Mathematical Sciences: Aspects of Discrete Tomography

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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.