Separability

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In image processing, the word separability describes the property that the impulse response of a two-dimensional filter can be represented by multiplying two one-dimensional operators. Thus, the two-dimensional convolution can be reduced to two one-dimensional operations by applying the second operator to the intermediate result of the first. In image processing, the original 2D filter is broken down into x and y kernels, which are then applied to the original image one after the other. A separation of a 3 × 3 matrix into two 1D vectors must look like this:

But it is also possible to use other input and output variables. A 5 × 5 filter can be separated into two 3 × 3 matrices.

The goal of the separation is to save computing time. The use of a 2D N × N filter requires read accesses and multiplications, as well as additions. As a result of the separation, the computational effort can be reduced to read accesses and multiplications and additions.

properties

A separable 3x3 matrix has the following properties:

  • Rank ( ) = dim ( SR (A)) = dim ( ZR (A)) = 1
  • ZR ( ) is orthogonal to NR ( ) =

Examples

1. A two-dimensional smoothing filter is separated in this example:

2. The Gaussian filter (soft focus)

3. The Sobel operator (edge ​​detection)

This also works with the Prewitt operator .

See also

The linear separability (classifiability) refers to mathematical relations and should not be confused with separability in image processing.