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Separable filter

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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 . [1]


1. A two-dimensional smoothing filter:

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

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

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.


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