Stable Infinite Dimensional Phase Retrieval

In a multitude of applications only the intensity of a wave-like signal can be measured while its phase remains unspecified. As a classical example we mention coherent diffraction imaging where an object is illuminated by a high-frequency monochromatic X-ray and what is measured is the intensity of the diffracted beam on an image plane. Assuming we can additionally restore the phase of the diffracted beam, it is possible to reconstruct the object of interest up to extremely high precision, thus forming the basis for the method of coherent diffraction imaging. This method has achieved remarkable successes, among which lies the discovery of the double helix structure of the DNA. However, similar problems also arise in a multitude of other application areas and they are in general known as `phase retrieval problems`. The numerical solution of such problems remains a big challenge, especially in the presence of measurement errors and/or noise, which in practice is unavoidable. As a matter of fact, these difficulties form a bottleneck for many applications related to phase retrieval. In recent breakthrough we could for the first time explain these problems by rigorously proving that in any phase retrieval problem even the slightest measurement error may in general lead to arbitrarily large reconstruction errors. In this project we plan to gain a complete mathematical understanding of the (in)stability properties of phase retrieval problems and to build a corresponding theory. Then, based on this understanding, we aim for the construction of a new class of algorithms, which cleverly bypass the unavoidable instabilities and consequently achieve a significantly improved reconstruction quality, even in the presence of noise. Applications of these new algorithmic methods range far beyond the area of applied mathematics and may potentially even lead to an increase of the best possible resolution of coherent diffraction imaging methods in nanoscale imaging.

Phase retrieval problems arise when a signal is to be reconstructed from measurements of its absolute value. A key example of such a problem is in coherent diffraction imaging. Among other things, these imaging methods are responsible for the discovery of the double helix structure of our DNA, and solving phase retrieval problems has been a key part in these developments. Unfortunately, most phase retrieval problems are currently solved with ad hoc methods without any guarantees on their correctness, in the sense that the algorithm always reconstructs the correct signal. The algorithmic reconstruction is further complicated by the fact that in most practical scenarios the measurements are very noisy. The construction of reliable and stable algorithms requires an improved understanding of the mathematical mechanisms underlying phase retrieval. In the past decades such an understanding could be achieved for certain stylized model problems. In this project we could for the first time achieve such an understanding for realistic infinite dimensional problems, particularly in the field of ptychography. Our results for the first time provide a full characterization of measurements that lead to a stable reconstruction. In order to establish this characterization we discovered a deep connection between phase retrieval problems and spectral clustering algorithms in data science. Using this connection, as well as deep results from Riemannian geometry and complex analysis, we could prove that the reconstruction is stable precisely if the measurements are connected in the sense of not consisting of more than one cluster. Our results allow for a number of practical conclusions. For example, we developed an algorithm that automatically partitions any given signal into parts that can be provably reconstructed in a stable fashion. Our results furthermore indicate how to construct novel experimental measurement setups to maximize the noise resilience of the phase reconstruction. Finally, we were able to develop the first algorithm capable of solving the ptychographic phase retrieval problem in a provably stable and efficient way.