RUI: Advanced Theories of Frames, Pseudoframes, and Optimality Issues with Applications

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This is a project to study several problems related to the theory and applications of frames and pseudoframes arising in the signal processing area. Frames and pseudoframes are basis-like systems. The use of frames and pseudoframes in signal processing comes naturally in problems such as irregular sampling, signal restoration/recovery, Gabor time-frequency representations, etc. The topics in this research program are largely linked to their practical values in applications. The first topic relates to the construction of affine frames with a generalized multiresolution structure (GMS) and applications, in which the investigator seeks to construct a GMS built upon affine pseudoframes for subspaces. The results of this study are expected to have practical values in terms of numerical efficiency and smooth filter design, and will be used in the study of applications in industrial problems such as VLSI design; The second topic is on pseudoframes for subspaces (PFFS) and their applications to irregular sampling and subspace signal processing problems. The key feature of PFFS is that (pseudoframe) elements are allowed to be outside of a concerned subspace, a freedom that permits the construction of simpler and implementable solutions such as sampling functions, which is otherwise difficult or impossible when restricted to the concerned subspace only; The third topic is on the role of various dual frames and their applications. The investigator will study optimality issues of frame applications using the dual frame formula derived by the PI. The results of the study serve as a foundation of applied optimal frame representations associated with (infinitely many) dual frames; The fourth topic focuses on refined time-frequency analysis via multi-Gabor expansions (MGE), which generalize the metaplectic representations that are useful in time-frequency (TF). The investigation will focus on theoretical and computational studies of refined and efficient TF representations of a signal using MGEs.