  Multi-energy computed tomography (CT) with photon counting detectors (PCDs)
enables spectral imaging as PCDs can assign the incoming photons to specific
energy channels. However, PCDs with many spectral channels drastically increase
the computational complexity of the CT reconstruction, and bespoke
reconstruction algorithms need fine-tuning to varying noise statistics.
\rev{Especially if many projections are taken, a large amount of data has to be
collected and stored. Sparse view CT is one solution for data reduction.
However, these issues are especially exacerbated when sparse imaging scenarios
are encountered due to a significant reduction in photon counts.} In this work,
we investigate the suitability of learning-based improvements to the
challenging task of obtaining high-quality reconstructions from sparse
measurements for a 64-channel PCD-CT. In particular, to overcome missing
reference data for the training procedure, we propose an unsupervised denoising
and artefact removal approach by exploiting different filter functions in the
reconstruction and an explicit coupling of spectral channels with the nuclear
norm. Performance is assessed on both simulated synthetic data and the openly
available experimental Multi-Spectral Imaging via Computed Tomography (MUSIC)
dataset. We compared the quality of our unsupervised method to iterative total
nuclear variation regularized reconstructions and a supervised denoiser trained
with reference data. We show that improved reconstruction quality can be
achieved with flexibility on noise statistics and effective suppression of
streaking artefacts when using unsupervised denoising with spectral coupling.
