Phase Retrieval: Uniqueness and Stability

@article{Grohs2019PhaseRU,
  title={Phase Retrieval: Uniqueness and Stability},
  author={Philipp Grohs and Sarah Koppensteiner and Martin Rathmair},
  journal={SIAM Rev.},
  year={2019},
  volume={62},
  pages={301-350}
}
The problem of phase retrieval, i.e., the problem of recovering a function from the magnitudes of its Fourier transform, naturally arises in various fields of physics, such as astronomy, radar, speech recognition, quantum mechanics and, perhaps most prominently, diffraction imaging. The mathematical study of phase retrieval problems possesses a long history with a number of beautiful and deep results drawing from different mathematical fields, such as harmonic analyis, complex analysis, or… 
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References

SHOWING 1-10 OF 101 REFERENCES

Phase retrieval techniques for radar ambiguity problems

The radar ambiguity function plays a central role in the theory of radar signals. Its absolute value (¦A(u)¦) measures the correlation between the signal u emitted by the radar transmitter and its…

Phase retrieval in infinite-dimensional Hilbert spaces

The main result of this paper states that phase retrieval in infinite-dimensional Hilbert spaces is never uniformly stable, in sharp contrast to the finite dimensional setting in which phase…

Phase Retrieval from Coded Diffraction Patterns

Saving phase: Injectivity and stability for phase retrieval

Phase Retrieval In The General Setting Of Continuous Frames For Banach Spaces

It is shown that the problem of phase retrieval is never uniformly stable in infinite dimensions and the stability properties cannot be improved by oversampling the underlying discrete frame.

The Question of Phase Retrieval in Optics

It is generally asserted that valuable phase information is irretrievably lost in the squaring operation required to obtain the intensity from the amplitude distribution in the image of a star formed…

Complex sequences with low periodic correlations

The upper and lower bound lies in the fact that they can be shown to agree for a large range of P, and M and R*(d)= co(R,‘, R/)(d) so that the upper and Lwer bounds agree.

The Direct Methods of X-ray Crystallography

The electron density function ρ(r) in a crystal determines its difaction pattern, that is, both the magnitudes and phases of its x-ray diffraction maxima, and conversely. If, however, as is always…

On the determination of the phase of a Fourier integral. I

Uniqueness results in an extension of Pauli's phase retrieval problem

...