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 cost of computing the operator.

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.

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