}_x = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} * \left ( \begin{bmatrix} -1 & 0 & 1 \end{bmatrix} * \mathbf{A} \right ) \quad \mbox{and} \quad \mathbf{G}_y = \begin{bmatrix} -1 \\ \ \ 0 \\ \ \ 1 \end{bmatrix} * \left ( \begin{bmatrix} 1 & 2 & 1 \end{bmatrix} * \mathbf{A} \right )$

In certain implementations, this separable computation may be advantageous since it implies fewer arithmetic computations for each image point.

Applying convolution K to pixel group P can be represented in pseudocode as:

N(x,y) = Sum of { K(i,j).P(x-i,y-j)}, for i,j running from -1 to 1. N(x,y) represents the new matrix resulted after applying the Convolution K to P. where P is pixel matrix.

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