Aspects of Discrete Tomography

研究者在离散层析成像中发展了新的理论结果（唯一性，存在性和复杂性理论），新的重建方法（用于吸收和标签分布）和应用（中子层析成像）。为了获得这些结果，应用了离散数学、概率论和计算机科学的方法，包括计算机建模和仿真研究。研究者研究的新问题有：(a)带吸收的离散层析成像；(b)估计用于二值层析成像的Gibbs先验参数；(c)从投影数据直接恢复标签分布；(d)中子断层成像重建的数学问题。粗略地说，断层扫描处理的问题是从有限数量的视图中识别物体，以及从有限数量的视图中区分物体。在数学上，这个想法是重建一个多维函数，这个函数只有几个低维的样本——投影——来描述。离散层析成像处理一种特殊类型的层析反演问题，其中从投影重建的函数是离散的（即，在其值范围内只有少数元素）。在特殊情况下，当函数是二进制的（只有两个可能的值），先验信息可以用来恢复它从很少（例如，只有2或3）的投影。离散层析成像在医学、分子生物学、电子显微镜、安全、无损检测等领域有着令人着迷的应用。在这个项目中，研究者研究了离散层析成像中的问题，开发了重建函数的新方法，并考虑了在中子层析成像中的应用。其他广泛的影响包括学生参与该项目的工作，与匈牙利的一位教职员工进行国际合作，并在纽约市立大学共同举办研究研讨会，以及关于离散断层扫描应用的研讨会。

Herman The investigator develops new theoretical results(uniqueness, existence, and complexity theories) in discretetomography), new reconstruction methods (for the case ofabsorption and for label distributions), and applications(neutron tomography). To obtain these results, methods ofdiscrete mathematics, probability theory, and computer scienceare applied, including modeling and simulation studies bycomputers. The new problems that the investigator studies are:(a) discrete tomography with absorption; (b) estimation of theparameters of Gibbs priors to be used in binary tomography; (c)direct recovery of label distributions from projection data; and(d) mathematical problems of reconstruction in neutrontomography. Roughly speaking, tomography deals with the problems ofrecognizing an object from a limited number of views of it, andof discriminating between objects from a limited number of viewsof them. Mathematically, the idea is to reconstruct amultidimensional function that describes, having only a fewlower-dimensional samples -- projections -- of the function.Discrete tomography deals with a special type of tomographicinverse problem in which the function to be reconstructed fromits projections is discrete (i.e., has only a few elements in itsrange of values). In the special case when the function is binary(only two possible values), prior information can be used torecover it from very few (e.g., only 2 or 3) projections.Discrete tomography has some fascinating applications, forexample, in medicine, molecular biology, electron microscopy,security, and nondestructive testing. In this project theinvestigator studies problems in discrete tomography, developsnew methods to reconstruct functions, and considers applicationsin neutron tomography. Other broad impacts include theinvolvement of students in the work of the project, aninternational collaboration with a faculty associate in Hungaryand jointly run research seminar at CUNY, and a workshop on theapplications of discrete tomography.