DEVELOPMENT OF A SPATIAL LIMITED ANGLE MODEL FOR COMPUTERIZED X RAY TOMOGRAPHY

我们的研究建议涉及到一种高度 可并行的、空间的、有限角度的计算机化X射线和 电子断层扫描，能够进行真实的空间重建 时间到了。该模型是基于氡的连续扩展。 有限域上向量空间上的变换。有限的角度 该模型的特点在三维环境中尤为重要 用透射式电子显微镜成像时，倾斜角度 被限制在/-60度。它在设计中也很重要 扫描设备用于X射线层析成像，因为它将允许 消除当前扫描仪的隧道形状。 拟议研究的第一阶段是关于X射线层析成像， 是双重的。首先，根据先前的研究结果，我们将完成 通过确定产生最小噪声的几何图形进行学习，以及 因此，我们将调整算法，以模拟 模型适用于空间情况，在顺序和并行情况下 编程。 其次，对重建和重建的质量进行了比较研究 速度将被执行，考虑到其他现有的 模特们。 第二阶段由本项目新的具体目标组成 到电子断层扫描，也就是应用我们的 利用透射电子显微镜收集的数据的算法， 遵循P.Schwander(AT&T Bell)最近开发的一项技术 实验室。)来确定晶体的原子结构。施万德氏病 方法打开了一个可以称为离散二进制的新字段 断层扫描，我们的算法似乎工作得很好。的确，以防万一 晶体由一种类型的原子组成，可以对晶体进行建模 作为仅由0‘S和1’S组成的三维数组，它是 可以只在两到四个方向上测量投影 某些区域轴。因此，启发式方法在于应用我们的 对可用投影进行重建变换以获得3维数据 值的数组。这些值中的每一个，从0.0到 1.0，给出原子存在于该位置的可能性。 在这个启发式算法之后开发的一些初步算法有 产生了有希望的重建。 这个项目的一个重要目标是吸引两位杰出的 数学或计算机专业的少数民族研究生 科学，到生物医学研究。在至少三年内，这些 学生将接受个性化的数学培训 体层摄影术，将被鼓励攻读生物医学博士学位。 数学，并将负责大部分的计算 模型的开发和实施。 这项提案的长期目标是创建一个多 纪律环境，涉及少数民族研究生， 数学家、计算机科学家、工程师和物理学家 在层析成像的各个方面进行生物医学研究时， 例如，基于我们的空间模型的X光CT扫描仪原型。

Our research proposal deals with the development of a highly parallelizable, spatial, limited angle model for computerized X-ray and electron tomography, capable to perform spatial reconstructions in real time. This model is based upon a continuous extension of the Radon transform on vector spaces over finite fields. The limited angle feature of this model is specially important in three-dimensional imaging with transmission electron microscopes where the tilting angle is limited to +/-60 degrees. It is also important in the design of scanning devices for X-ray tomography since it.would permit to eliminate the tunnel shape of current scanners. The first phase of the proposed research, regarding X-ray tomography, is two-fold. First, following previous findings, we shall complete the study by determining the geometry that yields minimum noise, and accordingly, we shall adjust the algorithms for simulating how the model works in the spatial case, under sequential and parallel programming. Secondly, a comparative study of the quality of the reconstructions and speed will be performed, taking into consideration other existing models. The second phase consist of a new specific aim of this project related to electron tomography, namely, to apply a modification of our algorithm to data collected by using transmission electron microscopy, following a recent technique developed by P. Schwander (AT&T Bell Labs.) to determine the atomic structure of crystals. Schwander's methods have open a new field which could be named Discrete Binary Tomography, in which our algorithms seem to work well. Indeed, in case that the crystal consists of one type of atom, a crystal can be model as a three dimensional array only of 0's and 1's of which it is possible to measure projection only in two to four directions along certain zone axes. Therefore, the heuristic consists in applying our reconstruction transform to the available projections obtaining a 3-dim array of values. Each of these values, measured on a scale from 0.0 to 1.0, gives the likelihood of the presence of an atom in that position. Some preliminary algorithms developed following this heuristic are yielding promising reconstructions. An important goal of this project is to attract two outstanding graduate minority students majoring in mathematics or in computer sciences, to biomedical research. For at least three years these students will receive personalized training in the mathematics of tomography, will be encouraged to pursue doctoral studies in bio- mathematics, and will be on charge of most of the computational development and implementation of the model. The long-term objectives of this proposal are to create a multi- disciplinary environment, involving graduate minority students, mathematicians, computer scientists, engineers and physicists, aimed at producing biomedical research in the various aspects of tomography, such as a prototype X-ray CT scanner based on our spatial model.