Nonlinear material decomposition using a regularized iterative scheme based on the Bregman distance
Résumé
In this work, we address the solution of a nonlinear inverse problem. Due to the nonlinearity of the forward operator, the functional to minimise may be nonconvex and standard approaches may fail. We are motivated by the existence of iterative regularized schemes that replace the initial functional with a surrogate functional with improved convergence properties and that can lead to a global minimum. In particular, efficient iterative schemes based on the use of the Bregman distance have been proposed [M Bachmayr, M Burger, Inv Probl, 10 (2009), pp. 105004].
The goal is to investigate the use of an iterative scheme based on the Bregman distance to solve the nonlinear material decomposition problem in spectral computerized tomography. Capitalising on a regularised Gauss Newton algorithm, the Bregman iterative scheme is created by replacing the regularization functional with its associated Bregman distance. First, we show that material decomposition is a nonconvex problem. Then, the proposed Bregman method is evaluated and compared to the Gauss Newton method. In particular, we focus on the convergence of the methods for different regularization parameters and initial guesses. Both methods are evaluated using data simulated in a realistic thorax phantom comprising three materials: soft tissue, bone and portal vein marked with gadolinium. Simulations include modelling of source spectrum and detector response functions, for four energy bins. Specific regularization is provided for each material: a smooth approximation of total variation for the marker and second order Tikhonov regularization for bone and soft tissue.
The reference Gauss Newton approach required precise selection of the regularization parameter to provide an optimal solution and did not converge when the initial guess was far from the solution. On the contrary, the use of the Bregman distance led to similar solutions independently of the initial estimate and was more robust to the selection of the regularization parameter.