The century-old problem known as “Phase retrieval” is the problem of recovering a complex signal from the magnitude of its Fourier transform. In the canonical phase retrieval setting, an electromagnetic field with an unknown phase and magnitude distribution undergoes a Fourier transform by propagating through free space or a lens and is then measured by a camera. Since a camera captures only intensity information, the phase retrieval problem must be solved to recover the original phase and magnitude profile. In realistic scenarios, this data may be noisy, and sometimes partial due to active blocking of the zero order diffraction (a common practice in some imaging applications). Current approaches to this problem either have large time complexity, require additional measurements or constraints, are unstable, and do not offer a guarantee for convergence1,2. In this seminar, I will present the latest updates on “Fast Phase Retrieval”3: the first and only algorithm capable of achieving deterministic recovery of a large class of complex objects from their oversampled Fourier transform in a polynomial number of arithmetic operations. Our algorithm achieves this recovery without requiring any manipulation of the object (such as adding a reference beam) nor requiring additional non-Fourier measurements, overcoming the strict limitations of existing algorithms. The stability to noise, O(Nlog(N)) arithmetic complexity and O(N) sample complexity, make it suitable for realistic, large-scale measurements. In many of these realistic scenarios, such as in coherent diffraction imaging, the central Fourier intensities are missing, typically blocked. This hinders the ability to recover the complex object. We develop a framework for recovering the blocked data by exploiting the object support and long range correlations. The same framework can interpolate blocked measurements and denoise measured intensities, even for large realistic datasets. We demonstrate the approach on both simulated and experimental diffraction data. Together, these methods allow, for the first time, fast, deterministic, and scalable recovery of complex objects from large (million pixel), noisy, and incomplete (blocked) Fourier-intensity measurements. References: 1. Jaganathan, K., Eldar, Y. C. & Hassibi, B. Phase retrieval: An overview of recent developments. arXiv:1510.07713. (2015) 2. Shechtman, Y. et al. Phase Retrieval with Application to Optical Imaging: A contemporary overview. IEEE Signal Process. Mag. 32, 87–109 (2015). 3. Brabec, C., Trajtenberg-Mills, S., Daniel, L. & Englund, D. Deterministic fast and stable phase retrieval in multiple dimensions. arXiv [eess.IV] (2024).

Dr. Sivan Trajtenberg Mills is an assistant professor at the EE department in Tel Aviv Univeristy, Israel. Her research is focused on development of optical tools and devices for quantum control. She was a postdoctoral researcher in the Quantum Photonics lab led by prof. Dirk Englund at the Massachusetts Institute of Technology (MIT). She received her Ph.D from Tel Aviv University, studying structured light in second order nonlinear interactions under supervision of prof. Ady Arie. Receiver of the 2025 Alon award for young faculty, Israel.