Aspects of Discrete Tomography

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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.