Joint sparse and low rank recovery algorithm for compressive hyperspectral imaging.

Compressive spectral imaging techniques encode and disperse a hyperspectral image (HSI) to sense its spatial and spectral information with few bidimensional (2D) multiplexed projections. Recovering the original HSI from the 2D projections is carried by traditional compressive sensing-based techniques that exploit the sparsity property of natural HSI as they are represented in a proper orthonormal basis. Nevertheless, HSIs also exhibit a low rank property inasmuch only a few numbers of spectral signatures are present in the images. Specifically, when an HSI is rearranged as a matrix whose columns represent vectorized 2D spatial images in a different wavelength, this matrix is said to be low rank. Therefore, this paper proposes an HSI recovering algorithm from compressed measurements involving a joint sparse and low rank optimization problem, which seeks to jointly minimize the ℓ-, ℓ-, and ℓ-norm, leading the solution to fit the given projections, and be simultaneously sparse and low rank. Several simulations, along different data sets and optical sensing architectures, show that when the low rank property is included in the inverse problem formulation, the reconstruction quality increases up to four (dB) in terms of peak signal to noise ratio.