Local Tomography and Microlocal Analysis

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The main goal of this project is to solve problems in tomography. Professor Quinto will develop and test local reconstruction algorithms and apply them to electron microscopy (ET), single photon emission tomography (SPECT), Sonar, and X-ray tomography. Professor Quinto is developing his fully three-dimensional tomography algorithm, and collaborators at Sidec and the Karolinska Institute, Stockholm, are testing it on electron microscope data of biological specimens. He is adapting and testing the algorithms on new scanning geometries (ways of taking the data) in ET and SPECT. With Andreas Rieder and Thomas Schuster, he is developing local algorithms for three-dimensional Sonar, and he is applying it to more general data acquisition geometries. Professor Quinto is developing X-ray tomography algorithms for novel X-ray source curves and applying them to medical tomography. In each case, he will develop pure mathematics to analyze what the algorithms do to boundaries or singularities, and he will improve the algorithms using this information. The PI will develop the microlocal analysis behind the physical models and his algorithms. Along with Hans Rullgard, he will quantify the strength of singularities of objects and their tomographic data. The principal investigator will solve specific mathematical problems and apply them to practical problems with scientists in industry. His electron microscopy project is undertaken jointly with biologists and mathematicians, and his algorithm is being used to help scientists better image and understand the structure of individual molecules. The X-ray tomography (CT) project is motivated by a novel X-ray CT scanner for the operating room, and his algorithm can be used to image the inside of the body as doctors operate. His research on measuring strength of singularities from Radon data could help researchers quantify what is visible from tomographic data, much as the principal investigator's previous research on qualitative singularity detection shows where these singularities are located in the body. The work in this proposal also supports the mathematical education of Prof. Quinto's students. He will use some of his NSF support to work with at least two undergraduate students on his proposed research projects.