Phaseless Signal Reconstruction in Infinite Dimensional and Redundant Signal Spaces
Final Report Abstract
The problem of recovering a signal from amplitude samples without any phase information, also known as phase retrieval, arises in many different practical applications such as X-ray crystallography, astronomical imaging or speech processing, to mention only a few. Recent years witnessed considerable progress in the theoretical treatment of the phase retrieval problem in finite dimensional signal spaces whereas there was only little research on the phase retrieval problem in infinite dimensional signal spaces and so this project focused on this aspect of the phase retrieval problem. In the first part of the project, we proposed a sampling scheme for band-limited functions and for functions in general shift-invariant signal spaces such that every signal in this spaces is uniquely determined by the amplitudes of these samples. For bandlimited functions, it could be shown that an average sampling rate of at least four times the Nyquist rate is sufficient for signal recovery whereas an average rate of four times the rate of innovation is sufficient for signals in shift invariant spaces. Moreover, we where able to provide a corresponding recovery algorithm. Additionally, we studied the phase retrieval problem in Hardy spaces, i.e. on spaces of analytic function inside the unit disk. Here, we could show that any function (without a singular part) in such a space is uniquely determined by its amplitude on the unit circle and on a second circle strictly inside the unit disk. The obtained results for bandlimited functions where specialized to finite dimensional spaces. This way, we obtained several sets of 4N − 4 deterministic measurement vectors for the phase retrieval problem in the N-dimensional Euclidean space CN . These sets of measurement vectors allow phase retrieval for almost all (up to a set of first category) vectors in CN and we provided a corresponding fast and efficient recovery algorithm. For the case where the measurements are contaminated by additive noise, we provided an upper bound on the corresponding recovery error showing that the proposed phase retrieval procedure is stable. Moreover, we could prove that these sets of measurement vectors satisfied the conditions for PhaseLift, allowing signal recovery also by a semi-definite program. In the second part of the project, we included prior information on the sparsity of the signals into the phase retrieval procedure. We showed how the sampling scheme for band-limited signals has to be adapted such that phase retrieval for sparse multiband signals is possible with an average sampling rate which is proportional to the sparsity of the signal. Specifically, we provided a sampling scheme and a corresponding recovery algorithm which is able to reconstruct sparse multiband signal from amplitude samples taken at a rate of at least eight times the Landau rate. In the finite dimensional setting, we showed that any k-sparse vectors in CN is uniquely determined by 8k − 4 specific phaseless measurements and we gave a general procedure how the corresponding measurement vectors can be constructed. Additionally, it was shown how a corresponding recovery algorithm is obtained. In all parts of the project, we performed extensive numerical simulations showing the applicability of the proposed recovery schemes and illustrating the derived error bounds. Moreover, comparisons with other known phase retrieval procedures (e.g. PhaseLift) showed a similar performance by a much lower computational complexity.
Publications
- Phaseless Signal Recovery in Infinite Dimensional Spaces using Structured Modulations, J. Fourier Anal. Appl. 20 (Dec. 2014), no. 6, 1212–1233
V. Pohl, F. Yang, and H. Boche
(See online at https://doi.org/10.1007/s00041-014-9352-3) - Fast Compressed Phase Retrieval from Fourier Measurements, Proc. 40th Intern. Conf. on Acoustics, Speech, and Signal Processing (ICASSP), Apr. 2015
C. Yapar, V. Pohl, and P. Boche
(See online at https://doi.org/10.1109/ICASSP.2015.7178575) - Phase Retrieval from Low-Rate Samples, Sampling Theory in Signal and Image Processing 14 (2015), no. 1, 71–99
V. Pohl, F. Yang, and H. Boche
- Modulated Wideband Converter for Compressive Phase Retrieval of Bandlimited Signals, Proc. 11th Intern. ITG Conf. on Systems, Communications and Coding (SCC), Feb. 2017
V. Pohl, P. Boche, and C. Yapar
- Phase Retrieval in Spaces of Analytic Functions in the Unit Disk, Proc. 12th Intern. Conf. on Sampling Theory and Applications (SampTA), July 2017, pp. 336–340
V. Pohl, N. Li, and H. Boche
(See online at https://doi.org/10.1109/SAMPTA.2017.8024411) - Limits of Calculating the Finite Hilbert Transform from Discrete Samples, Appl. Comput. Harmon. Anal 46 (Jan. 2019), no. 1, 66–93
H. Boche and V. Pohl
(See online at https://doi.org/10.1016/j.acha.2017.03.002)
