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 x M
m x n
l{O}(M ⋅ N ⋅ m ⋅ n)
l{O}(M ⋅ N ⋅ (m+n))
1. A two-dimensional smoothing filter:
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\begin{bmatrix}1\ 1\ 1\end{bmatrix}*
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\begin{bmatrix}1&1&1 \end{bmatrix} =
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\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:
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\begin{bmatrix}1\ 2\ 1\end{bmatrix}*
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\begin{bmatrix}1&2&1 \end{bmatrix} =
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\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.