This research addresses the fundamental science of measurement system design and recovery of large-scale structured signals of interest when (a) critical phase information cannot be measured, and (b) very few measurements are available. Also known in the technical literature as compressed phase-retrieval, this research has applications to a plethora of important scientific fields like optics, quantum physics, bio-medical imaging, astronomy, and material science. The inability to measure phase information renders the signal recovery problem particularly challenging. Most prior approaches are either computationally prohibitive and therefore hard to scale, or based on unproven heuristics that come with no performance guarantees. In contrast, this research fundamentally addresses the challenge of scale together with provable performance guarantees in the theory, design, experimental evaluation, and applications of compressed phase-retrieval systems.
This research addresses the theoretical and algorithmic foundations for next-generation compressed phase-retrieval systems. These are derived from a novel interdisciplinary mix of tools from coding theory, graph theory, statistical signal processing, as well as their integration into applications involving optical imaging and quantum information systems. The research revolves around a novel paradigm built on a family of sparse-graph codes that represent a radical departure from existing approaches based on either computationally intensive convex relaxation methods or greedy heuristics without performance guarantees. This research has the potential to revolutionize the design and applications of next-generation compressed phase retrieval systems, similar to how Low-Density-Parity-Check (LDPC) codes have revolutionized modern communication systems, with respect to (i) measurement cost (approaching the fundamental limits); (ii) computational and memory cost (enabling real-time or near-real-time processing); and (iii) provable performance guarantees (with respect to the problem scale and robustness to noise).
