  Spectral computed tomography (CT) can reconstruct spectral images from
different energy bins using photon counting detectors (PCDs). However, due to
the limited photons and counting rate in the corresponding spectral fraction,
the reconstructed spectral images usually suffer from severe noise. In this
paper, a fourth-order nonlocal tensor decomposition model for spectral CT image
reconstruction (FONT-SIR) method is proposed. Similar patches are collected in
both spatial and spectral dimensions simultaneously to form the basic tensor
unit. Additionally, principal component analysis (PCA) is applied to extract
latent features from the patches for a robust and efficient similarity measure.
Then, low-rank and sparsity decomposition is performed on the produced
fourth-order tensor unit, and the weighted nuclear norm and total variation
(TV) norm are used to enforce the low-rank and sparsity constraints,
respectively. The alternating direction method of multipliers (ADMM) is adopted
to optimize the objective function. The experimental results with our proposed
FONT-SIR demonstrates a superior qualitative and quantitative performance for
both simulated and real data sets relative to several state-of-the-art methods,
in terms of noise suppression and detail preservation.
