Pattern Recognition

Abstract

This chapter is the heart of the book and contains the main contributions. Here, we present a framework for pattern recognition on groups, based on Fourier invariants. Our aim is to give an effective procedure for discriminate functions up to the action of the left-regular representation of some group. In the first part of the chapter, we introduce the simplest Fourier-based invariants that we will focus on: the power spectrum and the bispectrum, and we show that the latter are weakly complete with respect to the left-regular representation, i.e., generic functions have the same bispectrum if and only if for some. In the second part of the chapter, we focus on the problem of discriminating functions on under the action of the semi-direct product, as given by its quasi-regular representation. For this aim, we will exploit the lifts presented in Chap. 3 by showing the bispectrum to be weakly complete for regular cyclic lifts but not for left-invariant ones. This yields us to consider stronger invariants, the rotational power spectrum and rotational bispectrum invariants. We then prove the main theorem of the chapter: Theorem 5.5, which states that, up to a centering operator, these invariants are weakly complete on left-invariant lifts of function in. Some stronger version of this theorem are also presented in the case of real functions and when. Finally, we conclude the chapter by presenting the extension of this theory to almost-periodic functions.