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 computational costs on an $N\times M$ image with a $m\times n$ filter from ${\mathcal {O}}(M\cdot N\cdot m\cdot n)$ down to ${\mathcal {O}}(M\cdot N\cdot (m+n))$.

Examples

1. A two-dimensional smoothing filter:

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

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. The Sobel operator, used commonly for edge detection:

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

This works also for the Prewitt operator.

In the examples, there is a cost of 3 multiply–accumulate operations for each vector which gives six total (horizontal and vertical). This is compared to the nine operations for the full 3x3 matrix.