  We consider a compressive hyperspectral imaging reconstruction problem, where
three-dimensional spatio-spectral information about a scene is sensed by a
coded aperture snapshot spectral imager (CASSI). The CASSI imaging process can
be modeled as suppressing three-dimensional coded and shifted voxels and
projecting these onto a two-dimensional plane, such that the number of acquired
measurements is greatly reduced. On the other hand, because the measurements
are highly compressive, the reconstruction process becomes challenging. We
previously proposed a compressive imaging reconstruction algorithm that is
applied to two-dimensional images based on the approximate message passing
(AMP) framework. AMP is an iterative algorithm that can be used in signal and
image reconstruction by performing denoising at each iteration. We employed an
adaptive Wiener filter as the image denoiser, and called our algorithm
"AMP-Wiener." In this paper, we extend AMP-Wiener to three-dimensional
hyperspectral image reconstruction, and call it "AMP-3D-Wiener." Applying the
AMP framework to the CASSI system is challenging, because the matrix that
models the CASSI system is highly sparse, and such a matrix is not suitable to
AMP and makes it difficult for AMP to converge. Therefore, we modify the
adaptive Wiener filter and employ a technique called damping to solve for the
divergence issue of AMP. Our approach is applied in nature, and the numerical
experiments show that AMP-3D-Wiener outperforms existing widely-used algorithms
such as gradient projection for sparse reconstruction (GPSR) and two-step
iterative shrinkage/thresholding (TwIST) given a similar amount of runtime.
Moreover, in contrast to GPSR and TwIST, AMP-3D-Wiener need not tune any
parameters, which simplifies the reconstruction process.
