Tomography and Microlocal Analysis

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The main goals of NSF grant DMS-1311558 are to use microlocal analysis and integral geometry to understand important problems in tomography and then to use this to develop and improve algorithms for those problems. Professor Quinto will develop and test algorithms for electron microscopy, X-ray tomography, and radar. Professor Quinto will adapt and test his three-dimensional electron microscope tomography (ET) algorithm with new scanning geometries (ways of obtaining the data) in ET, and he will refine the algorithm to address more aspects of the physical model. Long-term collaborators at the Royal Institute of Technology and the Karolinska Institute, Stockholm, will provide biological and realistic simulated data. Prof. Quinto will develop Sobolev seminorms that tell how strongly features in an object will show up in its tomographic data. He will use microlocal analysis to understand streak artifacts that occur in limited angle tomography and use this to develop algorithms that decrease these streaks. A large part of the PI's research in this project will involve novel methods in bistatic radar with general flight paths. He and his collaborators will develop the microlocal theory of radar for a broad range of flight paths. They will use this to find flight paths that provide better data, and he will develop and test algorithms for these flight paths.Tomography is the mathematics that images objects from indirect data including from X-ray CT scanners, electron microscopes, or radar. He will develop methods to image submicroscopic objects from electron microscopy data, the body from X-ray tomography data, and the terrain of the earth from radar data. His electron microscopy project is undertaken with biologists and mathematicians, and it has already been used to image and understand individual sub-microscopic structures. Prof. Quinto will use sophisticated mathematics to understand which features the specific data can image well and what features cannot be well-imaged. He will use this information to improve his algorithms. Professor Quinto is committed to undergraduate teaching, advising, and mentoring a broad range of students and young mathematicians, and his NSF project will directly support this through two undergraduate projects. The PI's many undergraduate research students help accomplish his research by developing algorithms and learning the underlying mathematics as they learn to become creative researchers.