On the connection between uniqueness from samples and stability in Gabor phase retrieval.

Area of Science:

• Signal Processing
• Harmonic Analysis
• Mathematical Physics

Background:

• Gabor phase retrieval aims to reconstruct signals from Gabor transform magnitudes.
• A potential link between discrete unique solvability and continuous stability was previously hypothesized.
• This connection is crucial for understanding the robustness of signal reconstruction.

Purpose of the Study:

• To investigate and disprove the hypothesized link between discrete Gabor phase retrieval uniqueness and continuous stability.
• To establish theoretical bounds on the conditions for unique signal recovery.
• To explore the relationship between instability in phase retrieval and spectral properties of related operators.

Main Methods:

• Mathematical analysis of Gabor transform properties.
• Construction of specific functions demonstrating non-uniqueness.
• Topological arguments using density in function spaces.
• Analysis of Laplacian eigenfunctions and their relation to instability.

Main Results:

• Proved that discrete Gabor phase retrieval uniqueness does not imply continuous stability.
• Demonstrated the existence of signals that break uniqueness from samples but maintain continuous stability.
• Established that counterexamples to unique recovery from samples are dense in the relevant function space.
• Developed an intuitive link between phase retrieval instability directions and Laplacian eigenfunctions.

Conclusions:

• The hypothesized link between discrete and continuous Gabor phase retrieval properties is invalid.
• Signal reconstruction from discrete Gabor magnitudes can be non-unique even when continuous stability holds.
• The density of non-unique recovery examples highlights challenges in practical phase retrieval.
• Instability in Gabor phase retrieval is connected to specific spectral properties of the Laplacian operator.