Continuous Analog of Accelerated OS‐EM Algorithm for Computed Tomography

Abstract

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 continuoustime 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.

Polychromatic CT Data Improvement with One‐Parameter Power Correction

“…for All ∈ [0, 500] do (12) ← LC( , 1 + /100); where X( ) is a sensitivity function of the detector. Therefore, (1) has been changed with additional integral through energies.…”

“…However, nowadays algorithm development work is in progress due to the fact that the objects with nonstandard attenuation properties [9] have to be measured or nonstandard experimental conditions are required (lowdose tomography) [10]. The works are focused also on an improvement of the algorithms properties like convergence [11,12] or the quality of the reconstruction [13].…”

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…”

“…It solves an initialvalue problem of differential equations consisting of state variables including parts of projections as well as image pixel values. A system of differential equations based on an extension of dynamical methods [10][11][12][13][14][15][16][17][18][19][20] is constructed. It is an approach that optimizes a cost function of unknown variables consisting of an image and a projection.…”

Tomographic Image Reconstruction Based on Minimization of Symmetrized Kullback-Leibler Divergence

“…The term "continuous analog" means that a numerical discretization of the differential equation is the same as the difference equation. The idea is based on the approach for solving a constrained tomographic inverse problem using nonlinear dynamical methods [22][23][24][25].…”