Microlocal Analysis of Elliptical Radon Transforms with Foci on a Line
نویسندگان
چکیده
In this paper, we take a microlocal approach to the study of an integral geometric problem involving integrals of a function on the plane over 2-dimensional sets of ellipses on the plane. We focus on two cases: (a) the family of ellipses where one focus is fixed at the origin and the other moves along the x-axis, and (b) the family of ellipses having a common offset geometry. For case (a), we will characterize the Radon transform as a Fourier integral operator associated to a fold and blowdown. This has implications on how the operator adds singularities, how backprojection reconstructions will show those singularities, and in comparison of the strengths of the original and added singularities in a Sobolev sense. For case (b) we show that this Radon transform has similar structure to case (a): it is a Fourier integral operator associated to a fold and blowdown. This case is related to previous results of authors one and three. We characterize singularities that are added by the reconstruction operator, and we present reconstructions from the authors’ algorithm that illustrate the microlocal properties. Venkateswaran P. Krishnan Tata Institute of Fundamental Research Centre for Applicable Mathematics, Bangalore 560065 India e-mail: [email protected] Howard Levinson 1759 Beechwood Blvd., Pittsburgh, PA 15217 USA e-mail: [email protected] Eric Todd Quinto Tufts University, Medford, MA 02155 USA e-mail: [email protected]
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تاریخ انتشار 2011