RUI: Development of Sparsity-Inducing Dual Frames and Applications

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Li1313490 The investigator studies the new notion of sparsity-inducing dual frames and their applications. Theme 1 defines sparsity-inducing dual frames of a non-exact frame. Preliminary studies show that if a signal has a sparse frame representation, then there is a sparsity-inducing dual frame (sparse dual) that produces the exact sparse coefficient when applied to the signal. Optimality issues of sparse duals and common sparse dual frames to classes of signals are examined. Theme 2 develops iterative algorithms for a sparse-dual-based analysis approach in sparse signal recovery. Improvement and convergence analysis of an alternative iterative algorithm are among the goals here. Theme 3 aims to understand the greatly improved sparse signal recovery capacity of the sparse-dual-based analysis method. This appears to be related to some geometric properties of the sparse hyperplane -- the affine subspace of the sparse representation fused with sparse dual information. One goal is to derive algorithms that adaptable to the geometric properties of the sparse hyperplane for greater effectiveness. Theme 4 develops the notion of sparse duals for practical scenarios where signals are merely approximately sparse with frames. Theme 5 develops "arbitrary" filter banks for multi-channel signal/image requisition and multi-channel communication applications. These systems exhibit arbitrary inter-channel relationships because they generally do not satisfy the commonly known multi-rate filter bank constraints rooted in wavelet theories. The challenge lies in actual constructions of realistic or FIR synthesis filters for these systems. Signal processing generally refers to making a signal easy to handle or of better quality. It is a common task in modern systems that transmit or store information. In this project, the principal novelties introduced to signal processing are the new notion of sparsity-inducing dual frames (sparse duals) and ideas of recovering underlying signals from much smaller samples of the signal. Sparsity-inducing dual frames are sets of optimal and basic signal components used in obtaining the sparsest (the smallest number of nonzero) signal decomposition coefficients. An advantage of sparse duals is that they can improve the recovery of a signal from a small number of samples of the signal. The investigator develops effective methods of sparse signal recovery. Results of the project could lead to simpler and more effective sampling devices, radar systems, commercial imaging techniques, geographic survey and mapping systems, wireless communication systems, multi-sensor/camera surveillance systems, and multi-channel medical image requisition methods. The project provides training opportunities for undergraduate students.