In this paper, we address the resolution of material decomposition, which is a nonlinear inverse problem encountered in spectral computed tomography (CT). The problem is usually solved in a variational framework but, due to the nonlinearity of the forward operator, the objective function may be nonconvex and standard approaches may fail. Regularized iterative schemes based on the Bregman distance have been suggested for improving global convergence properties. In this work, we analyze the convexity of the material decomposition problem and propose a regularized iterative scheme based on the Bregman distance to solve it. We evaluate our Bregman iterative algorithm and compare it with a regularized Gauss–Newton (GN) method using data simulated in a realistic thorax phantom.
First, we prove the existence of a convex set where the usual data fidelity term is convex. Interestingly, this set includes zero, making it a good initial guess for iterative minimization schemes. Using numerical simulations, we show that the data fidelity term can be nonconvex for large values of the decomposed materials. Second, the proposed Bregman iterative scheme is evaluated in different situations. It is observed to be robust to the selection of the initial guess, leading to the global minimum in all tested examples while the GN method fails to converge when the initial guess is not well chosen. Moreover, it is found to avoid the selection of the regularization parameter for little extra computation.
In conclusion, we have provided a suitable initialization strategy to solve the nonlinear material decomposition problem using convex optimization methods and evaluated a Bregman iterative scheme for this problem. The improvement in global convergence of Bregman iterative scheme combined with other interesting properties of the Bregman distance appears as a compelling strategy for nonlinear inverse problems.