A General Local Reconstruction Approach Based on a Truncated Hilbert Transform

Ye, Yangbo; Yu, Hengyong; Wei, Yuchuan; Wang, Ge

doi:https://doi.org/10.1155/2007/63634

International Journal of Biomedical Imaging

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Research Article | Open Access

Volume 2007 | Article ID 063634 | https://doi.org/10.1155/2007/63634

A General Local Reconstruction Approach Based on a Truncated Hilbert Transform

Yangbo Ye,¹Hengyong Yu,²Yuchuan Wei,³and Ge Wang^2,3

Academic Editor: Lizhi Sun

Received15 May 2007

Accepted22 May 2007

Published17 Jun 2007

Abstract

Exact image reconstruction from limited projection data has been a central topic in the computed tomography (CT) field. In this paper, we present a general region-of-interest/volume-of-interest (ROI/VOI) reconstruction approach using a truly truncated Hilbert transform on a line-segment inside a compactly supported object aided by partial knowledge on one or both neighboring intervals of that segment. Our approach and associated new data sufficient condition allows the most flexible ROI/VOI image reconstruction from the minimum account of data in both the fan-beam and cone-beam geometry. We also report primary numerical simulation results to demonstrate the correctness and merits of our finding. Our work has major theoretical potentials and innovative practical applications.

References

G. N. Hounsfield, “Computerized transverse axial scanning (tomography)—part I: description of system,” British Journal of Radiology, vol. 46, no. 552, pp. 1016–1022, 1973.
View at: Google Scholar
D. J. Brenner, C. D. Elliston, E. J. Hall, and W. E. Berdon, “Estimated risks of radiation-induced fatal cancer from pediatric CT,” American Journal of Roentgenology, vol. 176, no. 2, pp. 289–296, 2001.
View at: Google Scholar
W. Huda, J. V. Atherton, D. E. Ware, and W. A. Cumming, “An approach for the estimation of effective radiation dose at CT in pediatric patients,” Radiology, vol. 203, no. 2, pp. 417–422, 1997.
View at: Google Scholar
D. L. Parker, “Optimal short scan convolution reconstruction for fanbeam CT,” Medical Physics, vol. 9, no. 2, pp. 254–257, 1982.
View at: Publisher Site | Google Scholar
F. Noo, M. Defrise, R. Clackdoyle, and H. Kudo, “Image reconstruction from fan-beam projections on less than a short scan,” Physics in Medicine and Biology, vol. 47, no. 14, pp. 2525–2546, 2002.
View at: Publisher Site | Google Scholar
I. M. Gel'fand and M. I. Graev, “Crofton's function and inversion formulas in real integral geometry,” Functional Analysis and Its Applications, vol. 25, no. 1, pp. 1–5, 1991.
View at: Publisher Site | Google Scholar
F. Noo, R. Clackdoyle, and J. D. Pack, “A two-step Hilbert transform method for 2D image reconstruction,” Physics in Medicine and Biology, vol. 49, no. 17, pp. 3903–3923, 2004.
View at: Publisher Site | Google Scholar
F. G. Tricomi, “On the finite Hilbert transformation,” Quarterly Journal of Mathematics, vol. 2, no. 1, pp. 199–211, 1951.
View at: Publisher Site | Google Scholar
M. Defrise, F. Noo, R. Clackdoyle, and H. Kudo, “Truncated Hilbert transform and image reconstruction from limited tomographic data,” Inverse Problems, vol. 22, no. 3, pp. 1037–1053, 2006.
View at: Publisher Site | Google Scholar
A. Katsevich, “An improved exact filtered backprojection algorithm for spiral computed tomography,” Advances in Applied Mathematics, vol. 32, no. 4, pp. 681–697, 2004.
View at: Publisher Site | Google Scholar
A. Katsevich, “Theoretically exact filtered backprojection-type inversion algorithm for spiral CT,” SIAM Journal on Applied Mathematics, vol. 62, no. 6, pp. 2012–2026, 2002.
View at: Publisher Site | Google Scholar
A. Katsevich, “A general scheme for constructing inversion algorithms for cone beam CT,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 21, pp. 1305–1321, 2003.
View at: Publisher Site | Google Scholar
Y. Zou and X. Pan, “Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT,” Physics in Medicine and Biology, vol. 49, no. 6, pp. 941–959, 2004.
View at: Publisher Site | Google Scholar
Y. Ye, S. Zhao, H. Yu, and G. Wang, “Exact reconstruction for cone-beam scanning along nonstandard spirals and other curves,” in Developments in X-Ray Tomography IV, vol. 5535 of Proceedings of SPIE, pp. 293–300, Denver, Colo, USA, August 2004.
View at: Publisher Site | Google Scholar
S. Zhao, H. Yu, and G. Wang, “A family of analytic algorithms for cone-beam CT,” in Developments in X-Ray Tomography IV, vol. 5535 of Proceedings of SPIE, pp. 318–328, Denver, Colo, USA, August 2004.
View at: Publisher Site | Google Scholar
Y. Ye and G. Wang, “Filtered backprojection formula for exact image reconstruction from cone-beam data along a general scanning curve,” Medical Physics, vol. 32, no. 1, pp. 42–48, 2005.
View at: Publisher Site | Google Scholar
Y. Ye, S. Zhao, H. Yu, and G. Wang, “A general exact reconstruction for cone-beam CT via backprojection-filtration,” IEEE Transactions on Medical Imaging, vol. 24, no. 9, pp. 1190–1198, 2005.
View at: Publisher Site | Google Scholar
S. Zhao, H. Yu, and G. Wang, “A unified framework for exact cone-beam reconstruction formulas,” Medical Physics, vol. 32, no. 6, pp. 1712–1721, 2005.
View at: Publisher Site | Google Scholar
J. D. Pack and F. Noo, “Cone-beam reconstruction using 1D filtering along the projection of M-lines,” Inverse Problems, vol. 21, no. 3, pp. 1105–1120, 2005.
View at: Publisher Site | Google Scholar
J. D. Pack, F. Noo, and R. Clackdoyle, “Cone-beam reconstruction using the backprojection of locally filtered projections,” IEEE Transactions on Medical Imaging, vol. 24, no. 1, pp. 70–85, 2005.
View at: Publisher Site | Google Scholar
T. Zhuang, S. Leng, B. E. Nett, and G.-H. Chen, “Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data,” Physics in Medicine and Biology, vol. 49, no. 24, pp. 5489–5503, 2004.
View at: Publisher Site | Google Scholar
Y. Zou and X. Pan, “An extended data function and its generalized backprojection for image reconstruction in helical cone-beam CT,” Physics in Medicine and Biology, vol. 49, no. 22, pp. N383–N387, 2004.
View at: Publisher Site | Google Scholar
G. Wang, Y. Ye, and H. Yu, “General VOI/ROI Reconstruction Methods and Systems using a Truncated Hilbert Transform (Patent disclosure submitted to Virginia Tech. Intellectual Properties, May 2007),” 2007.
View at: Google Scholar
D. Youla and H. Webb, “Image restoration by the method of convex projections—part I: theory,” IEEE Transactions on Medical Imaging, vol. 1, no. 2, pp. 81–94, 1982.
View at: Google Scholar
A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, IEEE Press, New York, NY, USA, 1988.
K. Sourbelle, “Thorax Phantom,” http://www.imp.uni-erlangen.de/forbild/english/results/index.htm.
View at: Google Scholar

Copyright

Copyright © 2007 Yangbo Ye et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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