Theory, Algorithms, and Applications of Signal Processing with the Sparseness Constraint

稀疏约束信号处理的理论、算法和应用 本研究计画将探讨在几个重要应用领域中，具有稀疏性限制的信号处理问题所产生的理论与计算问题。研究计划包括使用优控制理论来开发和确定合适的多样性措施，其最小化导致稀疏的解决方案。然后，为了最小化这些措施，一类新的优化算法将被开发，分析和应用。算法的基础上的因子表示的梯度沿着与仿射尺度变换（AST）的内点优化理论的方法将是这项工作的出发点。为了便于更全面地理解这些方法，并开发对噪声具有鲁棒性的方法，将采用贝叶斯框架。多测量向量问题的重要扩展将被研究，大大扩展了应用范围。将开发学习算法，以调整特定应用环境所需的过完备字典，从而提高其整体效率。理论和算法的开发将由应用程序的要求指导。将特别注意使用脑磁图（MEG）（一种潜在的重要的大脑成像新模式）的信号表示和神经磁成像的应用。

CCR-9902961RaoTHEORY, ALGORITHMS, AND APPLICATIONS OF SIGNAL PROCESSING WITH THE SPARSENESS CONSTRAINT This research project will examine the theoretical and computational issues that arise in signal processing problems with the sparseness constraint in several important application domains. The research plan includes using majorization theory to develop and identify suitable diversity measures whose minimization leads to sparse solutions. Then, to minimize these measures, a new class of optimization algorithms will be developed, analyzed, and applied. Algorithms based on a factored representation for the gradient along with Affine Scaling Transformation (AST) based methods of interior point optimization theory will be the starting point of this work. To facilitate a more comprehensive understanding of the methods, and to develop methods robust to noise, a Bayesian framework will be employed. The important extension to the multiple measurement vector problem will be studied greatly expanding the range of applications. Learning algorithms will be developed to tune the required overcomplete dictionaries for specific application environments, thereby increasing their overall effectiveness. Theoretical and algorithmic development will be guided by the requirements of the applications. Particular attention will be given to the applications of signal representation and neuromagnetic imaging using Magnetoencephalography (MEG) (a potentially important new modality for the imaging of the brain).