"Frame theory and Phase Retrieval"Ghoreishi, DorsaFrames, like orthonormal bases, give a continuous, linear, and stable reconstruction formula for vectors in a Hilbert space. However, frames allow for redundancy, and this makes frames much more adaptable for theory and applications. Phase retrieval is one of the applications of frame theory that has been used in X-ray crystallography and coherent diffraction imaging where only the intensity of each linear measurement of a signal is available and the phase information is lost. Phase retrieval require the redundancy of a frame, and is not possible with a basis. Therefore, it is important to determine the exact amount of redundancy which is necessary to do phase retrieval using a frame. Stability of phase retrieval under error is of great importance. 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 with a certain stability bound, but discrete frames are better suited for computations. Because of this, it is important to determine when a continuous frame may be sampled to construct a frame with given frame bounds which satisfy phase retrieval and what are the tools one would pick to determine when a continuous frame may be randomly sampled to achieve such a frame with high probability. In this research, we used a unique approach to expanding the mathematical theory of phase retrieval by using a combination of techniques from frame theory, probability, and the geometry of Banach spaces. (Joint work with Daniel Freeman) |
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