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Title: Data consistency conditions for truncated fanbeam and parallel projections

Abstract

Purpose: In image reconstruction from projections, data consistency conditions (DCCs) are mathematical relationships that express the overlap of information between ideal projections. DCCs have been incorporated in image reconstruction procedures for positron emission tomography, single photon emission computed tomography, and x-ray computed tomography (CT). Building on published fanbeam DCCs for nontruncated projections along a line, the authors recently announced new DCCs that can be applied to truncated parallel projections in classical (two-dimensional) image reconstruction. These DCCs take the form of polynomial expressions for a weighted backprojection of the projections. The purpose of this work was to present the new DCCs for truncated parallel projections, to extend these conditions to truncated fanbeam projections on a circular trajectory, to verify the conditions with numerical examples, and to present a model of how DCCs could be applied with a toy problem in patient motion estimation with truncated projections. Methods: A mathematical derivation of the new parallel DCCs was performed by substituting the underlying imaging equation into the mathematical expression for the weighted backprojection and demonstrating the resulting polynomial form. This DCC result was extended to fanbeam projections by a substitution of parallel to fanbeam variables. Ideal fanbeam projections of a simple mathematical phantommore » were simulated and the DCCs for these projections were evaluated by fitting polynomials to the weighted backprojection. For the motion estimation problem, a parametrized motion was simulated using a dynamic version of the mathematical phantom, and both noiseless and noisy fanbeam projections were simulated for a full circular trajectory. The fanbeam DCCs were applied to extract the motion parameters, which allowed the motion contamination to be removed from the projections. A reconstruction was performed from the corrected projections. Results: The mathematical derivation revealed the anticipated polynomial behavior. The conversion to fanbeam variables led to a straight-forward weighted fanbeam backprojection which yielded the same function and therefore the same polynomial behavior as occurred in the parallel case. Plots of the numerically calculated DCCs showed polynomial behavior visually indistinguishable from the fitted polynomials. For the motion estimation problem, the motion parameters were satisfactorily recovered and ten times more accurately for the noise-free case. The reconstructed images showed that only a faint trace of the motion blur was still visible after correction from the noisy motion-contaminated projections. Conclusions: New DCCs have been established for fanbeam and parallel projections, and these conditions have been validated using numerical experiments with truncated projections. It has been shown how these DCCs could be applied to extract parameters of unwanted physical effects in tomographic imaging, even with truncated projections.« less

Authors:
 [1];  [2]
  1. Laboratoire Hubert Curien, CNRS and Université Jean Monnet (UMR5516), 18 rue du Professeur Benoit Lauras, 42000 Saint Etienne (France)
  2. TIMC-IMAG, CNRS and Université Joseph Fourier (UMR5525), Pavillon Taillefer, La Tronche, 38706 Grenoble (France)
Publication Date:
OSTI Identifier:
22413448
Resource Type:
Journal Article
Resource Relation:
Journal Name: Medical Physics; Journal Volume: 42; Journal Issue: 2; Other Information: (c) 2015 American Association of Physicists in Medicine; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
62 RADIOLOGY AND NUCLEAR MEDICINE; IMAGE PROCESSING; NOISE; PHANTOMS; POSITRON COMPUTED TOMOGRAPHY; SINGLE PHOTON EMISSION COMPUTED TOMOGRAPHY

Citation Formats

Clackdoyle, Rolf, E-mail: rolf.clackdoyle@univ-st-etienne.fr, and Desbat, Laurent. Data consistency conditions for truncated fanbeam and parallel projections. United States: N. p., 2015. Web. doi:10.1118/1.4905161.
Clackdoyle, Rolf, E-mail: rolf.clackdoyle@univ-st-etienne.fr, & Desbat, Laurent. Data consistency conditions for truncated fanbeam and parallel projections. United States. doi:10.1118/1.4905161.
Clackdoyle, Rolf, E-mail: rolf.clackdoyle@univ-st-etienne.fr, and Desbat, Laurent. Sun . "Data consistency conditions for truncated fanbeam and parallel projections". United States. doi:10.1118/1.4905161.
@article{osti_22413448,
title = {Data consistency conditions for truncated fanbeam and parallel projections},
author = {Clackdoyle, Rolf, E-mail: rolf.clackdoyle@univ-st-etienne.fr and Desbat, Laurent},
abstractNote = {Purpose: In image reconstruction from projections, data consistency conditions (DCCs) are mathematical relationships that express the overlap of information between ideal projections. DCCs have been incorporated in image reconstruction procedures for positron emission tomography, single photon emission computed tomography, and x-ray computed tomography (CT). Building on published fanbeam DCCs for nontruncated projections along a line, the authors recently announced new DCCs that can be applied to truncated parallel projections in classical (two-dimensional) image reconstruction. These DCCs take the form of polynomial expressions for a weighted backprojection of the projections. The purpose of this work was to present the new DCCs for truncated parallel projections, to extend these conditions to truncated fanbeam projections on a circular trajectory, to verify the conditions with numerical examples, and to present a model of how DCCs could be applied with a toy problem in patient motion estimation with truncated projections. Methods: A mathematical derivation of the new parallel DCCs was performed by substituting the underlying imaging equation into the mathematical expression for the weighted backprojection and demonstrating the resulting polynomial form. This DCC result was extended to fanbeam projections by a substitution of parallel to fanbeam variables. Ideal fanbeam projections of a simple mathematical phantom were simulated and the DCCs for these projections were evaluated by fitting polynomials to the weighted backprojection. For the motion estimation problem, a parametrized motion was simulated using a dynamic version of the mathematical phantom, and both noiseless and noisy fanbeam projections were simulated for a full circular trajectory. The fanbeam DCCs were applied to extract the motion parameters, which allowed the motion contamination to be removed from the projections. A reconstruction was performed from the corrected projections. Results: The mathematical derivation revealed the anticipated polynomial behavior. The conversion to fanbeam variables led to a straight-forward weighted fanbeam backprojection which yielded the same function and therefore the same polynomial behavior as occurred in the parallel case. Plots of the numerically calculated DCCs showed polynomial behavior visually indistinguishable from the fitted polynomials. For the motion estimation problem, the motion parameters were satisfactorily recovered and ten times more accurately for the noise-free case. The reconstructed images showed that only a faint trace of the motion blur was still visible after correction from the noisy motion-contaminated projections. Conclusions: New DCCs have been established for fanbeam and parallel projections, and these conditions have been validated using numerical experiments with truncated projections. It has been shown how these DCCs could be applied to extract parameters of unwanted physical effects in tomographic imaging, even with truncated projections.},
doi = {10.1118/1.4905161},
journal = {Medical Physics},
number = 2,
volume = 42,
place = {United States},
year = {Sun Feb 15 00:00:00 EST 2015},
month = {Sun Feb 15 00:00:00 EST 2015}
}
  • We report the results of two studies of aspects of the consistency of truncated nonlinear integral equation based theories of freezing: (i) We show that the self-consistent solutions to these nonlinear equations are unfortunately sensitive to the level of truncation. For the hard sphere system, if the Wertheim--Thiele representation of the pair direct correlation function is used, the inclusion of part but not all of the triplet direct correlation function contribution, as has been common, worsens the predictions considerably. We also show that the convergence of the solutions found, with respect to number of reciprocal lattice vectors kept in themore » Fourier expansion of the crystal singlet density, is slow. These conclusions imply great sensitivity to the quality of the pair direct correlation function employed in the theory. (ii) We show the direct correlation function based and the pair correlation function based theories of freezing can be cast into a form which requires solution of isomorphous nonlinear integral equations. However, in the pair correlation function theory the usual neglect of the influence of inhomogeneity of the density distribution on the pair correlation function is shown to be inconsistent to the lowest order in the change of density on freezing, and to lead to erroneous predictions.« less
  • The authors have developed a new imaging geometry--the asymmetric-fan-beam (AsF)--to expand the imaging field of view (FOV) for transmission imaging on current SPECT systems. The AsF geometry samples slightly more than one-half of the FOV in each projection and yields half-truncated projection data. Although each projection profile is not complete, the combined acquired data set meets the minimum sampling requirement because the other half of the FOV is sampled in the opposite projections after a full 360{degree} rotation of the detector system. To take advantage of the simple convolution-backprojection algorithm for reconstruction, the key is in the handling of themore » projection profile for convolution. They have investigated such a technique to process the truncated projection profile for reconstruction without the side effects caused by the truncation. This technique entails filling in the truncated portion of each profile with interpolated data derived from other projections. After convolution, only the corresponding half of the original projection profile is backprojected in reconstruction. This is done to minimize propagation of the errors of interpolation in the reconstruction. Reconstructed images from phantoms and human subjects demonstrate that this processing technique yields good quality transmission images.« less
  • Purpose: A dedicated cardiac hybrid single photon emission computed tomography (SPECT)/CT scanner that uses cadmium zinc telluride detectors and multiple pinhole collimators for stationary acquisition offers many advantages. However, the impact of the reconstruction system matrix (SM) dimension on the reconstructed image quality from truncated projections and 19 angular samples acquired on this scanner has not been extensively investigated. In this study, the authors aimed to investigate the impact of the dimensions of SM and the use of body contour derived from adjunctive CT imaging as an object support in reconstruction on this scanner, in relation to background extracardiac activity.more » Methods: The authors first simulated a generic SPECT/CT system to image four NCAT phantoms with various levels of extracardiac activity and compared the reconstructions using SM in different dimensions and with/without body contour as a support for quantitative evaluations. The authors then compared the reconstructions of 18 patient studies, which were acquired on a GE Discovery NM570c scanner following injection of different radiotracers, including {sup 99m}Tc-Tetrofosmin and {sup 123}I-mIBG, comparing the scanner’s default SM that incompletely covers the body with a large SM that incorporates a patient specific full body contour. Results: The simulation studies showed that the reconstructions using a SM that only partially covers the body yielded artifacts on the edge of the field of view (FOV), overestimation of activity and increased nonuniformity in the blood pool for the phantoms with higher relative levels of extracardiac activity. However, the impact on the quantitative accuracy in the high activity region, such as the myocardium, was subtle. On the other hand, an excessively large SM that enclosed the entire body alleviated the artifacts and reduced overestimation in the blood pool, but yielded slight underestimation in myocardium and defect regions. The reconstruction using the larger SM with body contour yielded the most quantitatively accurate results in all the regions of interest for a range of uptake levels in the extracardiac regions. In patient studies, the SM incorporating patient specific body contour minimized extracardiac artifacts, yielded similar myocardial activity, lower blood pool activity, and subsequently improved myocardium-to-blood pool contrast (p < 0.0001) by an average of 7% (range 0%–18%) across all the patients, compared to the reconstructions using the scanner’s default SM. Conclusions: Their results demonstrate that using a large SM that incorporates a CT derived body contour in the reconstruction could improve quantitative accuracy within the FOV for clinical studies with high extracardiac activity.« less
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