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 Gx and Gy 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.

Read more about this topic:  Sobel Operator

Famous quotes containing the words technical and/or details:

    “In effect, to follow, not to force the public inclination; to give a direction, a form, a technical dress, and a specific sanction, to the general sense of the community, is the true end of legislature.”
    —Edmund Burke (1729–1797)

    “Different persons growing up in the same language are like different bushes trimmed and trained to take the shape of identical elephants. The anatomical details of twigs and branches will fulfill the elephantine form differently from bush to bush, but the overall outward results are alike.”
    —Willard Van Orman Quine (b. 1908)