Geometric Tomography

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ABSTRACT - DMS 0203527Geometric tomography uses data concerning sections by planes andprojections on planes of geometric objects to obtain informationabout these objects. The latter include general compact sets,but often they are convex bodies, polytopes, star-shaped bodies,or finite sets. One advantage of this setting is that it becomesmore probable that inverse problems have a unique solution.Generally, the a priori knowledge that the unknown object is ofuniform density can be exploited to retrieve more informationthan would otherwise be possible. This can lead to algorithmsthat are more effective when few measurements are available, andless sensitive to measurement errors or noise. Geometrictomography has links to functional analysis, convex geometry,Minkowski geometry, and combinatorics. The project will continuethe development of geometric tomography. New directions includean extension of Lutwak's dual Brunn-Minkowski theory by theintroduction of moments of sets; work towards a coherent discreteBrunn-Minkowski theory that applies to finite sets, with volumereplaced by cardinality; and an investigation into thepossibility of a Brunn-Minkowski theory for capacity. In relatedapplications, the plan is to consolidate links between geometrictomography and a practical method called local stereology, and tosolve an inverse problem concerning finite sets with anapplication to medical imaging. New discoveries aboutintersection bodies are anticipated, with an application to thereconstruction of star bodies from the volumes of their sectionsthrough the origin. Also included is a program designed tostimulate undergraduate research.CAT scanners are machines that save lives daily. They takeX-rays in a number of different directions, and synthesize theinformation to create an image of a two-dimensional section ofpart of the body. The mathematics behind this process is calledcomputerized tomography. It is very successful, but not perfect;the reconstructed image is only approximate, and to get a betterpicture with the same procedure one has to take more X-rays,causing greater expense and likelihood of side effects. Ingeometric tomography, only homogeneous objects are allowed - thedensity of the object is the same everywhere inside it. Anexample from medicine would be a bone or a kidney. One can usethis information to find better reconstruction procedures. Thescope of geometric tomography is actually much wider. Anymeasurement involving sections of a homogeneous object by linesor planes or its shadows on lines or planes can be considered.Because of this, it has many links to other areas, both inmathematics (there is a large overlap with convex geometry, thegeometry of shapes without holes or dents) and outside. Forexample, a new technique called local stereology depends onmeasurements of planar sections of biological tissue; each sectionpasses through a fixed point, usually the nucleus of a cell, andthe measurements can be made optically rather than physically.This project continues the development of several aspects of themathematics of geometric tomography. Also included is a programdesigned to stimulate undergraduate research.