Local singularity reconstruction from integrals over curves in $\mathbb{R}^3$
Eric Todd Quinto Hans Rullgård
Inverse Problems & Imaging 2013, 7(2): 585-609 doi: 10.3934/ipi.2013.7.585
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
Inverse Problems & Imaging 2010, 4(4): 721-734 doi: 10.3934/ipi.2010.4.721
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.

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