How Does a Separable Filter Work?

1. What makes a filter separable?
2. What are the advantages of a separable filter?
3. Why is separability useful?
4. Is a box filter a separable filter?

What makes a filter separable?

What is a separable filter? A two-dimensional filter kernel is separable if it can be expressed as the outer product of two vectors. For example, let's look at a Sobel kernel. This kernel can be written as a matrix product of a column and a row vector.

What are the advantages of a separable filter?

The main advantage of separable filtering is quite clear; much reduced computational cost. In fact even the 2D-FFT algorithm makes use of it as the 2D-DFT kernel is separable.

Why is separability useful?

Separable filters are one of the most useful tools in image processing and they can turn algorithms from “theoretical and too expensive” to practical under the same computational constraints.

Is a box filter a separable filter?

Both, the Box filter and the Gaussian filter are separable: – First convolve each row with a 1D filter – Then convolve each column with a 1D filter.