skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Tomographic Reconstruction from a Few Views: A Multi-Marginal Optimal Transport Approach

Abstract

In this article, we focus on tomographic reconstruction. The problem is to determine the shape of the interior interface using a tomographic approach while very few X-ray radiographs are performed. We use a multi-marginal optimal transport approach. Preliminary numerical results are presented.

Authors:
 [1]; ;  [2];  [3]
  1. CEA Ile de France (France)
  2. Université d’Orléans, UFR Sciences, MAPMO, UMR 7349 (France)
  3. CEREMADE, UMR CNRS 7534, Université Paris IX Dauphine, Pl. de Lattre de Tassigny (France)
Publication Date:
OSTI Identifier:
22617102
Resource Type:
Journal Article
Resource Relation:
Journal Name: Applied Mathematics and Optimization; Journal Volume: 75; Journal Issue: 1; Other Information: Copyright (c) 2017 Springer Science+Business Media New York; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; IMAGE PROCESSING; IMAGES; INTERFACES; TOMOGRAPHY; TRANSPORT THEORY; X RADIATION; X-RAY RADIOGRAPHY

Citation Formats

Abraham, I., E-mail: isabelle.abraham@cea.fr, Abraham, R., E-mail: romain.abraham@univ-orleans.fr, Bergounioux, M., E-mail: maitine.bergounioux@univ-orleans.fr, and Carlier, G., E-mail: carlier@ceremade.dauphine.fr. Tomographic Reconstruction from a Few Views: A Multi-Marginal Optimal Transport Approach. United States: N. p., 2017. Web. doi:10.1007/S00245-015-9323-3.
Abraham, I., E-mail: isabelle.abraham@cea.fr, Abraham, R., E-mail: romain.abraham@univ-orleans.fr, Bergounioux, M., E-mail: maitine.bergounioux@univ-orleans.fr, & Carlier, G., E-mail: carlier@ceremade.dauphine.fr. Tomographic Reconstruction from a Few Views: A Multi-Marginal Optimal Transport Approach. United States. doi:10.1007/S00245-015-9323-3.
Abraham, I., E-mail: isabelle.abraham@cea.fr, Abraham, R., E-mail: romain.abraham@univ-orleans.fr, Bergounioux, M., E-mail: maitine.bergounioux@univ-orleans.fr, and Carlier, G., E-mail: carlier@ceremade.dauphine.fr. Wed . "Tomographic Reconstruction from a Few Views: A Multi-Marginal Optimal Transport Approach". United States. doi:10.1007/S00245-015-9323-3.
@article{osti_22617102,
title = {Tomographic Reconstruction from a Few Views: A Multi-Marginal Optimal Transport Approach},
author = {Abraham, I., E-mail: isabelle.abraham@cea.fr and Abraham, R., E-mail: romain.abraham@univ-orleans.fr and Bergounioux, M., E-mail: maitine.bergounioux@univ-orleans.fr and Carlier, G., E-mail: carlier@ceremade.dauphine.fr},
abstractNote = {In this article, we focus on tomographic reconstruction. The problem is to determine the shape of the interior interface using a tomographic approach while very few X-ray radiographs are performed. We use a multi-marginal optimal transport approach. Preliminary numerical results are presented.},
doi = {10.1007/S00245-015-9323-3},
journal = {Applied Mathematics and Optimization},
number = 1,
volume = 75,
place = {United States},
year = {Wed Feb 15 00:00:00 EST 2017},
month = {Wed Feb 15 00:00:00 EST 2017}
}
  • The main focus of this paper is reconstruction of tomographic phase-contrast image from a set of projections. We propose an efficient reconstruction algorithm for differential phase-contrast computed tomography that can considerably reduce the number of projections required for reconstruction. The key result underlying this research is a projection theorem that states that the second derivative of the projection set is linearly related to the Laplacian of the tomographic image. The proposed algorithm first reconstructs the Laplacian image of the phase-shift distribution from the second-derivative of the projections using total variation regularization. The second step is to obtain the phase-shift distributionmore » by solving a Poisson equation whose source is the Laplacian image previously reconstructed under the Dirichlet condition. We demonstrate the efficacy of this algorithm using both synthetically generated simulation data and projection data acquired experimentally at a synchrotron. The experimental phase data were acquired from a human coronary artery specimen using dark-field-imaging optics pioneered by our group. Our results demonstrate that the proposed algorithm can reduce the number of projections to approximately 33% as compared with the conventional filtered backprojection method, without any detrimental effect on the image quality.« less
  • Usually tomographic procedure requires a set of projections around the object under study and a mathematical processing of such projections through reconstruction algorithms. An accurate reconstruction requires a proper number of projections (angular sampling) and a proper number of elements in each projection (linear sampling). However in several practical cases it is not possible to fulfill these conditions leading to the so-called problem of few projections. In this case, iterative reconstruction algorithms are more suitable than analytic ones. In this work we present a program written in C++ that provides an environment for two iterative algorithm implementations, one algebraic andmore » the other statistical. The software allows the user a full definition of the acquisition and reconstruction geometries used for the reconstruction algorithms but also to perform projection and backprojection operations. A set of analysis tools was implemented for the characterization of the convergence process. We analyze the performance of the algorithms on numerical phantoms and present the reconstruction of experimental data with few projections coming from transmission X-ray and micro PIXE (Particle-Induced X-Ray Emission) images.« less
  • We present a method for tomographic reconstruction of objects containing several distinct materials, which is capable of accurately reconstructing a sample from vastly fewer angular projections than required by conventional algorithms. The algorithm is more general than many previous discrete tomography methods, as: (i) a priori knowledge of the exact number of materials is not required; (ii) the linear attenuation coefficient of each constituent material may assume a small range of a priori unknown values. We present reconstructions from an experimental x-ray computed tomography scan of cortical bone acquired at the SPring-8 synchrotron.
  • An ultrasonic signal processing technique is applied to multi-mode arrival time estimation from Lamb waveforms. The basic tool is a simplified time-scale projection called a dynamic wavelet fingerprint (DWFP) which enables direct observation of the variation of features of interest in non-stationary ultrasonic signals. The DWFP technique was used to automatically detect and evaluate each candidate through-transmitted Lamb mode. The area of the dynamic wavelet fingerprint was then used as a feature to distinguish false modes caused by noise and other interference from the true modes of interest. The set of estimated arrival times were then used as inputs formore » tomographic reconstruction. The Lamb wave tomography images generated with these estimated arrival times were able to indicate different defects in aluminum plates.« less