Compressive Covariance Matrix Estimation from a Dual-Dispersive Coded Aperture Spectral Imager

Abstract

Compressive covariance sampling (CCS) theory aims to recover the covariance matrix (CM) of a signal, instead of the signal itself, from a reduced set of random linear projections. Although several theoretical works demonstrate the CCS theory’s advantages in compressive spectral imaging tasks, a real optical implementation has no been proposed. Therefore, this paper proposes a compressive spectral sensing protocol for the dual-dispersive coded aperture spectral snapshot imager (DD-CASSI) to directly estimate the covariance matrix of the signal. Specifically, we propose a coded aperture design that allows recasting the vector sensing problem into matrix form, which enables to exploit the covariance matrix structure such as positive-semidefiniteness, low-rank, or Toeplitz. Additionally, a low-rank approximation of the image is reconstructed using a Principal Components Analysis (PCA) based method. In order to test the precision of the reconstruction, some spectral signatures of the image are captured with a spectrometer and compared with those obtained in the reconstruction using the covariance matrix. Results show the reconstructed spectrum is accurate with a spectral angle mapper (SAM) of less than 14°. RGB image composites of the spectral image also provide evidence of a correct color reconstruction.