Similarity methods in signal processing

A signal may contain information that is preserved by certain transformations of the signal. For example, the information phase-modulated signal is not altered by amplitude scaling of the signal. Many processing techniques have been developed to exploit such similarities. In the past, these algorithms have been developed in isolation without regard to common principles of invariance that tie them together. Similarity methods are presented as a unified method of designing processing algorithms invariant to specified transformations. These methods are based upon groups of continuous transformations known as local Lie groups and lead to a quasilinear partial differential equation. Solution of this partial differential equation specifies the form the signal processing operations must take. This form can then be applied using engineering judgment for algorithmic implementation. The paper presents an extended tutorial on Lie groups and similarity methods and quasilinear differential equations drawn from the mathematical literature. This is followed by several examples of signal processing interest that demonstrate that the similarity techniques may be applicable in certain kinds of signal processing problems.