Mathematical Sciences: Aspects of Discrete Tomography

赫尔曼9612077 研究人员研究如何确定功能定义在三维空间中的物理测量（因此只有近似）值的线积分，使用的想法从离散断层扫描。 在离散层析成像中，有一个域-它本身可以是离散的（如一组有序的整数对）或连续的（如欧几里德空间）-和一个未知函数，其范围已知是一个给定的离散集（通常是真实的数）。 离散层析成像的问题与确定的功能（也许只是部分或近似）从加权总和超过子集的域在离散的情况下，并从加权积分超过子空间的域在连续的情况下。 离散层析成像的基本方面是，知道函数的离散范围可以允许在没有这种知识的情况下无法确定的点处确定其值。 离散断层扫描充满了数学上令人着迷的问题（例如， 什么形状是由它们的X射线唯一确定的？）并且它具有许多有趣的应用（例如，电子显微镜和无损检测）。 研究人员研究如何确定三维物体的物理测量值（因此只有近似值）的基本上是平均值沿着一维切片的对象。 在数学上，这是一个从函数的线积分的近似值确定在普通三维空间中定义的函数的问题。 这种物理测量可以使用电子显微镜对生物大分子进行，或者使用高能X射线对工业物体进行。 这两个应用程序（以及许多其他应用程序）的共同点是，已知要确定的对象可能只有200多个可能值。 离散层析成像的显著特点是，通过利用这样的知识，我们可以成功地从数据中确定对象的结构，这些数据本身不足以在没有先验信息的情况下确定结构。 虽然该项目的重点是发展适当的数学理论（包括用于解决所研究的一般问题的具体实例的计算程序），这些将被开发，以便它们可以应用于通过利用对象的真正离散性质从电子显微镜图像组恢复生物大分子结构的领域。需要重建 这导致了以下结果：某些数据收集方法不允许我们确定一个完全未知的对象的结构（例如，电子显微镜在可以收集足够的数据之前破坏大分子）。 离散层析成像使我们能够克服这样的技术困难，在这些情况下，当对象是已知的，只有100多个值。 该项目已应用于医学，生物学和化学中的成像问题。

Herman 9612077 The investigator studies how to determine functions defined over three-dimensional space from physically measured (and hence only approximate) values of their line integrals, using ideas from discrete tomography. In discrete tomography there is a domain --- which may itself be discrete (such as a set of ordered pairs of integers) or continuous (such as Euclidean space) --- and an unknown function whose range is known to be a given discrete set (usually of real numbers). The problems of discrete tomography have to do with determining the function (perhaps only partially or approximately) from weighted sums over subsets of its domain in the discrete case and from weighted integrals over subspaces of its domain in the continuous case. The essential aspect of discrete tomography is that knowing the discrete range of the function may allow determining its value at points where without this knowledge it could not be determined. Discrete tomography is full of mathematically fascinating questions (e.g., what shapes are uniquely determined by their x-rays?) and it has many interesting applications (e.g., electron microscopy and nondestructive testing). The investigator studies how to determine three-dimensional objects from physically measured (and hence only approximate) values of what are essentially averages along one-dimensional slices of the objects. Mathematically, this is the problem of determining functions defined over ordinary three-dimensional space from approximate values of their line integrals. Such physical measurements may be taken of biological macromolecules using an electron microscope or of industrial objects using high energy x-rays. What is shared by these two (and many other) applications is that the object to be determined may be known to have only finitely many possible values. The distinguishing feature of discrete tomography is that by making use of such knowledge we may succeed in determining the structure of th e object from data that is not in itself sufficient for determining the structure without the prior information. While the emphasis of the project is on the development of an appropriate mathematical theory (including computational procedures for the solution of specific instances of the general problems under investigation), these will be developed so that they can be applied in the field of recovering the structure of biological macromolecules from sets of electron microscopic images by making use of the truly discrete nature of the objects to be reconstructed. What this boils done to is the following: certain methods of data collection do not allow us to determine the structure of a totally unknown object (e.g., the electron microscopes destroys a macromolecule before sufficient data can be collected). Discrete tomography allows us to overcome such a technical difficulty in those cases when the object is known to have only finitely many values. The project has applications to imaging problems in medicine, biology, and chemistry.