A Fast Iterative Method with Algebraic CT Image Reconstruction Technique Based on Dynamical System Theory

同步迭代重建技术（SIRT）是已知的用于重建X射线计算机断层摄影（CT）的图像的方法。使用代数重建技术（ART）和期望最大化（EM）公式的迭代去模糊方法具有针对滤波反投影过程减少伪影的优点，滤波反投影过程在实践中通常用于CT重建。由于高质量的重建，已经有很多研究改进SIRT程序。然而，每种SIRT都需要进行大量的迭代才能获得最终的重建图像，而SIRT的迭代公式可以看作是N维状态空间中重建网格元素的动力学系统，其中N为像素数。然后，系统的稳定不动点对应于要重建的图像。此外，在不动点附近的动态支配的SIRT过程的收敛速度。在这项研究中，我们从动力系统理论的观点调查系统的属性，并检查一个数值方法来解决系统的不动点，使用射击方法，如牛顿法。在研究期间，我们发现，所提出的方法产生了非常快的收敛重建。使用来自包括金属部件的体模的合成加性噪声投影数据的实验来说明效率。

The simultaneous iterative reconstruction technique (SIRT) is known as a method to reconstruct images of X-ray computerized tomography (CT). Iterative deblurring methods using the algebraic reconstruction technique (ART) and the expectation maximization (EM) formulation have advantages for reducing artifacts against the filtered backprojection procedure, which is commonly used for CT reconstruction in practice. Because of the high-quality reconstructions, there has been a lot of research improving the SIRT procedures. However, each type of SIRT requires a large number of iterations to obtain the final reconstruction image.On the other hand, the iterative formulae of SIRT can be considered as a dynamical system of reconstruction grid elements in the N-dimensional state space, where N is the number of pixels. Then a stable fixed point of the system corresponds to the image to be reconstructed. Moreover the dynamics in the neighborhood of the fixed point govern the convergence speed of the SIRT procedure. In this study, we investigate properties of the system from the viewpoint of dynamical system theory, and examine a numerical method for solving a fixed point of the system, using a shooting method such as Newton's method. In the study period, we found that the proposed method gives rise to the very fast convergence for the reconstruction. The efficiency was illustrated using experiments with synthetic additive noise projection data from a phantom including a metal part.