Linear transformations are often used to rearrange the data into a particular form. In particular they can be used to separate different components of an image, for example separating signal from interference or noise. In general, with a linear transform, each output value is a linear combination of all of the input pixel values:
It extends and generalises local linear filtering of Equation 8.4 by removing the restriction of the window and allowing a more general selection of weights.
Direct implementation of Equation 10.1 is very expensive. If the input image is , then each output value requires multiplication and additions. For an output, there are therefore operations. Many useful transforms are separable, in that they can be decomposed into separate, independent transforms on the rows and columns. In this case, Equation 10.1 simplifies to:
reducing the number of operations to . Separable transforms can be represented in matrix form as: ...