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

稀疏约束下信号处理的理论、算法和应用本研究项目将研究在几个重要应用领域中稀疏约束下信号处理问题中出现的理论和计算问题。研究计划包括使用多数化理论来开发和识别合适的多样性措施，其最小化导致稀疏解决方案。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。基于梯度因子表示的算法以及基于仿射尺度变换（AST）的内点优化理论方法将是这项工作的起点。为了促进对方法的更全面的理解，并开发对噪声具有鲁棒性的方法，将采用贝叶斯框架。对多测量向量问题的重要推广进行了研究，极大地扩展了应用范围。将开发学习算法来调优特定应用程序环境所需的过完整字典，从而提高其整体效率。理论和算法的发展将以应用程序的要求为指导。将特别关注信号表示和使用脑磁图（MEG）（一种潜在重要的脑成像新方式）的神经磁成像的应用。

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).