Analysis of Matrix Completion algorithms for spectral image estimation from compressive coded projections

The coded aperture snapshot spectral imaging (CASSI) system is an optical architecture designed to capture spectral images using the compressive sensing (CS) concepts. CASSI senses the spectral information of a three dimensional scene by using two-dimensional coded focal plane array (FPA) projections. The CASSI system improves the sensing speed and reduce the large amount of collected data given by conventional spectral imaging systems based on the Nyquist criterion. Compressive sensing reconstruction algorithms are commonly used to recover the underlying three dimensional source. However, CS assumes the signal is sparse, which is not always achievable. This work proposes the use of Matrix Completion (MC) theory as an alternative way to reconstruct the underlying three dimensional source from the compressive coded projections. The reconstruction is accomplished by solving a convex optimization problem, which relies on the nuclear-norm minimization of the measurement, subject to data constraints. Further, it is proposed and analyzed the impact of six different linear transformations to arrange the missing data, such that the degrees of freedom of the transformed matrix ease the completion process. Simulations show good quality of the reconstruction and it is observed that MC algorithms are much faster than conventional CS reconstruction algorithms.