On Compressive Sensing Related Algorithms And Applications

This thesis considers the classical compressed sensing problem:that is to recover a sparse signal from some (possibly noisy) linear measurements. In other words, we have the following linear measurements: b=Af+e. Here,f is the sparse signal that we need to estimate, A∈RMxN (M (?) N) is the known measurement matrix with more columns that rows, and e is the measurement noise, b is our obtained measurement. We are interested in how to recover signal f from measurement b.First, we propose the iterative support detection of lq algorithm to recover the signal f. Compared with classical lq minimization algorithm, it requires fewer measurements and the speed is fast. Iterative support detection ofalgorithm is generalized from classical iterative support detection algorithm. We demonstrate by experiments that the relative error of our algorithm can be significantly reduced compared with classicalminimization algorithm.Then, we investigate the dual frame based-analysis model to recover the signals under the assumption that signals are sparse (or compressible) in a general frame. Clas-sical analysis tells us that subgaussian random matrices with optimal number of mea-surements could guarantee accurate recovery of signals with high probability, while for general Weibull random matrix it requires more than optimal number of measurements. We conclude that Weibull random matrices with optimal number of measurements are enough for recovery. Our result is based on Foucart and general dual frame theory. Our result should be significant for existing and upcoming l1-analysis models for signal recovery.Last, we consider greedy-based algorithms to recover signals under the assumption that signals are sparse in a redundant dictionary. Greedy-based algorithm has the ad-vantage of speed. Our proposed algorithm is the signal space hard thresholding pursuit (SSHTP) algorithm. The classical signal space CoSaMP algorithm-a variant of CoSaMP algorithm-has great theoretical results and performance. Our proposed algorithms inher-it many merits of hard thresholding pursuit and can numerically outperform signal space CoSaMP algorithm significantly in both accuracy and speed.The above results are around the classical compressed sensing problem, that is the goal is to recover sparse signals. However, in practice our problem is not only restricted in signal recovery. This thesis considers the problem of detecting and recovering1-sparse signal in a compressed sensing system. Suppose the objective signal could be zero or nonzero. After making a limited number of measurements, our goal is to design a rule which could reliably test whether the signal is zero and recover its support simultaneously. We conclude that under the restricted isometry property of measurement matrices, the scale of measurement numbers and SNR are given to guarantee diminishing probabilities of error. Simulation results demonstrate that our method is comparable to detection based on traditional sensing.