Unknown pleasures: phase retrieval for time-frequency and time-scale structured data

Phase retrieval is an umbrella term given to various inverse problems in which one aims to recover signals from magnitude-only measurements. It enjoys a rich history: being linked to multiple Nobel prizes in diverse fields awarded over the past 100 years. It moreover is a beautiful mathematical problem in the sense that it is easy to pose while being hard to solve. In this thesis, we mainly consider phase retrieval problems with time-frequency and time-scale structured data which are inspired by applications in audio processing. In spite of its rich history, there are multiple deeply fundamental questions in phase retrieval which are only partially answered, as of now. Those include questions about two of the three favourite topics of every applied mathematician: the uniqueness and stability of solutions (existence is typically guaranteed by the problem formulation). The specific problem that phase retrieval with short-time Fourier transform magnitude measurements can or must be ill-conditioned was a particularly strong driving force for our research in the past four years. Indeed, the main aim of the first third of this thesis is to illustrate the ill-conditionedness of the discrete short-time Fourier transform phase retrieval problem and to propose a regularisation. The second third of the thesis deals with different novel uniqueness results for phase retrieval problems with time-frequency and time-scale structured data. In particular, questions about the uniqueness of sampled phase retrieval problems will be considered. The final third of the thesis examines the retrieval of entire functions from magnitude measurements on diverse subsets of the complex plane. The research in this final part of the thesis can naturally be linked to phase retrieval problems for the Gabor and the Cauchy wavelet transform. Using results from harmonic and functional analysis, from sampling theory and from complex analysis - in particular, the factorisation of entire functions - we prove multiple new results related to the stability and the uniqueness of different phase retrieval problems. We want to highlight the four most important findings here and refer the interested reader to the outlook at the end of the first section for a summary of further findings. First, we show that discrete short-time Fourier transform phase retrieval of bandlimited functions can be regularised using a semi-global phase retrieval regime as first proposed by Alaifari, Daubechies, Grohs and Yin for the Gabor transform phase retrieval problem. As signals in digital audio processing are discrete and bandlimited, this provides concrete evidence for the usefulness of semi-global phase retrieval regimes in audio processing applications. Secondly, we show that the phase retrieval problem of recovering real-valued bandlimited functions from samples of their short-time Fourier transform magnitudes enjoys uniqueness under mild assumptions on the window function. This is the first uniqueness result for sampled short-time Fourier transform phase retrieval and sparked research on sampled phase retrieval for time-frequency and time-scale structured data by a diverse cast of authors. Thirdly, we show that the phase retrieval problem of recovering real-valued bandlimited functions from the magnitude of their coefficients with respect to certain wavelet frames enjoys uniqueness. This is the first uniqueness result for sampled wavelet transform phase retrieval with complex-valued measurements. Finally, we provide a full classification of all the sources of non-uniqueness in the recovery of entire functions of finite order from magnitude measurements on two general lines in the complex plane. This classification has many implications for Gabor and Cauchy wavelet phase retrieval. Among these implications is our recent construction of counterexamples for the sampled Gabor phase retrieval problem