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### Open Access Journals

IPI

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.

IPI

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.

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