Interpolation-based reconstruction methods for tomographic imaging in 3D Positron Emission Tomography

Yingbo Li; Anton Kummert; Fritz Boschen; Hans Herzog

International Journal of Applied Mathematics and Computer Science (2008)

  • Volume: 18, Issue: 1, page 63-73
  • ISSN: 1641-876X

Abstract

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Positron Emission Tomography (PET) is considered a key diagnostic tool in neuroscience, by means of which valuable insight into the metabolism function in vivo may be gained. Due to the underlying physical nature of PET, 3D imaging techniques in terms of a 3D measuring mode are intrinsically demanded to assure satisfying resolutions of the reconstructed images. However, incorporating additional cross-plane measurements, which are specific for the 3D measuring mode, usually imposes an excessive amount of projection data and significantly complicates the reconstruction procedure. For this reason, interpolation-based reconstruction methods deserve a thorough investigation, whose crucial parts are the interpolating processes in the 3D frequency domain. The benefit of such approaches is apparently short reconstruction duration, which can, however, only be achieved at the expense of accepting the inaccuracies associated with the interpolating process. In the present paper, two distinct approaches to the realization of the interpolating procedure are proposed and analyzed. The first one refers to a direct approach based on linear averaging (inverse distance weighting), and the second one refers to an indirect approach based on two-dimensional convolution (gridding method). In particular, attention is paid to two aspects of the gridding method. The first aspect is the choice of the two-dimensional convolution function applied, and the second one is the correct discretization of the underlying continuous convolution. In this respect, the geometrical structure named the Voronoi diagram and its computational construction are considered. At the end, results of performed simulation studies are presented and discussed.

How to cite

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Yingbo Li, et al. "Interpolation-based reconstruction methods for tomographic imaging in 3D Positron Emission Tomography." International Journal of Applied Mathematics and Computer Science 18.1 (2008): 63-73. <http://eudml.org/doc/207865>.

@article{YingboLi2008,
abstract = {Positron Emission Tomography (PET) is considered a key diagnostic tool in neuroscience, by means of which valuable insight into the metabolism function in vivo may be gained. Due to the underlying physical nature of PET, 3D imaging techniques in terms of a 3D measuring mode are intrinsically demanded to assure satisfying resolutions of the reconstructed images. However, incorporating additional cross-plane measurements, which are specific for the 3D measuring mode, usually imposes an excessive amount of projection data and significantly complicates the reconstruction procedure. For this reason, interpolation-based reconstruction methods deserve a thorough investigation, whose crucial parts are the interpolating processes in the 3D frequency domain. The benefit of such approaches is apparently short reconstruction duration, which can, however, only be achieved at the expense of accepting the inaccuracies associated with the interpolating process. In the present paper, two distinct approaches to the realization of the interpolating procedure are proposed and analyzed. The first one refers to a direct approach based on linear averaging (inverse distance weighting), and the second one refers to an indirect approach based on two-dimensional convolution (gridding method). In particular, attention is paid to two aspects of the gridding method. The first aspect is the choice of the two-dimensional convolution function applied, and the second one is the correct discretization of the underlying continuous convolution. In this respect, the geometrical structure named the Voronoi diagram and its computational construction are considered. At the end, results of performed simulation studies are presented and discussed.},
author = {Yingbo Li, Anton Kummert, Fritz Boschen, Hans Herzog},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {tomographic reconstruction; three-dimensional positron emission tomography; Fourier slice theorem; frequency sample distribution; two-dimensional interpolation; inverse distance weighting; gridding method},
language = {eng},
number = {1},
pages = {63-73},
title = {Interpolation-based reconstruction methods for tomographic imaging in 3D Positron Emission Tomography},
url = {http://eudml.org/doc/207865},
volume = {18},
year = {2008},
}

TY - JOUR
AU - Yingbo Li
AU - Anton Kummert
AU - Fritz Boschen
AU - Hans Herzog
TI - Interpolation-based reconstruction methods for tomographic imaging in 3D Positron Emission Tomography
JO - International Journal of Applied Mathematics and Computer Science
PY - 2008
VL - 18
IS - 1
SP - 63
EP - 73
AB - Positron Emission Tomography (PET) is considered a key diagnostic tool in neuroscience, by means of which valuable insight into the metabolism function in vivo may be gained. Due to the underlying physical nature of PET, 3D imaging techniques in terms of a 3D measuring mode are intrinsically demanded to assure satisfying resolutions of the reconstructed images. However, incorporating additional cross-plane measurements, which are specific for the 3D measuring mode, usually imposes an excessive amount of projection data and significantly complicates the reconstruction procedure. For this reason, interpolation-based reconstruction methods deserve a thorough investigation, whose crucial parts are the interpolating processes in the 3D frequency domain. The benefit of such approaches is apparently short reconstruction duration, which can, however, only be achieved at the expense of accepting the inaccuracies associated with the interpolating process. In the present paper, two distinct approaches to the realization of the interpolating procedure are proposed and analyzed. The first one refers to a direct approach based on linear averaging (inverse distance weighting), and the second one refers to an indirect approach based on two-dimensional convolution (gridding method). In particular, attention is paid to two aspects of the gridding method. The first aspect is the choice of the two-dimensional convolution function applied, and the second one is the correct discretization of the underlying continuous convolution. In this respect, the geometrical structure named the Voronoi diagram and its computational construction are considered. At the end, results of performed simulation studies are presented and discussed.
LA - eng
KW - tomographic reconstruction; three-dimensional positron emission tomography; Fourier slice theorem; frequency sample distribution; two-dimensional interpolation; inverse distance weighting; gridding method
UR - http://eudml.org/doc/207865
ER -

References

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  10. Li Y., Kummert A. and Herzog H. (2006). Direct Fourier method in 3D PET using accurately determined frequency sample distribution, Proceedings of the 28th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, New York, USA. 
  11. Li Y., Kummert A., Li H. and Herzog H. (2006). Evaluation of the direct Fourier method for 3D-PET in the case of accurately determined projection data, Proceedings of the 11th IASTED International Conference on Signal and Image Processing, Honolulu, USA. 
  12. Matej S. and Lewitt R. M. (2001). 3D-FRP: Direct Fourier reconstruction with Fourier reprojection for fully 3-D PET, IEEE Transactions on Medical Imaging 48(4): 1378-1385. 
  13. Moon T.K. (1996). The expectation-maximization algorithm, IEEE Signal Processing Magazine 13(6): 47-60. 
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  15. Thevenaz P., Blu T. and Unser M. (2000). Interpolation revisited, IEEE Transactions on Medical Imaging 19(7): 739-758. 

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