We study the properties of a regularization method for inverse problems corrupted by Poisson noise with Kullback-Leibler divergence as data term. The regularization parameter is chosen according to a Morozov type principle. We show that this method of choice of the parameter is well-defined. This a posteriori choice leads to a convergent regularization method. Convergences rates are obtained for this a posteriori choice of the regularization parameter when some source condition is satisfied.
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. Index Terms-Alternating direction method of multipliers, spectral computed tomography, material decomposition, nonlinear inverse problem
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