Separable filter

A separable filter in image processing can be written as product of two more simple filters. Typically a 2-dimensional convolution operation is separated into two 1-dimensional filters. This reduces the cost of computing the operator.

Examples

1. A two-dimensional smoothing filter is separated in this sample:

${\frac {1}{3}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}*{\frac {1}{3}}{\begin{bmatrix}1&1&1\end{bmatrix}}={\frac {1}{9}}{\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}}$

2. Another two-dimensional smoothing filter with stronger weight in the middle:

${\frac {1}{4}}{\begin{bmatrix}1\\2\\1\end{bmatrix}}*{\frac {1}{4}}{\begin{bmatrix}1&2&1\end{bmatrix}}={\frac {1}{16}}{\begin{bmatrix}1&2&1\\2&4&2\\1&2&1\end{bmatrix}}$

3. Sobel operator (edge detection)

$\mathbf {G_{x}} ={\begin{bmatrix}\quad ~&\quad ~&\quad ~\\[-2.5ex]1&0&-1\\2&0&-2\\1&0&-1\end{bmatrix}}*A={\begin{bmatrix}1\\2\\1\end{bmatrix}}*{\begin{bmatrix}+1&0&-1\end{bmatrix}}*A$

This works also for Prewitt operator.