Uniqueness of results for multiple correlations of periodic functions

Abstract

In image processing, periodic functions arise in two ways: implicitly, in the discrete Fourier transform, and explicitly, in circular harmonic expansions. Multiple correlations, in particular the triple correlation, are used in both situations because they are invariant under image translation and because they are insensitive to additive Gaussian noise. The appeal of multiple correlations is further enhanced by several uniqueness theorems that show that most aperiodic functions of practical interest are determined uniquely up to a translation by their triple correlations. However, those theorems do not extend to cover periodic functions. We study the uniqueness properties of multiple correlations or, equivalently, higher-order-moment spectra, as their frequency-domain representations are called, for periodic functions. Our main result establishes that every real-valued periodic function that is square integrable on its fundamental domain is uniquely determined up to a translation by its infinite sequence of moment spectra. We provide a constructive proof by explicitly reconstructing a function’s Fourier coefficients (up to a linear phase term) from its moment spectra. Moreover, we derive a finite upper bound on the order of moment spectra that is required for characterizing a band-limited function. We show that the upper bound is tight for functions of one variable.