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The problem of tomographic image reconstruction can be reduced to an optimization problem of finding unknown pixel values subject to minimizing the difference between the measured and forward projections. Iterative image reconstruction algorithms provide significant improvements over transform methods in computed tomography. In this paper, we present an extended class of power-divergence measures (PDMs), which includes a large set of distance and relative entropy measures, and propose an iterative reconstruction algorithm based on the extended PDM (EPDM) as an objective function for the optimization strategy. For this purpose, we introduce a system of nonlinear differential equations whose Lyapunov function is equivalent to the EPDM. Then, we derive an iterative formula by multiplicative discretization of the continuous-time system. Since the parameterized EPDM family includes the Kullback–Leibler divergence, the resulting iterative algorithm is a natural extension of the maximum-likelihood expectation-maximization (MLEM) method. We conducted image reconstruction experiments using noisy projection data and found that the proposed algorithm outperformed MLEM and could reconstruct high-quality images that were robust to measured noise by properly selecting parameters.

Image reconstruction in computed tomography can be treated as an inverse problem, namely, obtaining pixel values of a tomographic image from measured projections. However, a seriously degraded image with artifacts is produced when a certain part of the projections is inaccurate or missing. A novel method for simultaneously obtaining a reconstructed image and an estimated projection by solving an initial-value problem of differential equations is proposed. A system of differential equations is constructed on the basis of optimizing a cost function of unknown variables for an image and a projection. Three systems described by nonlinear differential equations are constructed, and the stability of a set of equilibria corresponding to an optimized solution for each system is proved by using the Lyapunov stability theorem. To validate the theoretical result given by the proposed method, metal artifact reduction was numerically performed.

Binary tomography is the process of reconstructing a binary image from a finite number of projections. We present a novel method for solving binary tomographic inverse problems using a continuous-time image reconstruction (CIR) system described by nonlinear differential equations based on the minimization of a double Kullback- Leibler divergence. We prove theoretically that the divergence measure monotonically decreases in time. Moreover, we demonstrate numerically that the quality of the reconstructed images of the nonlinear CIR system is better than those from an iterative reconstruction method.

The maximum-likelihood expectation-maximization (ML-EM) algorithm is used for an iterative image reconstruction (IIR) method and performs well with respect to the inverse problem as cross-entropy minimization in computed tomography. For accelerating the convergence rate of the ML-EM, the ordered-subsets expectation-maximization (OS-EM) with a power factor is effective. In this paper, we propose a continuous analog to the power-based accelerated OS-EM algorithm. The continuous-time image reconstruction (CIR) system is described by nonlinear differential equations with piecewise smooth vector fields by a cyclic switching process. A numerical discretization of the differential equation by using the geometric multiplicative first-order expansion of the nonlinear vector field leads to an exact equivalent iterative formula of the power-based OS-EM. The convergence of nonnegatively constrained solutions to a globally stable equilibrium is guaranteed by the Lyapunov theorem for consistent inverse problems. We illustrate through numerical experiments that the convergence characteristics of the continuous system have the highest quality compared with that of discretization methods. We clarify how important the discretization method approximates the solution of the CIR to design a better IIR method.