Frequency subspace amplitude flow for phase retrieval.

Area of Science:

• Signal Processing
• Computational Imaging
• Applied Mathematics

Background:

• Reconstructing complex-valued signals from magnitude-only measurements is a fundamental challenge in various scientific fields.
• Existing methods often require additional constraints or exhibit limitations in sample complexity and flexibility.

Purpose of the Study:

• To introduce a novel and efficient algorithm, Frequency Subspace Amplitude Flow (FSAF), for reconstructing complex-valued signals from phaseless measurements.
• To demonstrate the superiority of FSAF over state-of-the-art methods in terms of accuracy and sample complexity.
• To highlight the flexibility and adaptability of FSAF to different signal types and noise models.

Main Methods:

• FSAF employs a two-stage approach: spectral method for low-frequency coefficient approximation and truncated conjugate gradient for refinement.
• The method utilizes an amplitude-based nonconvex formulation for signal reconstruction.
• It is designed for easy implementation and applicability to natural images without extra constraints.

Main Results:

• FSAF significantly improves the exact reconstruction of complex natural signals.
• It reduces sample complexity from 3.2n to 2.2n under the Gaussian model and from 5n to 3n under the coherent diffraction pattern (CDP) model.
• The method achieves these improvements without increasing computational complexity.

Conclusions:

• FSAF offers a powerful and flexible solution for phaseless signal reconstruction.
• The algorithm demonstrates state-of-the-art performance, particularly in reducing sample requirements.
• FSAF's adaptability makes it a valuable tool for diverse signal processing applications.