Abstract

In whole-core CT imaging, scanned data corresponding to the central portion of a cylindrical core often suffer from photon starvation, because increasing photon flux will cause overflow on some detector units under the restriction of detector dynamic range. Either photon starvation or data overflow will lead to increased noise or severe artifacts in the reconstructed CT image. In addition, cupping shaped beam hardening artifacts also appear in the whole-core CT image. In this paper, we present a method to design an attenuator for cone beam whole-core CT, which not only reduces the dynamic range requirement for high SNR data scanning, but also corrects beam hardening artifacts. Both simulation and real data are employed to verify our design method.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref] [PubMed]

2016 (1)

Y. Liu, A. M. Kiss, D. H. Larsson, F. Yang, and P. Pianetta, “To get the most out of high resolution X-ray tomography: A review of the post-reconstruction analysis,” Spectrochim. Acta B At. Spectrosc. 117, 29–41 (2016).
[Crossref]

2015 (2)

K. Kontson and R. J. Jennings, “Bowtie filters for dedicated breast CT: theory and computational implementation,” Med. Phys. 42(3), 1453–1462 (2015).
[Crossref] [PubMed]

K. Kontson and R. J. Jennings, “Bowtie filters for dedicated breast CT: Analysis of bowtie filter material selection,” Med. Phys. 42(9), 5270–5277 (2015).
[Crossref] [PubMed]

2014 (1)

F. Liu, Q. Yang, W. Cong, and G. Wang, “Dynamic bowtie filter for cone-beam/multi-slice CT,” PLoS One 9(7), e103054 (2014).
[Crossref] [PubMed]

2013 (6)

K. Taguchi and J. S. Iwanczyk, “Vision 20/20: Single photon counting X-ray detectors in medical imaging,” Med. Phys. 40(10), 100901 (2013).
[Crossref] [PubMed]

S. S. Hsieh and N. J. Pelc, “The feasibility of a piecewise-linear dynamic bowtie filter,” Med. Phys. 40(3), 031910 (2013).
[Crossref] [PubMed]

F. Liu, G. Wang, W. Cong, S. S. Hsieh, and N. J. Pelc, “Dynamic bowtie for fan-beam CT,” J. XRay Sci. Technol. 21(4), 579–590 (2013).
[PubMed]

I. Mori, Y. Machida, M. Osanai, and K. Iinuma, “Photon starvation artifacts of X-ray CT: their true cause and a solution,” Radiological Phys. Technol. 6(1), 130–141 (2013).
[Crossref] [PubMed]

V. Cnudde and M. N. Boone, “High-resolution X-ray computed tomography in geosciences: A review of the current technology and applications,” Earth Sci. Rev. 123, 1–17 (2013).
[Crossref]

F. Lück, D. Kolditz, M. Hupfer, and W. A. Kalender, “Effect of shaped filter design on dose and image quality in breast CT,” Phys. Med. Biol. 58(12), 4205–4223 (2013).
[Crossref] [PubMed]

2012 (3)

W. A. Kalender, M. Beister, J. M. Boone, D. Kolditz, S. V. Vollmar, and M. C. C. Weigel, “High-resolution spiral CT of the breast at very low dose: concept and feasibility considerations,” Eur. Radiol. 22(1), 1–8 (2012).
[Crossref] [PubMed]

M. Blessing, M. S. Bhagwat, Y. Lyatskaya, J. R. Bellon, J. Hesser, and P. Zygmanski, “Kilovoltage beam model for flat panel imaging system with bow-tie filter for scatter prediction and correction,” Phys. Med. 28(2), 134–143 (2012).
[Crossref] [PubMed]

B. M. Gramer, D. Muenzel, V. Leber, A. K. von Thaden, H. Feussner, A. Schneider, M. Vembar, N. Soni, E. J. Rummeny, and A. M. Huber, “Impact of iterative reconstruction on CNR and SNR in dynamic myocardial perfusion imaging in an animal model,” Eur. Radiol. 22(12), 2654–2661 (2012).
[Crossref] [PubMed]

2011 (4)

S. Bartolac, S. Graham, J. Siewerdsen, and D. Jaffray, “Fluence field optimization for noise and dose objectives in CT,” Med. Phys. 38(S1), S2–S17 (2011).
[Crossref] [PubMed]

S. E. McKenney, A. Nosratieh, D. Gelskey, K. Yang, S. Y. Huang, L. Chen, and J. M. Boone, “Experimental validation of a method characterizing bow tie filters in CT scanners using a real-time dose probe,” Med. Phys. 38(3), 1406–1415 (2011).
[Crossref] [PubMed]

R. B. Benítez and R. Ning, “Development of a beam hardening filter and characterization of the spatial resolution for a cone beam CT imaging system,” Proc. SPIE 91, 391–396 (2011).

G. J. Bootsma, F. Verhaegen, and D. A. Jaffray, “The effects of compensator and imaging geometry on the distribution of x-ray scatter in CBCT,” Med. Phys. 38(2), 897–914 (2011).
[Crossref] [PubMed]

2010 (1)

K. Taguchi, E. C. Frey, X. Wang, J. S. Iwanczyk, and W. C. Barber, “An analytical model of the effects of pulse pileup on the energy spectrum recorded by energy resolved photon counting x-ray detectors,” Med. Phys. 37(8), 3957–3969 (2010).
[Crossref] [PubMed]

2009 (2)

N. Mail, D. J. Moseley, J. H. Siewerdsen, and D. A. Jaffray, “The influence of bowtie filtration on cone-beam CT image quality,” Med. Phys. 36(1), 22–32 (2009).
[Crossref] [PubMed]

O. P. Wennberg, L. Rennan, and R. Basquet, “Computed tomography scan imaging of natural open fractures in a porous rock; geometry and fluid flow,” Geophys. Prospect. 57(2), 239–249 (2009).
[Crossref]

2006 (1)

J. M. Boone, A. L. C. Kwan, K. Yang, G. W. Burkett, K. K. Lindfors, and T. R. Nelson, “Computed tomography for imaging the breast,” J. Mammary Gland Biol. Neoplasia 11(2), 103–111 (2006).
[Crossref] [PubMed]

2005 (1)

E. Y. Sidky, L. Yu, X. Pan, Y. Zou, and M. Vannier, “A robust method of X-ray source spectrum estimation from transmission measurements: Demonstrated on computer simulated, scatter-free transmission data,” J. Appl. Phys. 97(12), 124701 (2005).
[Crossref]

2004 (2)

J. F. Barrett and N. Keat, “Artifacts in CT: Recognition and Avoidance,” Radiographics 24(6), 1679–1691 (2004).
[Crossref] [PubMed]

J. M. Boone, N. Shah, and T. R. Nelson, “A comprehensive analysis of DgN(CT) coefficients for pendant-geometry cone-beam breast computed tomography,” Med. Phys. 31(2), 226–235 (2004).
[Crossref] [PubMed]

2001 (1)

R. A. Ketcham and W. D. Carlson, “Acquisition, optimization and interpretation of X-ray computed tomographic imagery: applications to the geosciences,” Comput. Geosci. 27(4), 381–400 (2001).
[Crossref]

1995 (1)

K. H. Wedepohl, “The composition of the continental crust,” Geochim. Cosmochim. Acta 59(7), 1217–1232 (1995).
[Crossref]

Barber, W. C.

K. Taguchi, E. C. Frey, X. Wang, J. S. Iwanczyk, and W. C. Barber, “An analytical model of the effects of pulse pileup on the energy spectrum recorded by energy resolved photon counting x-ray detectors,” Med. Phys. 37(8), 3957–3969 (2010).
[Crossref] [PubMed]

Barrett, J. F.

J. F. Barrett and N. Keat, “Artifacts in CT: Recognition and Avoidance,” Radiographics 24(6), 1679–1691 (2004).
[Crossref] [PubMed]

Bartolac, S.

S. Bartolac, S. Graham, J. Siewerdsen, and D. Jaffray, “Fluence field optimization for noise and dose objectives in CT,” Med. Phys. 38(S1), S2–S17 (2011).
[Crossref] [PubMed]

Basquet, R.

O. P. Wennberg, L. Rennan, and R. Basquet, “Computed tomography scan imaging of natural open fractures in a porous rock; geometry and fluid flow,” Geophys. Prospect. 57(2), 239–249 (2009).
[Crossref]

Beister, M.

W. A. Kalender, M. Beister, J. M. Boone, D. Kolditz, S. V. Vollmar, and M. C. C. Weigel, “High-resolution spiral CT of the breast at very low dose: concept and feasibility considerations,” Eur. Radiol. 22(1), 1–8 (2012).
[Crossref] [PubMed]

Bellon, J. R.

M. Blessing, M. S. Bhagwat, Y. Lyatskaya, J. R. Bellon, J. Hesser, and P. Zygmanski, “Kilovoltage beam model for flat panel imaging system with bow-tie filter for scatter prediction and correction,” Phys. Med. 28(2), 134–143 (2012).
[Crossref] [PubMed]

Benítez, R. B.

R. B. Benítez and R. Ning, “Development of a beam hardening filter and characterization of the spatial resolution for a cone beam CT imaging system,” Proc. SPIE 91, 391–396 (2011).

Bhagwat, M. S.

M. Blessing, M. S. Bhagwat, Y. Lyatskaya, J. R. Bellon, J. Hesser, and P. Zygmanski, “Kilovoltage beam model for flat panel imaging system with bow-tie filter for scatter prediction and correction,” Phys. Med. 28(2), 134–143 (2012).
[Crossref] [PubMed]

Blessing, M.

M. Blessing, M. S. Bhagwat, Y. Lyatskaya, J. R. Bellon, J. Hesser, and P. Zygmanski, “Kilovoltage beam model for flat panel imaging system with bow-tie filter for scatter prediction and correction,” Phys. Med. 28(2), 134–143 (2012).
[Crossref] [PubMed]

Boone, J. M.

W. A. Kalender, M. Beister, J. M. Boone, D. Kolditz, S. V. Vollmar, and M. C. C. Weigel, “High-resolution spiral CT of the breast at very low dose: concept and feasibility considerations,” Eur. Radiol. 22(1), 1–8 (2012).
[Crossref] [PubMed]

S. E. McKenney, A. Nosratieh, D. Gelskey, K. Yang, S. Y. Huang, L. Chen, and J. M. Boone, “Experimental validation of a method characterizing bow tie filters in CT scanners using a real-time dose probe,” Med. Phys. 38(3), 1406–1415 (2011).
[Crossref] [PubMed]

J. M. Boone, A. L. C. Kwan, K. Yang, G. W. Burkett, K. K. Lindfors, and T. R. Nelson, “Computed tomography for imaging the breast,” J. Mammary Gland Biol. Neoplasia 11(2), 103–111 (2006).
[Crossref] [PubMed]

J. M. Boone, N. Shah, and T. R. Nelson, “A comprehensive analysis of DgN(CT) coefficients for pendant-geometry cone-beam breast computed tomography,” Med. Phys. 31(2), 226–235 (2004).
[Crossref] [PubMed]

Boone, M. N.

V. Cnudde and M. N. Boone, “High-resolution X-ray computed tomography in geosciences: A review of the current technology and applications,” Earth Sci. Rev. 123, 1–17 (2013).
[Crossref]

Bootsma, G. J.

G. J. Bootsma, F. Verhaegen, and D. A. Jaffray, “The effects of compensator and imaging geometry on the distribution of x-ray scatter in CBCT,” Med. Phys. 38(2), 897–914 (2011).
[Crossref] [PubMed]

Burkett, G. W.

J. M. Boone, A. L. C. Kwan, K. Yang, G. W. Burkett, K. K. Lindfors, and T. R. Nelson, “Computed tomography for imaging the breast,” J. Mammary Gland Biol. Neoplasia 11(2), 103–111 (2006).
[Crossref] [PubMed]

Carlson, W. D.

R. A. Ketcham and W. D. Carlson, “Acquisition, optimization and interpretation of X-ray computed tomographic imagery: applications to the geosciences,” Comput. Geosci. 27(4), 381–400 (2001).
[Crossref]

Chen, L.

S. E. McKenney, A. Nosratieh, D. Gelskey, K. Yang, S. Y. Huang, L. Chen, and J. M. Boone, “Experimental validation of a method characterizing bow tie filters in CT scanners using a real-time dose probe,” Med. Phys. 38(3), 1406–1415 (2011).
[Crossref] [PubMed]

Cnudde, V.

V. Cnudde and M. N. Boone, “High-resolution X-ray computed tomography in geosciences: A review of the current technology and applications,” Earth Sci. Rev. 123, 1–17 (2013).
[Crossref]

Cong, W.

F. Liu, Q. Yang, W. Cong, and G. Wang, “Dynamic bowtie filter for cone-beam/multi-slice CT,” PLoS One 9(7), e103054 (2014).
[Crossref] [PubMed]

F. Liu, G. Wang, W. Cong, S. S. Hsieh, and N. J. Pelc, “Dynamic bowtie for fan-beam CT,” J. XRay Sci. Technol. 21(4), 579–590 (2013).
[PubMed]

Feussner, H.

B. M. Gramer, D. Muenzel, V. Leber, A. K. von Thaden, H. Feussner, A. Schneider, M. Vembar, N. Soni, E. J. Rummeny, and A. M. Huber, “Impact of iterative reconstruction on CNR and SNR in dynamic myocardial perfusion imaging in an animal model,” Eur. Radiol. 22(12), 2654–2661 (2012).
[Crossref] [PubMed]

Frey, E. C.

K. Taguchi, E. C. Frey, X. Wang, J. S. Iwanczyk, and W. C. Barber, “An analytical model of the effects of pulse pileup on the energy spectrum recorded by energy resolved photon counting x-ray detectors,” Med. Phys. 37(8), 3957–3969 (2010).
[Crossref] [PubMed]

Gelskey, D.

S. E. McKenney, A. Nosratieh, D. Gelskey, K. Yang, S. Y. Huang, L. Chen, and J. M. Boone, “Experimental validation of a method characterizing bow tie filters in CT scanners using a real-time dose probe,” Med. Phys. 38(3), 1406–1415 (2011).
[Crossref] [PubMed]

Graham, S.

S. Bartolac, S. Graham, J. Siewerdsen, and D. Jaffray, “Fluence field optimization for noise and dose objectives in CT,” Med. Phys. 38(S1), S2–S17 (2011).
[Crossref] [PubMed]

Gramer, B. M.

B. M. Gramer, D. Muenzel, V. Leber, A. K. von Thaden, H. Feussner, A. Schneider, M. Vembar, N. Soni, E. J. Rummeny, and A. M. Huber, “Impact of iterative reconstruction on CNR and SNR in dynamic myocardial perfusion imaging in an animal model,” Eur. Radiol. 22(12), 2654–2661 (2012).
[Crossref] [PubMed]

Hesser, J.

M. Blessing, M. S. Bhagwat, Y. Lyatskaya, J. R. Bellon, J. Hesser, and P. Zygmanski, “Kilovoltage beam model for flat panel imaging system with bow-tie filter for scatter prediction and correction,” Phys. Med. 28(2), 134–143 (2012).
[Crossref] [PubMed]

Hsieh, S. S.

F. Liu, G. Wang, W. Cong, S. S. Hsieh, and N. J. Pelc, “Dynamic bowtie for fan-beam CT,” J. XRay Sci. Technol. 21(4), 579–590 (2013).
[PubMed]

S. S. Hsieh and N. J. Pelc, “The feasibility of a piecewise-linear dynamic bowtie filter,” Med. Phys. 40(3), 031910 (2013).
[Crossref] [PubMed]

Huang, S. Y.

S. E. McKenney, A. Nosratieh, D. Gelskey, K. Yang, S. Y. Huang, L. Chen, and J. M. Boone, “Experimental validation of a method characterizing bow tie filters in CT scanners using a real-time dose probe,” Med. Phys. 38(3), 1406–1415 (2011).
[Crossref] [PubMed]

Huber, A. M.

B. M. Gramer, D. Muenzel, V. Leber, A. K. von Thaden, H. Feussner, A. Schneider, M. Vembar, N. Soni, E. J. Rummeny, and A. M. Huber, “Impact of iterative reconstruction on CNR and SNR in dynamic myocardial perfusion imaging in an animal model,” Eur. Radiol. 22(12), 2654–2661 (2012).
[Crossref] [PubMed]

Hupfer, M.

F. Lück, D. Kolditz, M. Hupfer, and W. A. Kalender, “Effect of shaped filter design on dose and image quality in breast CT,” Phys. Med. Biol. 58(12), 4205–4223 (2013).
[Crossref] [PubMed]

Iinuma, K.

I. Mori, Y. Machida, M. Osanai, and K. Iinuma, “Photon starvation artifacts of X-ray CT: their true cause and a solution,” Radiological Phys. Technol. 6(1), 130–141 (2013).
[Crossref] [PubMed]

Iwanczyk, J. S.

K. Taguchi and J. S. Iwanczyk, “Vision 20/20: Single photon counting X-ray detectors in medical imaging,” Med. Phys. 40(10), 100901 (2013).
[Crossref] [PubMed]

K. Taguchi, E. C. Frey, X. Wang, J. S. Iwanczyk, and W. C. Barber, “An analytical model of the effects of pulse pileup on the energy spectrum recorded by energy resolved photon counting x-ray detectors,” Med. Phys. 37(8), 3957–3969 (2010).
[Crossref] [PubMed]

Jaffray, D.

S. Bartolac, S. Graham, J. Siewerdsen, and D. Jaffray, “Fluence field optimization for noise and dose objectives in CT,” Med. Phys. 38(S1), S2–S17 (2011).
[Crossref] [PubMed]

Jaffray, D. A.

G. J. Bootsma, F. Verhaegen, and D. A. Jaffray, “The effects of compensator and imaging geometry on the distribution of x-ray scatter in CBCT,” Med. Phys. 38(2), 897–914 (2011).
[Crossref] [PubMed]

N. Mail, D. J. Moseley, J. H. Siewerdsen, and D. A. Jaffray, “The influence of bowtie filtration on cone-beam CT image quality,” Med. Phys. 36(1), 22–32 (2009).
[Crossref] [PubMed]

Jennings, R. J.

K. Kontson and R. J. Jennings, “Bowtie filters for dedicated breast CT: theory and computational implementation,” Med. Phys. 42(3), 1453–1462 (2015).
[Crossref] [PubMed]

K. Kontson and R. J. Jennings, “Bowtie filters for dedicated breast CT: Analysis of bowtie filter material selection,” Med. Phys. 42(9), 5270–5277 (2015).
[Crossref] [PubMed]

Kalender, W. A.

F. Lück, D. Kolditz, M. Hupfer, and W. A. Kalender, “Effect of shaped filter design on dose and image quality in breast CT,” Phys. Med. Biol. 58(12), 4205–4223 (2013).
[Crossref] [PubMed]

W. A. Kalender, M. Beister, J. M. Boone, D. Kolditz, S. V. Vollmar, and M. C. C. Weigel, “High-resolution spiral CT of the breast at very low dose: concept and feasibility considerations,” Eur. Radiol. 22(1), 1–8 (2012).
[Crossref] [PubMed]

Keat, N.

J. F. Barrett and N. Keat, “Artifacts in CT: Recognition and Avoidance,” Radiographics 24(6), 1679–1691 (2004).
[Crossref] [PubMed]

Ketcham, R. A.

R. A. Ketcham and W. D. Carlson, “Acquisition, optimization and interpretation of X-ray computed tomographic imagery: applications to the geosciences,” Comput. Geosci. 27(4), 381–400 (2001).
[Crossref]

Kiss, A. M.

Y. Liu, A. M. Kiss, D. H. Larsson, F. Yang, and P. Pianetta, “To get the most out of high resolution X-ray tomography: A review of the post-reconstruction analysis,” Spectrochim. Acta B At. Spectrosc. 117, 29–41 (2016).
[Crossref]

Kolditz, D.

F. Lück, D. Kolditz, M. Hupfer, and W. A. Kalender, “Effect of shaped filter design on dose and image quality in breast CT,” Phys. Med. Biol. 58(12), 4205–4223 (2013).
[Crossref] [PubMed]

W. A. Kalender, M. Beister, J. M. Boone, D. Kolditz, S. V. Vollmar, and M. C. C. Weigel, “High-resolution spiral CT of the breast at very low dose: concept and feasibility considerations,” Eur. Radiol. 22(1), 1–8 (2012).
[Crossref] [PubMed]

Kontson, K.

K. Kontson and R. J. Jennings, “Bowtie filters for dedicated breast CT: Analysis of bowtie filter material selection,” Med. Phys. 42(9), 5270–5277 (2015).
[Crossref] [PubMed]

K. Kontson and R. J. Jennings, “Bowtie filters for dedicated breast CT: theory and computational implementation,” Med. Phys. 42(3), 1453–1462 (2015).
[Crossref] [PubMed]

Kwan, A. L. C.

J. M. Boone, A. L. C. Kwan, K. Yang, G. W. Burkett, K. K. Lindfors, and T. R. Nelson, “Computed tomography for imaging the breast,” J. Mammary Gland Biol. Neoplasia 11(2), 103–111 (2006).
[Crossref] [PubMed]

Larsson, D. H.

Y. Liu, A. M. Kiss, D. H. Larsson, F. Yang, and P. Pianetta, “To get the most out of high resolution X-ray tomography: A review of the post-reconstruction analysis,” Spectrochim. Acta B At. Spectrosc. 117, 29–41 (2016).
[Crossref]

Leber, V.

B. M. Gramer, D. Muenzel, V. Leber, A. K. von Thaden, H. Feussner, A. Schneider, M. Vembar, N. Soni, E. J. Rummeny, and A. M. Huber, “Impact of iterative reconstruction on CNR and SNR in dynamic myocardial perfusion imaging in an animal model,” Eur. Radiol. 22(12), 2654–2661 (2012).
[Crossref] [PubMed]

Lindfors, K. K.

J. M. Boone, A. L. C. Kwan, K. Yang, G. W. Burkett, K. K. Lindfors, and T. R. Nelson, “Computed tomography for imaging the breast,” J. Mammary Gland Biol. Neoplasia 11(2), 103–111 (2006).
[Crossref] [PubMed]

Liu, F.

F. Liu, Q. Yang, W. Cong, and G. Wang, “Dynamic bowtie filter for cone-beam/multi-slice CT,” PLoS One 9(7), e103054 (2014).
[Crossref] [PubMed]

F. Liu, G. Wang, W. Cong, S. S. Hsieh, and N. J. Pelc, “Dynamic bowtie for fan-beam CT,” J. XRay Sci. Technol. 21(4), 579–590 (2013).
[PubMed]

Liu, Y.

Y. Liu, A. M. Kiss, D. H. Larsson, F. Yang, and P. Pianetta, “To get the most out of high resolution X-ray tomography: A review of the post-reconstruction analysis,” Spectrochim. Acta B At. Spectrosc. 117, 29–41 (2016).
[Crossref]

Lück, F.

F. Lück, D. Kolditz, M. Hupfer, and W. A. Kalender, “Effect of shaped filter design on dose and image quality in breast CT,” Phys. Med. Biol. 58(12), 4205–4223 (2013).
[Crossref] [PubMed]

Lyatskaya, Y.

M. Blessing, M. S. Bhagwat, Y. Lyatskaya, J. R. Bellon, J. Hesser, and P. Zygmanski, “Kilovoltage beam model for flat panel imaging system with bow-tie filter for scatter prediction and correction,” Phys. Med. 28(2), 134–143 (2012).
[Crossref] [PubMed]

Machida, Y.

I. Mori, Y. Machida, M. Osanai, and K. Iinuma, “Photon starvation artifacts of X-ray CT: their true cause and a solution,” Radiological Phys. Technol. 6(1), 130–141 (2013).
[Crossref] [PubMed]

Mail, N.

N. Mail, D. J. Moseley, J. H. Siewerdsen, and D. A. Jaffray, “The influence of bowtie filtration on cone-beam CT image quality,” Med. Phys. 36(1), 22–32 (2009).
[Crossref] [PubMed]

McKenney, S. E.

S. E. McKenney, A. Nosratieh, D. Gelskey, K. Yang, S. Y. Huang, L. Chen, and J. M. Boone, “Experimental validation of a method characterizing bow tie filters in CT scanners using a real-time dose probe,” Med. Phys. 38(3), 1406–1415 (2011).
[Crossref] [PubMed]

Mori, I.

I. Mori, Y. Machida, M. Osanai, and K. Iinuma, “Photon starvation artifacts of X-ray CT: their true cause and a solution,” Radiological Phys. Technol. 6(1), 130–141 (2013).
[Crossref] [PubMed]

Moseley, D. J.

N. Mail, D. J. Moseley, J. H. Siewerdsen, and D. A. Jaffray, “The influence of bowtie filtration on cone-beam CT image quality,” Med. Phys. 36(1), 22–32 (2009).
[Crossref] [PubMed]

Muenzel, D.

B. M. Gramer, D. Muenzel, V. Leber, A. K. von Thaden, H. Feussner, A. Schneider, M. Vembar, N. Soni, E. J. Rummeny, and A. M. Huber, “Impact of iterative reconstruction on CNR and SNR in dynamic myocardial perfusion imaging in an animal model,” Eur. Radiol. 22(12), 2654–2661 (2012).
[Crossref] [PubMed]

Nelson, T. R.

J. M. Boone, A. L. C. Kwan, K. Yang, G. W. Burkett, K. K. Lindfors, and T. R. Nelson, “Computed tomography for imaging the breast,” J. Mammary Gland Biol. Neoplasia 11(2), 103–111 (2006).
[Crossref] [PubMed]

J. M. Boone, N. Shah, and T. R. Nelson, “A comprehensive analysis of DgN(CT) coefficients for pendant-geometry cone-beam breast computed tomography,” Med. Phys. 31(2), 226–235 (2004).
[Crossref] [PubMed]

Ning, R.

R. B. Benítez and R. Ning, “Development of a beam hardening filter and characterization of the spatial resolution for a cone beam CT imaging system,” Proc. SPIE 91, 391–396 (2011).

Nosratieh, A.

S. E. McKenney, A. Nosratieh, D. Gelskey, K. Yang, S. Y. Huang, L. Chen, and J. M. Boone, “Experimental validation of a method characterizing bow tie filters in CT scanners using a real-time dose probe,” Med. Phys. 38(3), 1406–1415 (2011).
[Crossref] [PubMed]

Osanai, M.

I. Mori, Y. Machida, M. Osanai, and K. Iinuma, “Photon starvation artifacts of X-ray CT: their true cause and a solution,” Radiological Phys. Technol. 6(1), 130–141 (2013).
[Crossref] [PubMed]

Pan, X.

E. Y. Sidky, L. Yu, X. Pan, Y. Zou, and M. Vannier, “A robust method of X-ray source spectrum estimation from transmission measurements: Demonstrated on computer simulated, scatter-free transmission data,” J. Appl. Phys. 97(12), 124701 (2005).
[Crossref]

Pelc, N. J.

S. S. Hsieh and N. J. Pelc, “The feasibility of a piecewise-linear dynamic bowtie filter,” Med. Phys. 40(3), 031910 (2013).
[Crossref] [PubMed]

F. Liu, G. Wang, W. Cong, S. S. Hsieh, and N. J. Pelc, “Dynamic bowtie for fan-beam CT,” J. XRay Sci. Technol. 21(4), 579–590 (2013).
[PubMed]

Pianetta, P.

Y. Liu, A. M. Kiss, D. H. Larsson, F. Yang, and P. Pianetta, “To get the most out of high resolution X-ray tomography: A review of the post-reconstruction analysis,” Spectrochim. Acta B At. Spectrosc. 117, 29–41 (2016).
[Crossref]

Rennan, L.

O. P. Wennberg, L. Rennan, and R. Basquet, “Computed tomography scan imaging of natural open fractures in a porous rock; geometry and fluid flow,” Geophys. Prospect. 57(2), 239–249 (2009).
[Crossref]

Rummeny, E. J.

B. M. Gramer, D. Muenzel, V. Leber, A. K. von Thaden, H. Feussner, A. Schneider, M. Vembar, N. Soni, E. J. Rummeny, and A. M. Huber, “Impact of iterative reconstruction on CNR and SNR in dynamic myocardial perfusion imaging in an animal model,” Eur. Radiol. 22(12), 2654–2661 (2012).
[Crossref] [PubMed]

Schneider, A.

B. M. Gramer, D. Muenzel, V. Leber, A. K. von Thaden, H. Feussner, A. Schneider, M. Vembar, N. Soni, E. J. Rummeny, and A. M. Huber, “Impact of iterative reconstruction on CNR and SNR in dynamic myocardial perfusion imaging in an animal model,” Eur. Radiol. 22(12), 2654–2661 (2012).
[Crossref] [PubMed]

Shah, N.

J. M. Boone, N. Shah, and T. R. Nelson, “A comprehensive analysis of DgN(CT) coefficients for pendant-geometry cone-beam breast computed tomography,” Med. Phys. 31(2), 226–235 (2004).
[Crossref] [PubMed]

Sidky, E. Y.

E. Y. Sidky, L. Yu, X. Pan, Y. Zou, and M. Vannier, “A robust method of X-ray source spectrum estimation from transmission measurements: Demonstrated on computer simulated, scatter-free transmission data,” J. Appl. Phys. 97(12), 124701 (2005).
[Crossref]

Siewerdsen, J.

S. Bartolac, S. Graham, J. Siewerdsen, and D. Jaffray, “Fluence field optimization for noise and dose objectives in CT,” Med. Phys. 38(S1), S2–S17 (2011).
[Crossref] [PubMed]

Siewerdsen, J. H.

N. Mail, D. J. Moseley, J. H. Siewerdsen, and D. A. Jaffray, “The influence of bowtie filtration on cone-beam CT image quality,” Med. Phys. 36(1), 22–32 (2009).
[Crossref] [PubMed]

Soni, N.

B. M. Gramer, D. Muenzel, V. Leber, A. K. von Thaden, H. Feussner, A. Schneider, M. Vembar, N. Soni, E. J. Rummeny, and A. M. Huber, “Impact of iterative reconstruction on CNR and SNR in dynamic myocardial perfusion imaging in an animal model,” Eur. Radiol. 22(12), 2654–2661 (2012).
[Crossref] [PubMed]

Taguchi, K.

K. Taguchi and J. S. Iwanczyk, “Vision 20/20: Single photon counting X-ray detectors in medical imaging,” Med. Phys. 40(10), 100901 (2013).
[Crossref] [PubMed]

K. Taguchi, E. C. Frey, X. Wang, J. S. Iwanczyk, and W. C. Barber, “An analytical model of the effects of pulse pileup on the energy spectrum recorded by energy resolved photon counting x-ray detectors,” Med. Phys. 37(8), 3957–3969 (2010).
[Crossref] [PubMed]

Vannier, M.

E. Y. Sidky, L. Yu, X. Pan, Y. Zou, and M. Vannier, “A robust method of X-ray source spectrum estimation from transmission measurements: Demonstrated on computer simulated, scatter-free transmission data,” J. Appl. Phys. 97(12), 124701 (2005).
[Crossref]

Vembar, M.

B. M. Gramer, D. Muenzel, V. Leber, A. K. von Thaden, H. Feussner, A. Schneider, M. Vembar, N. Soni, E. J. Rummeny, and A. M. Huber, “Impact of iterative reconstruction on CNR and SNR in dynamic myocardial perfusion imaging in an animal model,” Eur. Radiol. 22(12), 2654–2661 (2012).
[Crossref] [PubMed]

Verhaegen, F.

G. J. Bootsma, F. Verhaegen, and D. A. Jaffray, “The effects of compensator and imaging geometry on the distribution of x-ray scatter in CBCT,” Med. Phys. 38(2), 897–914 (2011).
[Crossref] [PubMed]

Vollmar, S. V.

W. A. Kalender, M. Beister, J. M. Boone, D. Kolditz, S. V. Vollmar, and M. C. C. Weigel, “High-resolution spiral CT of the breast at very low dose: concept and feasibility considerations,” Eur. Radiol. 22(1), 1–8 (2012).
[Crossref] [PubMed]

von Thaden, A. K.

B. M. Gramer, D. Muenzel, V. Leber, A. K. von Thaden, H. Feussner, A. Schneider, M. Vembar, N. Soni, E. J. Rummeny, and A. M. Huber, “Impact of iterative reconstruction on CNR and SNR in dynamic myocardial perfusion imaging in an animal model,” Eur. Radiol. 22(12), 2654–2661 (2012).
[Crossref] [PubMed]

Wang, G.

F. Liu, Q. Yang, W. Cong, and G. Wang, “Dynamic bowtie filter for cone-beam/multi-slice CT,” PLoS One 9(7), e103054 (2014).
[Crossref] [PubMed]

F. Liu, G. Wang, W. Cong, S. S. Hsieh, and N. J. Pelc, “Dynamic bowtie for fan-beam CT,” J. XRay Sci. Technol. 21(4), 579–590 (2013).
[PubMed]

Wang, X.

K. Taguchi, E. C. Frey, X. Wang, J. S. Iwanczyk, and W. C. Barber, “An analytical model of the effects of pulse pileup on the energy spectrum recorded by energy resolved photon counting x-ray detectors,” Med. Phys. 37(8), 3957–3969 (2010).
[Crossref] [PubMed]

Wedepohl, K. H.

K. H. Wedepohl, “The composition of the continental crust,” Geochim. Cosmochim. Acta 59(7), 1217–1232 (1995).
[Crossref]

Weigel, M. C. C.

W. A. Kalender, M. Beister, J. M. Boone, D. Kolditz, S. V. Vollmar, and M. C. C. Weigel, “High-resolution spiral CT of the breast at very low dose: concept and feasibility considerations,” Eur. Radiol. 22(1), 1–8 (2012).
[Crossref] [PubMed]

Wennberg, O. P.

O. P. Wennberg, L. Rennan, and R. Basquet, “Computed tomography scan imaging of natural open fractures in a porous rock; geometry and fluid flow,” Geophys. Prospect. 57(2), 239–249 (2009).
[Crossref]

Yang, F.

Y. Liu, A. M. Kiss, D. H. Larsson, F. Yang, and P. Pianetta, “To get the most out of high resolution X-ray tomography: A review of the post-reconstruction analysis,” Spectrochim. Acta B At. Spectrosc. 117, 29–41 (2016).
[Crossref]

Yang, K.

S. E. McKenney, A. Nosratieh, D. Gelskey, K. Yang, S. Y. Huang, L. Chen, and J. M. Boone, “Experimental validation of a method characterizing bow tie filters in CT scanners using a real-time dose probe,” Med. Phys. 38(3), 1406–1415 (2011).
[Crossref] [PubMed]

J. M. Boone, A. L. C. Kwan, K. Yang, G. W. Burkett, K. K. Lindfors, and T. R. Nelson, “Computed tomography for imaging the breast,” J. Mammary Gland Biol. Neoplasia 11(2), 103–111 (2006).
[Crossref] [PubMed]

Yang, Q.

F. Liu, Q. Yang, W. Cong, and G. Wang, “Dynamic bowtie filter for cone-beam/multi-slice CT,” PLoS One 9(7), e103054 (2014).
[Crossref] [PubMed]

Yu, L.

E. Y. Sidky, L. Yu, X. Pan, Y. Zou, and M. Vannier, “A robust method of X-ray source spectrum estimation from transmission measurements: Demonstrated on computer simulated, scatter-free transmission data,” J. Appl. Phys. 97(12), 124701 (2005).
[Crossref]

Zou, Y.

E. Y. Sidky, L. Yu, X. Pan, Y. Zou, and M. Vannier, “A robust method of X-ray source spectrum estimation from transmission measurements: Demonstrated on computer simulated, scatter-free transmission data,” J. Appl. Phys. 97(12), 124701 (2005).
[Crossref]

Zygmanski, P.

M. Blessing, M. S. Bhagwat, Y. Lyatskaya, J. R. Bellon, J. Hesser, and P. Zygmanski, “Kilovoltage beam model for flat panel imaging system with bow-tie filter for scatter prediction and correction,” Phys. Med. 28(2), 134–143 (2012).
[Crossref] [PubMed]

Comput. Geosci. (1)

R. A. Ketcham and W. D. Carlson, “Acquisition, optimization and interpretation of X-ray computed tomographic imagery: applications to the geosciences,” Comput. Geosci. 27(4), 381–400 (2001).
[Crossref]

Earth Sci. Rev. (1)

V. Cnudde and M. N. Boone, “High-resolution X-ray computed tomography in geosciences: A review of the current technology and applications,” Earth Sci. Rev. 123, 1–17 (2013).
[Crossref]

Eur. Radiol. (2)

W. A. Kalender, M. Beister, J. M. Boone, D. Kolditz, S. V. Vollmar, and M. C. C. Weigel, “High-resolution spiral CT of the breast at very low dose: concept and feasibility considerations,” Eur. Radiol. 22(1), 1–8 (2012).
[Crossref] [PubMed]

B. M. Gramer, D. Muenzel, V. Leber, A. K. von Thaden, H. Feussner, A. Schneider, M. Vembar, N. Soni, E. J. Rummeny, and A. M. Huber, “Impact of iterative reconstruction on CNR and SNR in dynamic myocardial perfusion imaging in an animal model,” Eur. Radiol. 22(12), 2654–2661 (2012).
[Crossref] [PubMed]

Geochim. Cosmochim. Acta (1)

K. H. Wedepohl, “The composition of the continental crust,” Geochim. Cosmochim. Acta 59(7), 1217–1232 (1995).
[Crossref]

Geophys. Prospect. (1)

O. P. Wennberg, L. Rennan, and R. Basquet, “Computed tomography scan imaging of natural open fractures in a porous rock; geometry and fluid flow,” Geophys. Prospect. 57(2), 239–249 (2009).
[Crossref]

J. Appl. Phys. (1)

E. Y. Sidky, L. Yu, X. Pan, Y. Zou, and M. Vannier, “A robust method of X-ray source spectrum estimation from transmission measurements: Demonstrated on computer simulated, scatter-free transmission data,” J. Appl. Phys. 97(12), 124701 (2005).
[Crossref]

J. Mammary Gland Biol. Neoplasia (1)

J. M. Boone, A. L. C. Kwan, K. Yang, G. W. Burkett, K. K. Lindfors, and T. R. Nelson, “Computed tomography for imaging the breast,” J. Mammary Gland Biol. Neoplasia 11(2), 103–111 (2006).
[Crossref] [PubMed]

J. XRay Sci. Technol. (1)

F. Liu, G. Wang, W. Cong, S. S. Hsieh, and N. J. Pelc, “Dynamic bowtie for fan-beam CT,” J. XRay Sci. Technol. 21(4), 579–590 (2013).
[PubMed]

Med. Phys. (10)

K. Taguchi and J. S. Iwanczyk, “Vision 20/20: Single photon counting X-ray detectors in medical imaging,” Med. Phys. 40(10), 100901 (2013).
[Crossref] [PubMed]

K. Taguchi, E. C. Frey, X. Wang, J. S. Iwanczyk, and W. C. Barber, “An analytical model of the effects of pulse pileup on the energy spectrum recorded by energy resolved photon counting x-ray detectors,” Med. Phys. 37(8), 3957–3969 (2010).
[Crossref] [PubMed]

N. Mail, D. J. Moseley, J. H. Siewerdsen, and D. A. Jaffray, “The influence of bowtie filtration on cone-beam CT image quality,” Med. Phys. 36(1), 22–32 (2009).
[Crossref] [PubMed]

G. J. Bootsma, F. Verhaegen, and D. A. Jaffray, “The effects of compensator and imaging geometry on the distribution of x-ray scatter in CBCT,” Med. Phys. 38(2), 897–914 (2011).
[Crossref] [PubMed]

K. Kontson and R. J. Jennings, “Bowtie filters for dedicated breast CT: theory and computational implementation,” Med. Phys. 42(3), 1453–1462 (2015).
[Crossref] [PubMed]

K. Kontson and R. J. Jennings, “Bowtie filters for dedicated breast CT: Analysis of bowtie filter material selection,” Med. Phys. 42(9), 5270–5277 (2015).
[Crossref] [PubMed]

S. S. Hsieh and N. J. Pelc, “The feasibility of a piecewise-linear dynamic bowtie filter,” Med. Phys. 40(3), 031910 (2013).
[Crossref] [PubMed]

S. Bartolac, S. Graham, J. Siewerdsen, and D. Jaffray, “Fluence field optimization for noise and dose objectives in CT,” Med. Phys. 38(S1), S2–S17 (2011).
[Crossref] [PubMed]

S. E. McKenney, A. Nosratieh, D. Gelskey, K. Yang, S. Y. Huang, L. Chen, and J. M. Boone, “Experimental validation of a method characterizing bow tie filters in CT scanners using a real-time dose probe,” Med. Phys. 38(3), 1406–1415 (2011).
[Crossref] [PubMed]

J. M. Boone, N. Shah, and T. R. Nelson, “A comprehensive analysis of DgN(CT) coefficients for pendant-geometry cone-beam breast computed tomography,” Med. Phys. 31(2), 226–235 (2004).
[Crossref] [PubMed]

Phys. Med. (1)

M. Blessing, M. S. Bhagwat, Y. Lyatskaya, J. R. Bellon, J. Hesser, and P. Zygmanski, “Kilovoltage beam model for flat panel imaging system with bow-tie filter for scatter prediction and correction,” Phys. Med. 28(2), 134–143 (2012).
[Crossref] [PubMed]

Phys. Med. Biol. (1)

F. Lück, D. Kolditz, M. Hupfer, and W. A. Kalender, “Effect of shaped filter design on dose and image quality in breast CT,” Phys. Med. Biol. 58(12), 4205–4223 (2013).
[Crossref] [PubMed]

PLoS One (1)

F. Liu, Q. Yang, W. Cong, and G. Wang, “Dynamic bowtie filter for cone-beam/multi-slice CT,” PLoS One 9(7), e103054 (2014).
[Crossref] [PubMed]

Proc. SPIE (1)

R. B. Benítez and R. Ning, “Development of a beam hardening filter and characterization of the spatial resolution for a cone beam CT imaging system,” Proc. SPIE 91, 391–396 (2011).

Radiographics (1)

J. F. Barrett and N. Keat, “Artifacts in CT: Recognition and Avoidance,” Radiographics 24(6), 1679–1691 (2004).
[Crossref] [PubMed]

Radiological Phys. Technol. (1)

I. Mori, Y. Machida, M. Osanai, and K. Iinuma, “Photon starvation artifacts of X-ray CT: their true cause and a solution,” Radiological Phys. Technol. 6(1), 130–141 (2013).
[Crossref] [PubMed]

Spectrochim. Acta B At. Spectrosc. (1)

Y. Liu, A. M. Kiss, D. H. Larsson, F. Yang, and P. Pianetta, “To get the most out of high resolution X-ray tomography: A review of the post-reconstruction analysis,” Spectrochim. Acta B At. Spectrosc. 117, 29–41 (2016).
[Crossref]

Other (1)

J. H. Hubbell and S. M. Seltzer, Radiation and biomolecular physics division, PML NIST [Online]. Available: http://physics.nist.gov/xaamdi .

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Figures (20)

Fig. 1
Fig. 1 A schematic diagram of a whole-core CT scanner.
Fig. 2
Fig. 2 The projection curves with respect to sample thickness.
Fig. 3
Fig. 3 Photographs of the whole-core CT: (a) Appearance; (b) Internal structure.
Fig. 4
Fig. 4 The simulated X-ray spectrum.
Fig. 5
Fig. 5 The mass attenuation coefficient of three materials
Fig. 6
Fig. 6 The curve of r ( t ) with different initial conditions.
Fig. 7
Fig. 7 Two orthogonal sections of the attenuator with r 0 = 1 mm: (a) the cross section; (b) the longitudinal section.
Fig. 8
Fig. 8 Two orthogonal sections of the attenuator with r 0 = 7 mm: (a) the cross section; (b) the longitudinal section.
Fig. 9
Fig. 9 The distribution of detected photon counts: (a) the result of the attenuator with r 0 = 1 mm; (b) the result of the attenuator with r 0 = 7 mm.
Fig. 10
Fig. 10 The images reconstructed from data (a) without attenuator, (b) with r 0 = 1 mm attenuator, and (c) with r 0 = 7 mm attenuator.
Fig. 11
Fig. 11 Profiles along the lines in Fig. 10.
Fig. 12
Fig. 12 The (a) digital model and (b) the object of the manufactured attenuator.
Fig. 13
Fig. 13 The CT images of the artificial core sample. (a) and (c) are two orthogonal sections reconstructed from data scanned without attenuator (C/W = 0.0335/0.017); (b) and (d) are two orthogonal sections reconstructed from data scanned with the manufactured attenuator (C/W = 0.026/0.008).
Fig. 14
Fig. 14 Profiles along the lines in Figs. 13(a) and (b).
Fig. 15
Fig. 15 The CT image of nature core sample: (a) and (c) are two orthogonal section of the reconstruction image without attenuator (C/W = 0.0335/0.017); (b) and (d) are two orthogonal section of the reconstruction image with the manufactured attenuator (C/W = 0.026/0.008).
Fig. 16
Fig. 16 Profiles along the lines in Figs. 15(a) and (b).
Fig. 17
Fig. 17 The images of aluminum-air sample reconstructed from data (a) without attenuator, (b) with r 0 = 1 mm attenuator, and (c) with r 0 = 7 mm attenuator.
Fig. 18
Fig. 18 Profiles along the lines in Fig. 17.
Fig. 19
Fig. 19 The images of aluminum-copper sample reconstructed from data (a) without attenuator, (b) with r 0 = 1 mm attenuator, and (c) with r 0 = 7 mm attenuator.
Fig. 20
Fig. 20 Profiles along the lines in Fig. 19.

Tables (2)

Tables Icon

Table 1 The dynamic range of detected photon counts.

Tables Icon

Table 2 The SNR of the ROIs in Fig. 13 and Fig. 15

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

I ( L ) = E min E max S ( E ) γ ( E , L ) e ν ( E ) r ( L ) e x L μ ( x , E ) d L d E + σ ( L ) ,
p ( t , r ) = ln E min E max S ¯ ( E , L ) e ν ( E ) r e μ ¯ ( E ) t d E E min E max S ¯ ( E , L ) e ν ( E ) r d E ,
S ¯ ( E , L ) e ν ( E ) r E min E max S ¯ ( E , L ) e ν ( E ) r d E .
p ( t , r ( t ) ) = μ ¯ ( E 0 ) t ,
F ( t , r ) = def p ( t , r ) μ ¯ ( E 0 ) t = 0.
F ( t 0 , r 0 ) = p ( t 0 , r 0 ) μ ¯ ( E 0 ) t 0 = 0.
F ( 0 , 1 ) ( t , r ) = p ( 0 , 1 ) ( t , r ) < 0 , for all 0 < t < t 0 , r 0 < r
d r d t = F ( 1 , 0 ) ( t , r ) F ( 0 , 1 ) ( t , r ) = p ( 1 , 0 ) ( t , r ) μ ¯ ( E 0 ) p ( 0 , 1 ) ( t , r ) ,
p ( 1 , 0 ) ( t , r ) = E min E max S ¯ ( E ) e ν ( E ) r μ ¯ ( E ) t μ ¯ ( E ) d E E min E max S ¯ ( E ) e ν ( E ) r μ ¯ ( E ) t d E ,
p ( 0 , 1 ) ( t , r ) = E min E max S ¯ ( E ) e ν ( E ) r ν ( E ) d E E min E max S ¯ ( E ) e ν ( E ) r d E + E min E max S ¯ ( E ) e ν ( E ) r μ ¯ ( E ) t ν ( E ) d E E min E max S ¯ ( E ) e ν ( E ) r μ ¯ ( E ) t d E .
{ d r d t = p ( 1 , 0 ) ( t , r ) μ ¯ ( E 0 ) p ( 0 , 1 ) ( t , r ) , 0 < t < t 0 , r 0 < r r ( t 0 ) = r 0 .
t ( ξ , η ) = ( ( R + D ) 2 + η 2 ) ( 4 R 2 ξ 2 + t 0 2 ( ( R + D ) 2 + η 2 ) ) ( R + D ) ( R + D ) 2 + ξ 2 + η 2 ,
( x , y , z ) = ( D , ξ , η ) + ( R , 0 , 0 ) ( D , ξ , η ) | ( R , 0 , 0 ) ( D , ξ , η ) | r ( t ( ξ , η ) ) .
{ u = ( 1 r ( t ( ξ , η ) ) ( R + D ) 2 + ξ 2 + η 2 ) ξ v = ( 1 r ( t ( ξ , η ) ) ( R + D ) 2 + ξ 2 + η 2 ) η w = ( R + D ) r ( t ( ξ , η ) ) ( R + D ) 2 + ξ 2 + η 2 ,
p real = ln E min E max S ¯ ( E , L ) e ν ( E ) r ( t 1 + t 2 ) e μ ¯ ( E ) t 1 d E E min E max S ¯ ( E , L ) e ν ( E ) r ( t 1 + t 2 ) d E ,
p ideal = ln E min E max S ¯ ( E , L ) e ν ( E ) r ( t 1 ) e μ ¯ ( E ) t 1 d E E min E max S ¯ ( E , L ) e ν ( E ) r ( t 1 ) d E .
a b T ( τ ) μ ( τ ) d τ a b T ( σ ) ν ( σ ) d σ < a b T ( τ ) d τ a b T ( σ ) ν ( σ ) μ ( σ ) d σ .
M 1 = a b T ( τ ) μ ( τ ) d τ a b T ( σ ) ν ( σ ) d σ = a b d τ a b T ( τ ) T ( σ ) μ ( τ ) ν ( σ ) d σ ,
M 2 = a b T ( τ ) d τ a b T ( σ ) μ ( σ ) ν ( σ ) d σ = a b d τ a b T ( τ ) T ( σ ) μ ( σ ) ν ( σ ) d σ .
M 1 M 2 = a b d τ a τ T ( τ ) T ( σ ) ν ( σ ) ( μ ( τ ) μ ( σ ) ) d σ + a b d τ τ b T ( τ ) T ( σ ) ν ( σ ) ( μ ( τ ) μ ( σ ) ) d σ .
a b d τ τ b T ( τ ) T ( σ ) ν ( σ ) ( μ ( τ ) μ ( σ ) ) d σ = a b d τ a τ T ( τ ) T ( σ ) ν ( τ ) ( μ ( τ ) μ ( σ ) ) d σ .
M 1 M 2 = a b d τ a τ T ( τ ) T ( σ ) ( ν ( τ ) ν ( σ ) ) ( μ ( τ ) μ ( σ ) ) d σ .
F ( 0 , 1 ) ( t , r ) = E min E max S ¯ ( E ) e ν ( E ) r ν ( E ) d E E min E max S ¯ ( E ) e ν ( E ) r d E + E min E max S ¯ ( E ) e ν ( E ) r μ ¯ ( E ) t ν ( E ) d E E min E max S ¯ ( E ) e ν ( E ) r μ ¯ ( E ) t d E .
f ( t ) = E min E max T ( E ) ν ( E ) d E E min E max T ( E ) μ ¯ ( E ) d E ( E min E max T ( E ) d E ) 2 E min E max T ( E ) d E E min E max T ( E ) μ ¯ ( E ) ν ( E ) d E ( E min E max T ( E ) d E ) 2 .
E min E max T ( E ) ν ( E ) d E E min E max T ( E ) μ ¯ ( E ) d E < E min E max T ( E ) d E E min E max T ( E ) μ ¯ ( E ) ν ( E ) d E .

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