  Compressive covariance estimation has arisen as a class of techniques whose
aim is to obtain second-order statistics of stochastic processes from
compressive measurements. Recently, these methods have been used in various
image processing and communications applications, including denoising, spectrum
sensing, and compression. Notice that estimating the covariance matrix from
compressive samples leads to ill-posed minimizations with severe performance
loss at high compression rates. In this regard, a regularization term is
typically aggregated to the cost function to consider prior information about a
particular property of the covariance matrix. Hence, this paper proposes an
algorithm based on the projected gradient method to recover low-rank or
Toeplitz approximations of the covariance matrix from compressive measurements.
The algorithm divides the compressive measurements into data subsets projected
onto different subspaces and accurately estimates the covariance matrix by
solving a single optimization problem assuming that each data subset contains
an approximation of the signal statistics. Furthermore, gradient filtering is
included at every iteration of the proposed algorithm to minimize the
estimation error. The error induced by the proposed splitting approach is
analytically derived along with the convergence guarantees of the proposed
method. The algorithm estimates the covariance matrix of hyperspectral images
from synthetic and real compressive samples. Extensive simulations show that
the proposed algorithm can effectively recover the covariance matrix of
hyperspectral images from compressive measurements (8-15% approx). Moreover,
simulations and theoretical results show that the filtering step reduces the
recovery error up to twice the number of eigenvectors. Finally, an optical
implementation is proposed, and real measurements are used to validate the
theoretical findings.
