Shidong Li

Ningning Han, Shidong Li, Zhanjie Song et al.

Iterative algorithms based on thresholding, feedback and null space tuning (NST+HT+FB) for sparse signal recovery are exceedingly effective and fast, particularly for large scale problems. The core algorithm is shown to converge in finitely many steps under a (preconditioned) restricted isometry condition. In this paper, we present a new perspective to analyze the algorithm, which turns out that the efficiency of the algorithm can be further elaborated by an estimate of the number of iterations for the guaranteed convergence. The convergence condition of NST+HT+FB is also improved. Moreover, an adaptive scheme (AdptNST+HT+FB) without the knowledge of the sparsity level is proposed with its convergence guarantee. The number of iterations for the finite step of convergence of the AdptNST+HT+FB scheme is also derived. It is further shown that the number of iterations can be significantly reduced by exploiting the structure of the specific sparse signal or the random measurement matrix.

Peter G. Casazza, Gitta Kutyniok, Shidong Li

Let $\{W_i\}_{i\in I}$ be a (redundant) sequence of subspaces each being endowed with a weight $v_i$, and let $\mathcal{H}$ be the closed linear span of the $W_i$'s, a composite Hilbert space. Provided that $\{(W_i,v_i)\}_{i \in I}$ satisfies a certain property which controls the weighted overlaps of the subspaces, it is called a {\em fusion frame}. These systems contain conventional frames as a special case, however they go far ``beyond frame theory''. In case each subspace $W_i$ is equipped with a frame system $\{f_{ij}\}_{j \in J_i}$ by which it is spanned, we refer to $\{(W_i,v_i,\{f_{ij}\}_{j \in J_i})\}_{i \in I}$ as a {\em fusion frame system}. In this paper, we describe a weighted and distributed processing procedure that fuse together information in all subspaces $W_i$ of a fusion frame system to obtain the global information in $\mathcal{H}$. The weighted and distributed processing technique described in fusion frames is not only a natural fit in distributed processing systems such as sensor networks, but also an efficient scheme for parallel processing of very large frame systems. We further provide an extensive study of the robustness of fusion frame systems.