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.

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