Tomographic Image Reconstruction from Limited View Angle Projection Data

在计算机层析成像(CT)的许多应用中，从有限视角投影、中空投影和稀疏观察少数投影等不完全投影重建图像一直是人们关注的问题。首先，从线性算子谱分解的角度考虑图像重建问题，将Radon变换及其逆变换表示为完备的奇异系统。基于这一框架，我推导出了重建图像中由于有限数目的投影而产生的模糊性以及由于观测噪声而导致的重建分辨率或精度的评估公式。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：基于深度数据冗余度的有限视角投影重建算法、基于投影数据因果关系的中空投影重建算法和基于投影数据一致性条件的小视角投影重建算法。通过对模型图像和实际CT数据的计算机仿真，验证了这些算法的性能和实用局限性，从重建图像质量和计算量的角度来看，所提算法的性能均优于传统方法。

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.