Optimal compressive sensing systems for signal acquisition, reconstruction and processing

简而言之，压缩感知（CS）系统通过纠正少量的“投影”来间接获取感兴趣的信号，而不是以所谓的奈奎斯特率均匀采样，这对于许多应用中遇到的宽带信号来说可能过高。这种新的信号采集范式彻底改变了传统的数字数据采集方式。拟议研究的目标有三个方面。首先，我们的目标是开发最优的CS系统，它需要比以往更少的测量次数，但却包含完整的数据信息。这需要减少传感设备的数量（或者在软件实现的情况下降低软件复杂性），从而提高处理效率并降低成本。这一目标将通过探索和最大化测量子系统和信号稀疏表示的过完备字典之间的不相干的新度量来实现。其次，我们研究了从有限数量的测量中恢复感兴趣数据的新方法，相对于现有算法具有更好的精度。我们感兴趣的是大规模数据，因此我们必须开发快速算法，使其在实时数据处理中有用。为此，我们将开发一类求解Lp型（p < 1）混合凸-非凸问题的近端梯度算法。第三，所开发的算法将应用于数字信号处理问题，如医学成像中的去噪、去模糊和分割问题，以及无线信道估计和消息恢复问题等通信问题。由于涉及大量数据（医学成像）或由于其实时性（无线通信应用），这些问题在各自的领域都是当前的、重要的和具有技术挑战性的问题。当我们提出的算法开始在解决这些问题和其他相关问题方面发挥关键作用，并具有令人满意的处理速度和性能时，信息处理社区将认为我们的研究工作是重要的。

In a nutshell, a compressive sensing (CS) system acquires a signal of interest indirectly by correcting a small number of its "projections" rather than evenly sampling it at the so-called Nyquist rate which can be prohibitively high for broadband signals encountered in many applications. This new signal acquisition paradigm has revolutionized the way digital data are traditionally acquired. The objectives of the proposed research are threefold. First, we aim at developing optimal CS systems that require fewer-than-ever number of measurements that yet contain complete information of the data. This entails reduced number of sensing devices (or lower software complexity in the case of software implementation), hence improving processing efficiency and reducing cost. This goal will be achieved by exploring and maximizing a new measure for the incoherence between the measurement subsystem and an overcomplete dictionary for sparse representation of signals. Second, we investigate new methods to recover the data of interest from the limited number of measurements with better accuracy relative to existing algorithms. Our interest is in large-scale data, hence we must develop fast algorithms for them to be useful in real-time data processing. To this end, we shall develop a class of proximal-gradient algorithms for solving Lp type (with p < 1) mixed convex-nonconvex problems. Third, the algorithms developed are to be applied to problems in digital signal processing such as de- nosing, de-blurring and segmentation in medical imaging and communications such as wireless channel estimation and message recovery problems. These problems are current, significant and technically challenging in their respective fields either because of their involvement in large amount of data (medical imaging) or because of their real-time nature (wireless communication applications). The information processing communities will regard our research endeavors as significant when the proposed algorithms begin to play a crucial role in solving these and other related problems with satisfactory processing speed and performance.