  Natural images tend to mostly consist of smooth regions with individual
pixels having highly correlated spectra. This information can be exploited to
recover hyperspectral images of natural scenes from their incomplete and noisy
measurements. To perform the recovery while taking full advantage of the prior
knowledge, we formulate a composite cost function containing a square-error
data-fitting term and two distinct regularization terms pertaining to spatial
and spectral domains. The regularization for the spatial domain is the sum of
total-variation of the image frames corresponding to all spectral bands. The
regularization for the spectral domain is the l1-norm of the coefficient matrix
obtained by applying a suitable sparsifying transform to the spectra of the
pixels. We use an accelerated proximal-subgradient method to minimize the
formulated cost function. We analyze the performance of the proposed algorithm
and prove its convergence. Numerical simulations using real hyperspectral
images exhibit that the proposed algorithm offers an excellent recovery
performance with a number of measurements that is only a small fraction of the
hyperspectral image data size. Simulation results also show that the proposed
algorithm significantly outperforms an accelerated proximal-gradient algorithm
that solves the classical basis-pursuit denoising problem to recover the
hyperspectral image.
