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).