Homometry and the Phase Retrieval Problem

Summary

This chapter studies in depth the notion of homometry, i.e. having identical internal shape, as seen from Fourier space, where homometry can be seen at a glance by the size (or magnitude) of the Fourier coefficients. Finding homometric distributions is then a question of choosing the phases of these coefficients, hence this problem is often called phase retrieval in the literature. Such a choice of phases is summed up in the objects called spectral units, which connect homometric sets together. I included the original proof of the one difficult theorem of this book (Theorem 2.10), which non-mathematicians are quite welcome to skip. Some generalisations of the hexachord theorem are given, followed by the few easy results on higher-order homometry which deserve some room in this book because they rely heavily on DFT machinery. An original method for phase-retrieval with singular distributions (the difficult case) is also given. Some knowledge of basic linear algebra may help in this chapter.