Maximum-likelihood expectation-maximization (ML-EM) method and multiplicative algebraic reconstruction technique (MART), which are well-known iterative image reconstruction algorithms, produce relatively highquality performance but each of which has an advantage and disadvantage. In this paper, in order to compensate for both disadvantages, we present a novel iterative algorithm constructed by a nonautonomous iterative system derived from the minimization of an α-skew Kullback–Leibler divergence, which is considered as a combined objective function for ML-EM and MART. We confirmed effectiveness of the proposed hybrid method through numerical experiments.
We propose a hybrid dynamical system as a continuous analog to the block-iterative multiplicative algebraic reconstruction technique (BI-MART), which is a well-known iterative image reconstruction algorithm for computed tomography. The hybrid system is described by a switched nonlinear system with a piecewise smooth vector field or differential equation and, for consistent inverse problems, the convergence of non-negatively constrained solutions to a globally stable equilibrium is guaranteed by the Lyapunov theorem. Namely, we can prove theoretically that a weighted Kullback-Leibler divergence measure can be a common Lyapunov function for the switched system. We show that discretizing the differential equation by using the first-order approximation (Euler's method) based on the geometric multiplicative calculus leads to the same iterative formula of the BI-MART with the scaling parameter as a time-step of numerical discretization. The present paper is the first to reveal that a kind of iterative image reconstruction algorithm is constructed by the discretization of a continuous-time dynamical system for solving tomographic inverse problems. Iterative algorithms with not only the Euler method but also the Runge-Kutta methods of lower-orders applied for discretizing the continuous-time system can be used for image reconstruction. A numerical example showing the characteristics of the discretized iterative methods is presented.
In clinical X-ray computed tomography (CT), filtered back-projection as a transform method and iterative reconstruction such as the maximum-likelihood expectation-maximization (ML-EM) method are known methods to reconstruct tomographic images. As the other reconstruction method, we have presented a continuous-time image reconstruction (CIR) system described by a nonlinear dynamical system, based on the idea of continuous methods for solving tomographic inverse problems. Recently, we have also proposed a multiplicative CIR system described by differential equations based on the minimization of a weighted Kullback–Leibler divergence. We prove theoretically that the divergence measure decreases along the solution to the CIR system, for consistent inverse problems. In consideration of the noisy nature of projections in clinical CT, the inverse problem belongs to the category of ill-posed problems. The performance of a noise-reduction scheme for a new (previously developed) CIR system was investigated by means of numerical experiments using a circular phantom image. Compared to the conventional CIR and the ML-EM methods, the proposed CIR method has an advantage on noisy projection with lower signal-to-noise ratios in terms of the divergence measure on the actual image under the same common measure observed via the projection data. The results lead to the conclusion that the multiplicative CIR method is more effective and robust for noise reduction in CT compared to the ML-EM as well as conventional CIR methods.
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