Phase Retrieval from Very Few Measurements
Document Type
Article
Publication Date
5-15-2014
Abstract
In many applications, signals are measured according to a linear process, but the phases of these measurements are often unreliable or not available. To reconstruct the signal, one must perform a process known as phase retrieval. This paper focuses on completely determining signals with as few intensity measurements as possible, and on efficient phase retrieval algorithms from such measurements. For the case of complex M-dimensional signals, we construct a measurement ensemble of size 4M-4 which yields injective intensity measurements; this is conjectured to be the smallest such ensemble. For the case of real signals, we devise a theory of "almost" injective intensity measurements, and we characterize such ensembles. Later, we show that phase retrieval from M+1 almost injective intensity measurements is NP-hard, indicating that computationally efficient phase retrieval must come at the price of measurement redundancy.
DOI
10.1016/j.laa.2014.02.011
Source Publication
Linear Algebra and its Applications
Recommended Citation
Fickus, M., Mixon, D. G., Nelson, A. A., & Wang, Y. (2014). Phase retrieval from very few measurements. Linear Algebra and Its Applications, 449, 475–499. https://doi.org/10.1016/j.laa.2014.02.011
Comments
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Reviewed at MR3191879.
Previous version (Pre-print): arXiv:1307.7176 [math.FA]
Date submitted to arXiv [v1]: 26 Jul 2013