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. 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.