Filtered gradient compressive sensing reconstruction algorithm for sparse and structured measurement matrices

摘要

Compressive sensing state-of-the-art proposes random Gaussian and Bernoulli as measurement matrices. Nevertheless, often the design of the measurement matrix is subject to physical constraints, and therefore it is frequently not possible that the matrix follows a Gaussian or Bernoulli distribution. Examples of these limitations are the structured and sparse matrices of the compressive X-Ray, and compressive spectral imaging systems. A standard algorithm for recovering sparse signals consists in minimizing an objective function that includes a quadratic error term combined with a sparsity-inducing regularization term. This problem can be solved using the iterative algorithms for solving linear inverse problems. This class of methods, which can be viewed as an extension of the classical gradient algorithm, is attractive due to its simplicity. However, current algorithms are slow for getting a high quality image reconstruction because they do not exploit the structured and sparsity characteristics of the compressive measurement matrices. This paper proposes the development of a gradient-based algorithm for compressive sensing reconstruction by including a filtering step that yields improved quality using less iterations. This algorithm modifies the iterative solution such that it forces to converge to a filtered version of the residual A~Ty, where y is the measurement vector and A is the compressive measurement matrix. We show that the algorithm including the filtering step converges faster than the unfiltered version. We design various filters that are motivated by the structure of A~Ty. Extensive simulation results using various sparse and structured matrices highlight the relative performance gain over the existing iterative process.