Aspects of Discrete Tomography

Cuny Graduate School University Center, New York NY

Investigators

Abstract

Herman The investigator develops new theoretical results (uniqueness, existence, and complexity theories) in discrete tomography), new reconstruction methods (for the case of absorption and for label distributions), and applications (neutron tomography). To obtain these results, methods of discrete mathematics, probability theory, and computer science are applied, including modeling and simulation studies by computers. The new problems that the investigator studies are: (a) discrete tomography with absorption; (b) estimation of the parameters 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 neutron tomography. Roughly speaking, tomography deals with the problems of recognizing an object from a limited number of views of it, and of discriminating between objects from a limited number of views of them. Mathematically, the idea is to reconstruct a multidimensional function that describes, having only a few lower-dimensional samples -- projections -- of the function. Discrete tomography deals with a special type of tomographic inverse problem in which the function to be reconstructed from its projections is discrete (i.e., has only a few elements in its range of values). In the special case when the function is binary (only two possible values), prior information can be used to recover it from very few (e.g., only 2 or 3) projections. Discrete tomography has some fascinating applications, for example, in medicine, molecular biology, electron microscopy, security, and nondestructive testing. In this project the investigator studies problems in discrete tomography, develops new methods to reconstruct functions, and considers applications in neutron tomography. Other broad impacts include the involvement of students in the work of the project, an international collaboration with a faculty associate in Hungary and jointly run research seminar at CUNY, and a workshop on the applications of discrete tomography.