Tomographic Image Reconstruction from Limited View Angle Projection Data

在CT的许多应用中，对有限视角投影、空心投影、稀疏投影等不完全投影的图像重建问题一直受到人们的关注。本课题的目的是研究基于不完全投影的图像重建算法，首先从线性算子谱分解的角度考虑图像重建问题，并将Radon变换及其逆变换用完备奇异系统表示。基于这个框架，我推导出公式的模糊性的重建图像由于有限数量的投影和分辨率或精度的重建，由于观察噪声的评估。明确了不完全投影重建的数学问题，提出了新的不完全观测重建算法：基于投影数据冗余的有限视角投影重建算法、基于投影数据因果关系的空心投影重建算法和基于投影数据一致性条件的少量投影重建算法。通过对体模图像和实际CT数据的计算机仿真，证实了这些算法的性能和实际局限性，从重建图像质量和计算量上看，所提算法的性能上级传统方法。

In many application of computed tomography(CT), the image reconstruction from incomplete projections, such as limited view angle projections, hollow projections and sparsely viewed small number of projections, has attracted much attentions. The aim of this project is to develop the image reconstruction algorithm from such incomplete projections.At first, the image reconstruction problem was considered from a view poit of spectral decomposition of linear operator, and Radon transform and its inversion were represented by complete singular systems. Based on this framework, I derived formulations for evaluation of an ambiguity in the reconstructed image due to the finite number of projections and of resolution or accuracy of reconstruction due to observation noise. The mathematical problem of reconstruction from incomplete projection was made clear.New reconstruction algorithms were developed for incomplete observations: algorithm for limited view angle projections based on the redundancy of profection data, algorithm for hollow projections based on causality of projection data and algorithm for a small number view of projections based on consistency condition of projection data. The performance and practical limitation of these algorithms were confirmed by computer simulation of phantom image and actual CT data.The performances of proposed algorithm were superior to those of conventional methods from the view point of reconstructed image quality and computational amount.