Separable filter

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

But it is also possible to use other input and output sizes. For example, a 5 × 5 filter in two 3 × 3 matrices are separated.

The goal of separation is a saving of computing time. The use of a 2D N × N filter requires read access and multiplications and additions. By separating the amount of calculation for read accesses and multiplications and additions can be reduced.

Properties

A separable 3x3 matrix has the following property:

  • Rank () = dim ( SR ) = dim ( ZR ) = 1
  • ZR () is orthogonal to the NO ()

Examples

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

2 The Gaussian filter ( blur )

3 The Sobel operator ( edge detection )

This also works with the Prewitt operator.

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