Compressive imaging with complex wavelet transform and turbo AMP reconstruction

We extend the “turbo” belief propagation framework for compressive imaging to the dual-tree complex wavelet transform (DT-CWT) to exploit both sparsity and dependency across scales. Due to the near shift-invariance property and the improved angular resolution of DT-CWT, better reconstruction can be expected when incorporating with the compressed sensing (CS) algorithms. Two types priors to form the hidden Markov tree structure for the DT-CWT coefficients are considered. One models the real and imaginary components of DT-CWT separately while the other assumes the shared hidden states between the two. Simulation with natural images confirm an improved performance when iterating between the CS reconstruction and the DT-CWT HMT

Dynamic Compressive Sensing of Time-Varying Signals via Approximate Message Passing

In this work the dynamic compressive sensing (CS) problem of recovering sparse, correlated, time-varying signals from sub-Nyquist, non-adaptive, linear measurements is explored from a Bayesian perspective. While there has been a handful of previously proposed Bayesian dynamic CS algorithms in the literature, the ability to perform inference on high-dimensional problems in a computationally efficient manner remains elusive. In response, we propose a probabilistic dynamic CS signal model that captures both amplitude and support correlation structure, and describe an approximate message passing algorithm that performs soft signal estimation and support detection with a computational complexity that is linear in all problem dimensions. The algorithm, DCS-AMP, can perform either causal filtering or non-causal smoothing, and is capable of learning model parameters adaptively from the data through an expectation-maximization learning procedure. We provide numerical evidence that DCS-AMP performs within 3 dB of oracle bounds on synthetic data under a variety of operating conditions. We further describe the result of applying DCS-AMP to two real dynamic CS datasets, as well as a frequency estimation task, to bolster our claim that DCS-AMP is capable of offering state-of-the-art performance and speed on real-world high-dimensional problems.Comment: 32 pages, 7 figure

Compressive Sensing and Recovery of Structured Sparse Signals

In the recent years, numerous disciplines including telecommunications, medical imaging, computational biology, and neuroscience benefited from increasing applications of high dimensional datasets. This calls for efficient ways of data capturing and data processing. Compressive sensing (CS), which is introduced as an efficient sampling (data capturing) method, is addressing this need. It is well-known that the signals, which belong to an ambient high-dimensional space, have much smaller dimensionality in an appropriate domain. CS taps into this principle and dramatically reduces the number of samples that is required to be captured to avoid any distortion in the information content of the data. This reduction in the required number of samples enables many new applications that were previously infeasible using classical sampling techniques. Most CS-based approaches take advantage of the inherent low-dimensionality in many datasets. They try to determine a sparse representation of the data, in an appropriately chosen basis using only a few significant elements. These approaches make no extra assumptions regarding possible relationships among the significant elements of that basis. In this dissertation, different ways of incorporating the knowledge about such relationships are integrated into the data sampling and the processing schemes. We first consider the recovery of temporally correlated sparse signals and show that using the time correlation model. The recovery performance can be significantly improved. Next, we modify the sampling process of sparse signals to incorporate the signal structure in a more efficient way. In the image processing application, we show that exploiting the structure information in both signal sampling and signal recovery improves the efficiency of the algorithm. In addition, we show that region-of-interest information can be included in the CS sampling and recovery steps to provide a much better quality for the region-of-interest area compared the rest of the image or video. In spectrum sensing applications, CS can dramatically improve the sensing efficiency by facilitating the coordination among spectrum sensors. A cluster-based spectrum sensing with coordination among spectrum sensors is proposed for geographically disperse cognitive radio networks. Further, CS has been exploited in this problem for simultaneous sensing and localization. Having access to this information dramatically facilitates the implementation of advanced communication technologies as required by 5G communication networks

Compressed sensing with approximate message passing: measurement matrix and algorithm design

Compressed sensing (CS) is an emerging technique that exploits the properties of a sparse or compressible signal to efficiently and faithfully capture it with a sampling rate far below the Nyquist rate. The primary goal of compressed sensing is to achieve the best signal recovery with the least number of samples. To this end, two research directions have been receiving increasing attention: customizing the measurement matrix to the signal of interest and optimizing the reconstruction algorithm. In this thesis, contributions in both directions are made in the Bayesian setting for compressed sensing. The work presented in this thesis focuses on the approximate message passing (AMP) schemes, a new class of recovery algorithm that takes advantage of the statistical properties of the CS problem. First of all, a complete sample distortion (SD) framework is presented to fundamentally quantify the reconstruction performance for a certain pair of measurement matrix and recovery scheme. In the SD setting, the non-optimality region of the homogeneous Gaussian matrix is identified and the novel zeroing matrix is proposed with an improved performance. With the SD framework, the optimal sample allocation strategy for the block diagonal measurement matrix are derived for the wavelet representation of natural images. Extensive simulations validate the optimality of the proposed measurement matrix design. Motivated by the zeroing matrix, we extend the seeded matrix design in the CS literature to the novel modulated matrix structure. The major advantage of the modulated matrix over the seeded matrix lies in the simplicity of its state evolution dynamics. Together with the AMP based algorithm, the modulated matrix possesses a 1-D performance prediction system, with which we can optimize the matrix configuration. We then focus on a special modulated matrix form, designated as the two block matrix, which can also be seen as a generalization of the zeroing matrix. The effectiveness of the two block matrix is demonstrated through both sparse and compressible signals. The underlining reason for the improved performance is presented through the analysis of the state evolution dynamics. The final contribution of the thesis explores improving the reconstruction algorithm. By taking the signal prior into account, the Bayesian optimal AMP (BAMP) algorithm is demonstrated to dramatically improve the reconstruction quality. The key insight for its success is that it utilizes the minimum mean square error (MMSE) estimator for the CS denoising. However, the prerequisite of the prior information makes it often impractical. A novel SURE-AMP algorithm is proposed to address the dilemma. The critical feature of SURE-AMP is that the Stein’s unbiased risk estimate (SURE) based parametric least square estimator is used to replace the MMSE estimator. Given the optimization of the SURE estimator only involves the noisy data, it eliminates the need for the signal prior, thus can accommodate more general sparse models