Elliptical Radon Transforms in Image Reconstruction

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The principal investigator of this project and his collaborator study elliptical Radon transforms (ERT), which play an important role in bi-static models of various imaging modalities such as near-field ultrasound tomography, synthetic aperture radar, geophysical exploration imaging, and sonar. Bi-static data acquisition geometries involve separate emitter and receiver to image a medium using signals such as acoustic or electromagnetic waves. This approach offers several advantages. In the case of radar imaging the receivers are passive; hence separating their motion from the active emitters allows receivers alone to be flown in an unsafe environment. In sonar, data collection involves the receivers towed behind the boat transmitting the sound source. In ultrasound tomography separate emitter and receiver can provide data sets, for example, of high angular resolution. Assuming constant speed of wave propagation in the medium, the mathematical model for all these modalities is expressed through the ERT defined by integrals of an unknown function over ellipsoids or ellipses. The investigators develop image reconstruction techniques for several imaging modalities based on such elliptical Radon transforms. The principal mathematical problems they address include: development of microlocal analysis of ERT in various practically important data acquisition geometries, development of exact and approximate inversion formulas and algorithms for ERT, description of injectivity sets, and range characterization of ERT, i.e. the characterization of consistency conditions for the data for the aforementioned imaging modalities. The investigators test the results of their research in laboratory ultrasound experiments performed by their collaborators in radiology and biomedical engineering.The principal investigator and his collaborator solve image reconstruction problems that are of great importance in several fields including ultrasonic reflectivity imaging in medicine, geophysical exploration, sonar and synthetic aperture radar imaging. Many of these problems have similar mathematical representation, and the investigators develop solutions as well as derive numerical algorithms for such problems. The goals and novel results of this project provide a positive impact on healthcare technologies (e.g. early detection of cancer), national security (e.g. improved radar systems), economics (e.g. improved geophysical exploration, or industrial nondestructive testing), and for many other fields of modern life, where the use of remote sensing and imaging has become an indispensable tool. The investigators work with their collaborators to test the results of their research in laboratory ultrasound experiments. The investigators advise and mentor students on research topics directly related to this proposal, and train future specialists of the field. They are committed to graduate and undergraduate teaching and engage students from underrepresented groups in the form of introductory workshops, tutorials and short courses. The investigators broadly disseminate the results of the project through publications in selected scientific journals and presentations at conferences.