Local singularity reconstruction from integrals over curves in $\mathbb{R}^3$
Eric Todd Quinto Hans Rullgård
We define a general curvilinear Radon transform in $\mathbb{R}^3$, and we develop its microlocal properties. We prove that singularities can be added (or masked) in any backprojection reconstruction method for this transform. We use the microlocal properties of the transform to develop a local backprojection reconstruction algorithm that decreases the effect of the added singularities and reconstructs the shape of the object. This work was motivated by new models in electron microscope tomography in which the electrons travel over curves such as helices or spirals, and we provide reconstructions for a specific transform motivated by this electron microscope tomography problem.
keywords: microlocal analysis. Electron microscopy Fourier integral operator curvilinear Radon transform
Local Sobolev estimates of a function by means of its Radon transform
Hans Rullgård Eric Todd Quinto
In this article, we will define local and microlocal Sobolev seminorms and prove local and microlocal inverse continuity estimates for the Radon hyperplane transform in these seminorms. The relation between the Sobolev wavefront set of a function $f$ and of its Radon transform is well-known [18]. However, Sobolev wavefront is qualitative and therefore the relation in [18] is qualitative. Our results will make the relation between singularities of a function and those of its Radon transform quantitative. This could be important for practical applications, such as tomography, in which the data are smooth but can have large derivatives.
keywords: Sobolev seminorms limited data tomography Radon transform wavefront set singularity detection.
Common midpoint versus common offset acquisition geometry in seismic imaging
Raluca Felea Venkateswaran P. Krishnan Clifford J. Nolan Eric Todd Quinto
We compare and contrast the qualitative nature of backprojected images obtained in seismic imaging when common offset data are used versus when common midpoint data are used. Our results show that the image obtained using common midpoint data contains artifacts which are not present with common offset data. Although there are situations where one would still want to use common midpoint data, this result points out a shortcoming that should be kept in mind when interpreting the images.
keywords: Fourier integral operators fold and blowdown singularities geophysics. microlocal analysis Seismic imaging paired Lagrangians
Microlocal aspects of common offset synthetic aperture radar imaging
Venkateswaran P. Krishnan Eric Todd Quinto
In this article, we analyze the microlocal properties of the linearized forward scattering operator $F$ and the reconstruction operator $F^{*}F$ appearing in bistatic synthetic aperture radar imaging. In our model, the radar source and detector travel along a line a fixed distance apart. We show that $F$ is a Fourier integral operator, and we give the mapping properties of the projections from the canonical relation of $F$, showing that the right projection is a blow-down and the left projection is a fold. We then show that $F^{*}F$ is a singular FIO belonging to the class $I^{3,0}$.
keywords: Fourier Integral Operators Microlocal Analysis SAR Imaging Scattering. Radar

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