Frame Theory and Phase Retrieval

Saint Louis University, Saint Louis MO

Investigators

Abstract

Frames give a continuous, linear, and stable method for reconstructing a signal from linear measurements. However, there are many situations where physical limitations result in the loss of important aspects of those measurements. This imposes constraints on which frames can be used for the analysis of a signal, and different algorithms are required for reconstruction based only on partial information. For example, phase retrieval is applied in X-ray crystallography and coherent diffraction imaging where scientists are only able to identify the magnitude (or intensity) of each linear measurement of a signal. A different scenario occurs when using sensors with a fixed range, such as a pixel in a digital camera. In this case, any measurement with an intensity above the range saturates the sensor which then outputs the maximum value. In both situations, we can formalize the physical limitations imposed by the measurement process as applying a non-linear operator to a sequence of linear measurements. Although these non-linear operators are very simple, the loss of linearity can cause significant difficulty for signal reconstruction in high dimensions which becomes further confounded in the presence of error. These kinds of reconstruction scenarios arise naturally in many circumstances, and researchers in a variety of disciplines have developed solutions for specific applications. This makes the mathematical foundation for solving these types of inverse questions particularly important, and has led to significant research being devoted to the mathematics of phase retrieval in particular. The investigators and their students are working on a unique approach to expanding the mathematical theory of phase retrieval and saturation recovery by using a combination of techniques from frame theory, probability, and the geometry of Banach spaces. Both phase retrieval and saturation recovery require the redundancy of a frame, and are not possible with a basis. One component of the project concerns identifying the exact amount of redundancy which is necessary to do phase retrieval or saturation recovery using a frame or fusion frame. The second component considers a generalization of the phase retrieval scenario to the setting of subspaces of Banach lattices where the goal is to identify a vector in a subspace from its absolute value. This connection allows for established techniques in Banach lattices to prove new theorems about phase retrieval, and also opens a new line of inquiry in the theory of Banach lattices itself. It is not only important for phase retrieval to be possible, but for phase retrieval to be stable under error. The best known methods for constructing frames for high dimensional spaces which do stable phase retrieval are random constructions which achieve a certain stability bound with high probability. It is much easier to construct continuous frames which do stable phase retrieval with a certain stability bound, but discrete frames are better suited for computations. Because of this, important parts of the project involve both (1) determining when a continuous frame may be sampled to construct a frame with given frame bounds which does phase retrieval with a given stability bound and (2) using probabilistic methods to determine when a continuous frame may be randomly sampled to achieve such a frame with high probability. The final component of the project introduces phase retrieval for vector bundles over manifolds. That is, instead of recovering a single vector up to a phase factor from the magnitude of its frame coefficients, the goal is to use a continuously moving frame to recover a section of a vector bundle up to an equivalence relation. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.