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 image with a filter from down to . {{Cite web|url=https://www.cv-foundation.org/openaccess/content_cvpr_2013/papers/Rigamonti_Learning_Separable_Filters_2013_CVPR_paper.pdf|title=Learning Separable Filters|access-date=2021-01-06|page=3|archive-url=https://web.archive.org/web/20200709094810/https://www.cv-foundation.org/openaccess/content_cvpr_2013/papers/Rigamonti_Learning_Separable_Filters_2013_CVPR_paper.pdf|archive-date=2020-07-09}}
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.
Another notable example of a separable filter is the Gaussian blur whose performance can be greatly improved the bigger the convolution window becomes.
References
{{Reflist}}