# 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 ${\displaystyle N\times M}$ image with a ${\displaystyle m\times n}$ filter from ${\displaystyle {\mathcal {O}}(M\cdot N\cdot m\cdot n)}$ down to ${\displaystyle {\mathcal {O}}(M\cdot N\cdot (m+n))}$. [1]

## Examples

1. A two-dimensional smoothing filter:

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

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

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

3. The Sobel operator, used commonly for edge detection:

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

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.

Another notable example of a separable filter is the Gaussian blur whose performance can be greatly improved the bigger the convolution window becomes.

## References

1. ^ "Learning Separable Filters" (PDF). p. 3. Archived from the original (PDF) on 2020-07-09. Retrieved 2021-01-06.