An ADMM Algorithm for Constrained Material Decomposition in Spectral CT

Thanks to photon-counting detectors, spectral computerized tomography records energy-resolved data from which the chemical composition of a sample can be recovered. This problem, referred to as material decomposition, can be formulated as a nonlinear inverse problem. In previous work, we proposed to decompose the projection images using a regularized Gauss-Newton algorithm. To reduce further the ill-posedness of the problem, we propose here to consider equality and inequality constraints that are based on physical priors. In particular, we impose the positivity of the solutions as well the total mass in each projection image. In practice, we first decompose the projection images for each projection angle independently. Then, we reconstruct the sample slices from the decomposed projection images using a standard filtered-back projection algorithm. The constrained material decomposition problem is solved by the alternating direction method of multipliers (ADMM). We compare the proposed ADMM algorithm to the unconstrained Gauss-Newton algorithm in a numerical thorax phantom. Including constraints reduces the cross-talk between materials in both the decomposed projections and the reconstructed slices.