Local singularity reconstruction from integrals over curves in $\mathbb{R}^3$

Eric Todd Quinto; Hans Rullgård

doi:10.3934/ipi.2013.7.585

Article Contents

2013, Volume 7, Issue 2: 585-609. Doi: 10.3934/ipi.2013.7.585

This issue Previous Article Total variation and wavelet regularization of orientation distribution functions in diffusion MRI Next Article Constructing continuous stationary covariances as limits of the second-order stochastic difference equations

Local singularity reconstruction from integrals over curves in $\mathbb{R}^3$

Eric Todd Quinto^1, and
Hans Rullgård^2,

1.
Department of Mathematics, Tufts University, Medford, MA 02155
2.
Department of Mathematics, Stockholm University, 10691 Stockholm

Received: December 2011

Revised: September 2012

Published online: May 2013

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Abstract

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:

Mathematics Subject Classification: Primary: 44A12, 92C55, 35S30; Secondary: 58J40.

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