Sobel and Prewitt operators decomposition

Question

I was reading in the wikipedia page of Sobel operator and Prewitt operator that is possible to decompose these two operators (quote form the Formulation paragraph):

"as the products of an averaging and a differentiation kernel, they compute the gradient with smoothing."

I know that this means that I can rewrite and simplify a Prewitt mask (for example) in this way:

\begin{align*} \begin{bmatrix} +1 & 0 & -1 \\ +1 & 0 & -1 \\ +1 & 0 & -1 \end{bmatrix} = \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} \begin{bmatrix} +1 & 0 & -1 \end{bmatrix} \end{align*}

But I don't understand what are in these cases the averaging and the differentiation kernel? And, why it is written that they can compute the gradient with smoothing?

Answer 1

The part: \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} acts on vertical pixels \begin{bmatrix} x_{1,\cdot}\\ x_{2,\cdot}\\ x_{3,\cdot} \end{bmatrix} as a weighted sum with equal weights, and if divided by $3$ like a 3-point average $(x_{1,\cdot}+x_{2,\cdot}+x_{3,\cdot})/3$.

Then, \begin{bmatrix} +1 & 0 & -1 \end{bmatrix} acts on a row of horizontal pixels \begin{bmatrix} x_{\cdot,1}& x_{\cdot,2}& x_{\cdot,3} \end{bmatrix} like the discrete 3-point derivative: $(x_{\cdot,1}-x_{\cdot,3})/2$ up to the $2$ factor. Here, up to a global $6$ factor, it acts like a smoother vertically combined with a gradient horizontally. The same reasoning works when you switch directions. This notion relates to the separability of a 2D mask into the product of two 1D operators.

Additional sources:

• What's logic behind the constuction of Sobel's filter in image processing?
• How to find out if a transform matrix is separable?