Abstract
AbstractThe phase retrieval problem has a long and rich history with applications in physics and engineering such as crystallography, astronomy, and laser optics. Usually, the phase retrieval consists in recovering a real-valued or complex-valued signal from the intensity measurements of its Fourier transform. If the complete phase information in frequency domain is lost then the problem of signal reconstruction is severelly ill-posed and possesses many non-trivial ambiguities. Therefore, it can only be solved using appropriate additional signal information. We restrict ourselves to one-dimensional discrete-time phase retrieval from Fourier intensities and particularly consider signals with finite support. In the first part of this section, we study the structure of the arising ambiguities of the phase retrieval problem and show how they can be characterized using the given Fourier intensity. Employing these observations, in the second part, we study different kinds of a priori assumptions on the signal, where we are especially interested in their ability to reduce the non-trivial ambiguities or even to ensure uniqueness of the solution. In particular, we consider the assumption of non-negativity of the solution signal, additional magnitudes or phases of some signal components in time domain, or additional intensities of interference measurements in frequency domain. Finally, we transfer our results to phase retrieval problems where the intensity measurements arise, for example, from the Fresnel or fractional Fourier transform.
Highlights
As a consequence of this characterization, we show how ambiguities of the discrete-time phase retrieval problem are related to the true solution signal
As we have seen in Theorem 24.3, the solution set of the discrete-time phase retrieval problem usually consists of a vast number of non-trivially different solutions that strongly differ in shape and form
There is no simple way to decide from the given intensity data whether the a priori condition of non-negativity is helpful for reducing the number of ambiguities of the discrete-time phase retrieval problem
Summary
In the classical phase retrieval problem, one is usually faced with the recovery of a complex-valued signal from intensity measurements of its Fourier transform. The obtained restricted onedimensional phase retrieval problem with a priori magnitude information in time domain can be efficiently solved by multilevel Gauß–Newton methods [6, 15, 16] While these numerical methods work well in certain cases, their stability strongly depends on the given Fourier intensities [17]. 24.2 and 24.3, we introduce the one-dimensional discrete-time phase retrieval problem in more detail and derive a characterization of the entire solution set by factorizing the autocorrelation function—the squared Fourier intensity—suitably Using this characterization, we show that each ambiguity is caused by rotation, time-shift, and conjugation and reflection of the factors in a convolution representation of the true signal. 24.4–24.6—we exploit our findings on the solution set to investigate different a priori assumptions and additional information about the signal with respect to their capability to ensure a unique recovery of the unknown true signal. Due to the structure of these transforms the characterization of the solution set and uniqueness guarantees can be transferred to the new setting
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
