Tomography and Integral Geometry

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ABSTRACT: Professor Quinto will pursue problems in tomographyand the mathematical analysis of Radon transforms. He will developand refine tomographic singularity detection algorithms and apply themto electron microscopy. Professor Quinto has successfully tested hislimited data tomography algorithm on electron microscope data, and hewill use these results as a basis for further refinement with the goalof incorporating scattering and other effects. He will continueresearch on the approximate inverse in three-dimensional SONAR.Professor Quinto plans to prove uniqueness theorems for a model ofsynthetic aperture RADAR, research which is based on his puremathematical work. Within pure mathematics, the principalinvestigator plans to prove uniqueness and support theorems forspherical Radon transforms. He and Professor Agranovsky will usethese results to characterize stationary sets of solutions of the waveequation. Stationary sets, where the solution is zero for all time,are very important and difficult to characterize. He plans to proveuniqueness theorems for Radon transforms on spheres on real-analyticmanifolds and use these results to begin to characterize stationarysets on rank-one symmetric spaces. He is proving Morera theorems forthe distinguished boundary of polydisks, and he plans to generalizethem to complex manifolds. He hopes to prove inverse continuity foran important class of local limited data problems or clarify when theydo not hold.This research encompasses both applied and puremathematics: tomography and integral geometry. The pure research willbe used to develop, understand, and justify the applied algorithms,and the applied problems will motivate much of the pure research. Acomputed tomography algorithm will be developed for electronmicroscopy and tested jointly with colleagues at the KarolinskaInstitute in Sweden. Our goal is to produce accurate pictures ofviruses and small molecules using an electron microscope. Jointlywith a colleague and an undergraduate student, he will develop analgorithm for emission tomography, a type of tomography that locatesmetabolic processes. He will develop pure mathematics that will showhow well the algorithms work and where their limitations might occur.These pure mathematical underpinnings are required to ensure that his(or any other) algorithms are as effective as they can be. SONAR datacan be modeled as averages over spheres (the spherical wave fronts ofthe sound wave), and Professor Quinto will jointly develop newalgorithms for SONAR. This pure mathematics is intriguing in its ownright. Professor Quinto will prove theorems about spherical averageswhich will be used to understand the wave equation (the equation thatdescribes how sound and light behave). The wave equation describesthe motion of a drum head, and he will specify which points on thedrum do not move at all. Finally, he will prove theorems he willapply to complex and harmonic analysis, fields in pure mathematics.