| The rapid development of information technology in recent decades has led to an increasing demand for data acquisition systems, such as audio recorders and imaging devices. Moreover, large amounts of data must be stored or transmitted, and the time or power consumption required for transmission must be reduced. Although many types of data are high dimensional, they can often be represented efficiently when being projected to another space, which implies that the data is sparse or compressible in this projected space. Compressive sensing is an emerging field that employs the sparsity property of data and acquires the data in a compressive fashion. Our goal is to reconstruct the original data from those compressive measurements. Generalized approximate message passing (GAMP) and approximate message passing (AMP) are iterative compressive signal reconstruction algorithms that enjoy many mathematical and practical advantages. In this dissertation we utilize these advantages and develop fast and robust compressive signal reconstruction algorithms.;This dissertation contains two parts. In the first part, we explore compressive signal re-construction algorithms that achieve application-oriented reconstruction quality. In order to evaluate the quality of reconstruction, standard error metrics such as square error are usually evaluated, and algorithms are developed to minimize these error metrics. However, reconstruction algorithms that minimize standard error metrics may not be robust, because in some applications they may not achieve satisfactory reconstruction quality. Therefore, we propose a compressive signal reconstruction algorithm that modifies the last iteration of GAMP and minimizes a broad range of user-defined error metrics, which provides great flexibility in achieving application-oriented performance.;Infinity norm error, also known as worst-case error, is an example of such non-standard error metrics. By minimizing the infinity norm error, we ensure that the reconstructed error for each signal component is modest. We explore some interesting theoretical properties of the worst-case error of signal reconstruction procedures.;In the second part of this dissertation, we study compressive imaging problems. Compressive imaging is an important application of compressive sensing, where the signals of interest are images. First, we consider a general compressive imaging problem where the imaging process is modeled by a random matrix; we then look into practical hyperspectral imaging problems, where the imaging process is modeled by a highly structured matrix that is far from random. Considering that AMP is an iterative signal reconstruction framework that performs scalar denoising at each iteration, we apply efficient and robust image denoisers within AMP iterations. Numerical results show that our algorithms for random matrix imaging and hyperspectral imaging are both fast and robust, and improve over the state of the art in terms of runtime and reconstruction quality. |
