Continuous analog of multiplicative algebraic reconstruction technique for computed tomography

Abstract

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.

Tomographic Inverse Problem with Estimating Missing Projections

“…The second and third systems, which are respectively inspired by continuous analogs [19,22] of the multiplicative algebraic reconstruction technique [23,24] and the maximum-likelihood expectation-maximization [3,25] algorithm, are given by…”

Continuous Analog of Accelerated OS‐EM Algorithm for Computed Tomography

“…Reconstructed image pixels were obtained by using the initial value problem of differential equations describing the dynamical system, for example, a continuous-time image reconstruction (CIR) system. The proposed hybrid dynamical system has a different vector field from that of our previously presented continuous system [20][21][22]. We indicated that discretizing the differential equations using the geometric multiplicative 2 Mathematical Problems in Engineering first-order expansion of the nonlinear vector field leads to the exact same iterative formula of the power-based OS-EM with the scaling parameter as a step size of discretization.…”

Multivalued Discrete Tomography Using Dynamical System That Describes Competition

“…Conventional methods for discrete tomography include the iterative reconstruction method involving an iterative algorithm and image segmentation [3], an optimization algorithm based on minimizing an energy function to discretize multiple intensity values [4], and various other methods [5][6][7][8]. In this paper, we propose a dynamical method based on the continuous-time optimization approach with nonlinear differential equations [9][10][11][12][13] that are capable of obtaining a desired tomographic image through convergence to a limit set of the differential equations. Our method utilizes the competition dynamics of generalized Lotka-Volterra systems [14] to solve the problem of multivalued discrete tomography.…”