Contributors: Imagerie Tomographique et Radiothérapie; Centre de Recherche en Acquisition et Traitement de l'Image pour la Santé (CREATIS); Université Jean Monnet Saint-Étienne (UJM)-Hospices Civils de Lyon (HCL)-Institut National des Sciences Appliquées de Lyon (INSA Lyon); Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université Claude Bernard Lyon 1 (UCBL); Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Jean Monnet Saint-Étienne (UJM)-Hospices Civils de Lyon (HCL)-Institut National des Sciences Appliquées de Lyon (INSA Lyon); Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Institut National de la Santé et de la Recherche Médicale (INSERM); ANR-11-LABX-0063,PRIMES,Physique, Radiobiologie, Imagerie Médicale et Simulation(2011); ANR-17-CE19-0011,SALTO,Développent et évaluation de méthodes de CT spectral pour la détection précoce de l'arthrose(2017); European Project: 701915,H2020,H2020-MSCA-IF-2015,SUCCESS(2016)
نبذة مختصرة : International audience ; In this paper, we address the resolution of material decomposition, which is a nonlinear inverse problem encountered in spectral computerized 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 analyse 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 to 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 minimisation 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.
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