30 March 2016 Continuous analog of multiplicative algebraic reconstruction technique for computed tomography
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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.
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Kiyoko Tateishi, Kiyoko Tateishi, Yusaku Yamaguchi, Yusaku Yamaguchi, Omar M. Abou Al-Ola, Omar M. Abou Al-Ola, Takeshi Kojima, Takeshi Kojima, Tetsuya Yoshinaga, Tetsuya Yoshinaga, } "Continuous analog of multiplicative algebraic reconstruction technique for computed tomography", Proc. SPIE 9783, Medical Imaging 2016: Physics of Medical Imaging, 97834Q (30 March 2016); doi: 10.1117/12.2214598; https://doi.org/10.1117/12.2214598

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