Direct Methods of 3 Dimensional Image Reconstruction from Cone-Beam Projections.

计算机断层扫描对医学诊断技术产生了深远的影响。在未来，不断增加的计算能力将使直接从高效收集的投影重建三维图像成为可能。在这项研究项目中，我们从以下几个角度为这一最终目标发展了理论基础。(1)收集投影最有效的扫描(采样)方案是什么？采样点的排列方式给了我们足够的信息来唯一地估计图像？如何从这些采样数据中重建图像？(2)即使在如此高效的采样模式下，数据之间也会存在一定的冗余。在这个项目中，我们解决了以下问题：(1)作为一种有效的跟踪方案，我们使用了锥束扫描方案，并推导了在该方案中采样足以唯一确定图像的条件。此外，我们还利用级数展开的方法推导了采样的重建公式，其中解析地讨论了采样点的最优排列。(2)提出了利用数据中的冗余度在不降低图像分辨率的情况下消除噪声的投影滤波方法。最后，作为利用冗余的另一种方法，我们提出了一种基于模拟编码思想的脉冲噪声抵消技术。

Computerized Tomography has had a profound impact on medical diagnosis techniques. In the future, increasing computational power will make it possible to reconstruct 3 dimensional images directly from efficiently collected projections. In this research project we developped a theoretical basis for this ultimate goal from the following viewpoints.(1) What is the most efficient scanning (sampling) schema to collect projections? And what way of arrangement of the sampling points gives us enough imformation to estimate the image uniquely? How can we reconstruct the image from these sampled data?(2) Even in such an efficient sampling schema, there will be some redundancy among the data. How can we use the redundancy to make the reconstruction method robust and stable against the noise.In this project we have snswered above questions as follows.(1) As the efficient acsnning schema, we used the cone beam scanning schema and derived the conditions for the samplings in this schema to be enough to determine the image uniquely. In addition we developped the reconstruction formula for samplings using a series expansion method in which the optimal arrangement of the sample points was analytically discussed.(2) We developped projection filter which uesd the redundancy in data to cancel the noise without degrading the resolution of the image. Finally, as another way of making use of the redunfancy, we decelopped an impulsive noise canceling technique based on the idea of analog coding.