Frame Theory And Its Applications

Since its introduction in the early 1950’s, Hilbert space frame theory has become an active area of research due to its applications in engineering and physics, including in speech recognition, optical imaging, and X-ray crystallography. Frames, like orthonor-mal 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 in which only the intensity of each linear measurement of a signal is available and the phase information is lost. In 2006, Balan, Casazza, and Edidin introduced a more powerful notion of phase retrieval using the magnitude of frame coefficients. Closely related to the subject of phase retrieval is weak phase retrieval. Weakening the con-ditions of phase retrieval, in which we have fewer measurements, still satisfies most of the properties of phase retrieval. In other words it is not “weak” at all. In this talk, we give an overview of phase retrieval and weak phase retrieval. In addition, a review of current phase retrieval algorithms will be discussed; however, an algorithm for weak phase retrieval has yet to be established.