  We present a simple but novel hybrid approach to hyperspectral data cube
reconstruction from computed tomography imaging spectrometry (CTIS) images that
sequentially combines neural networks and the iterative Expectation
Maximization (EM) algorithm. We train and test the ability of the method to
reconstruct data cubes of $100\times100\times25$ and $100\times100\times100$
voxels, corresponding to 25 and 100 spectral channels, from simulated CTIS
images generated by our CTIS simulator. The hybrid approach utilizes the
inherent strength of the Convolutional Neural Network (CNN) with regard to
noise and its ability to yield consistent reconstructions and make use of the
EM algorithm's ability to generalize to spectral images of any object without
training. The hybrid approach achieves better performance than both the CNNs
and EM alone for seen (included in CNN training) and unseen (excluded from CNN
training) cubes for both the 25- and 100-channel cases. For the 25 spectral
channels, the improvements from CNN to the hybrid model (CNN + EM) in terms of
the mean-squared errors are between 14-26%. For 100 spectral channels, the
improvements between 19-40% are attained with the largest improvement of 40%
for the unseen data, to which the CNNs are not exposed during the training.
