Exploiting Algebraic and Structural Complexity for Single Snapshot Computed Tomography Hyperspectral Imaging Systems

This paper presents strategies for spectral de- noising of hyperspectral images and 3-D data cube reconstruction from a limited number of tomographic measurements, arising in single snapshot imaging systems. For de-noising the main idea is to exploit the incoherency between the algebraic complexity measure, namely the low rank of the noise-free hyperspectral data cube, and the sparsity structure of the spectral noise. In particular, the non-noisy spectral data, when stacked across the spectral dimension, exhibits low-rank due to a small number of species. On the other hand, under the same representation, the spectral noise exhibits a banded structure. Motivated by this we show that the de-noised spectral data and the unknown spectral noise and the respective bands can be simultaneously estimated through the use of a low-rank and simultaneous sparse minimization operation without prior knowledge of the noisy bands. This result is novel for hyperspectral imaging applications and we compare our results with several existing methods for noisy band recovery. For recovery under limited tomographic projections we exploit both the low algebraic and structural complexity of the data cube via joint rank penalization plus Total Variation/wavelet domain sparsity, which is novel for single snapshot hyperspectral imaging systems. We combine these two approaches for simultaneous spectral de-noising and data cube recovery under limited measurements. We perform extensive simulations and our main result indicates that exploiting both low algebraic and structural complexity has a superior performance compared to exploiting only the structural complexity. To address the computational challenges associated with the resulting optimization problem we adapt several recent developments in the area of convex optimization, specifically employing splitting and proximal point based methods.