Abstract

High-dynamic-range (HDR) X-ray CT imaging is effective in detecting some complex structures. For previous low-dynamic-range (LDR) imaging detectors, multi-energy LDR image sequence fusion can extend the dynamic range, but the efficiency is decreased. However, with the application of HDR imaging devices, traditional fixed-energy X-ray imaging can cause incongruity within energy, dynamic range, and the equivalent thickness of the workpiece at different projection angles. Then, the projection has a blurred edge, and the CT image quality is poor because of scattering and the inadequate dose. In this paper, a new HDR X-ray CT imaging method with energy self-adaptation between scanning angles for HDR imaging devices is studied. Low-energy prescanning is used to determine the initial scanning energy and obtain the edge contour information with an attenuating effect on scattering. By establishing a mathematical model between the gray level of the projection and the transmission voltage, the transmission energy at each angle is adjusted adaptively. Then, the prescanning and energy self-adaption scanning projections are fused to obtain the complete projection of the complex workpiece. Finally, a conventional reconstruction algorithm is used to reconstruct the HDR CT image. The experimental results show that the proposed imaging method can achieve HDR CT imaging of complex structures with high reconstruction quality, clear edge details, and high completeness.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  20. J. T. Wei, P. Chen, and J. X. Pan, “Multi-voltage Digital Radiography images fusion based on gray consistency,” Nondestructive Evaluation testing: New Technology & Application IEEE. 209–211 (2013).
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    [Crossref]

2018 (2)

A. Alpers and P. Gritzmann, “On double-resolution imaging in discrete tomography,” SIAM J. Discrete Math. 32(2), 1369–1399 (2018).
[Crossref]

X. G. Yang, V. D. Andrade, W. Scullin, E. L. Dyer, N. Kasthuri, F. D. Carlo, and D. Gürsoy, “Low-dose x-ray tomography through a deep convolutional neural network,” Sci. Rep. 8(1), 1–13 (2018).
[Crossref]

2017 (3)

Y. M. Guo, L. Zeng, C. X. Wang, and L. L. Zhang, “Image reconstruction model for the exterior problem of computed tomography based on weighted directional total variation,” Appl Math Model. 52, 358–377 (2017).
[Crossref]

Y. H. Li, Y. Han, and P. Chen, “X-ray energy self-adaption high dynamic range (HDR) imaging based on linear constraints with variable energy,” IEEE Photonics J. 10(1), 1 (2017).
[Crossref]

M. A. Haidekker, D. K. Morrison, A. Sharma, and E. Burke, “Enhanced dynamic range x-ray imaging,” Comput. Biol. Med. 82(C), 40–48 (2017).
[Crossref]

2016 (2)

J. T. Wei, Y. Han, and P. Chen, “Improved contrast of materials based on multi-voltage images decomposition in X-ray CT,” Meas. Sci. Technol. 27(2), 025402 (2016).
[Crossref]

P. Chen and Y. Han, “Varying-energy CT imaging method based on EM-TV,” Meas. Sci. Technol. 27(11), 114004 (2016).
[Crossref]

2015 (1)

P. Chen, Y. Han, and J. X. Pan, “High-Dynamic-Range CT Reconstruction Based on Varying Tube-Voltage Imaging,” PLoS One 10(11), e0141789 (2015).
[Crossref]

2014 (4)

A. Mirone, E. Brun, and P. Coan, “A dictionary learning approach with overlap for the low dose computed tomography reconstruction and its vectorial application to differential phase tomography,” PLoS One 9(12), e114325 (2014).
[Crossref]

D. L. Chiffre, S. Carmignato, J. P. Kruth, R. Schmitt, and A. Weckenmann, “Industrial applications of computed tomography,” CIRP Ann. 63(2), 655–677 (2014).
[Crossref]

B. Liu, Y. Han, J. X. Pan, and P. Chen, “Multi-energy image sequence fusion based on variable energy X-ray imaging,” J. X-Ray Sci. Technol. 22(2), 241–251 (2014).
[Crossref]

M. J. Schrapp and G. T. Herman, “Data fusion in X-ray computed tomography using a superiorization approach,” Rev. Sci. Instrum. 85(5), 053701 (2014).
[Crossref]

2010 (1)

T. Y. Niu and L. Zhu, “Overview of X-ray Scatter in Cone-beam Computed Tomography and Its Correction Methods,” Curr. Med. Imaging Rev. 6(2), 82–89 (2010).
[Crossref]

2008 (1)

M. C. Chen, H. M. Li, Z. Y. Chen, and J. Shen, “An examination of mass thickness measurements with X-ray sources,” Appl. Radiat. Isot. 66(10), 1387–1391 (2008).
[Crossref]

2007 (1)

J. Rinkel, L. Gerfault, F. Estève, and J. M. Dinten, “A new method for x-ray scatter correction: first assessment on a cone-beam CT experimental setup,” Phys. Med. Biol. 52(15), 4633–4652 (2007).
[Crossref]

1985 (1)

H. Kanamori, N. Nakamori, K. Inoue, and E. Takenaka, “Effects of scattered X-rays on CT images,” Phys. Med. Biol. 30(3), 239–249 (1985).
[Crossref]

1984 (1)

L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. 1(6), 612–619 (1984).
[Crossref]

1982 (1)

P. C. Johns and M. Yaffe, “Scattered radiation in fan beam imaging systems,” Med. Phys. 9(2), 231–239 (1982).
[Crossref]

1976 (1)

A. Macovski, R. E. Alvarez, J. L. H. Chan, J. P. Stonestrom, and L. M. Zatz, “Energy dependent reconstruction in X-ray computerized tomography,” Comput. Biol. Med. 6(4), 325–336 (1976).
[Crossref]

Alpers, A.

A. Alpers and P. Gritzmann, “On double-resolution imaging in discrete tomography,” SIAM J. Discrete Math. 32(2), 1369–1399 (2018).
[Crossref]

Alvarez, R. E.

A. Macovski, R. E. Alvarez, J. L. H. Chan, J. P. Stonestrom, and L. M. Zatz, “Energy dependent reconstruction in X-ray computerized tomography,” Comput. Biol. Med. 6(4), 325–336 (1976).
[Crossref]

Andrade, V. D.

X. G. Yang, V. D. Andrade, W. Scullin, E. L. Dyer, N. Kasthuri, F. D. Carlo, and D. Gürsoy, “Low-dose x-ray tomography through a deep convolutional neural network,” Sci. Rep. 8(1), 1–13 (2018).
[Crossref]

Brun, E.

A. Mirone, E. Brun, and P. Coan, “A dictionary learning approach with overlap for the low dose computed tomography reconstruction and its vectorial application to differential phase tomography,” PLoS One 9(12), e114325 (2014).
[Crossref]

Burke, E.

M. A. Haidekker, D. K. Morrison, A. Sharma, and E. Burke, “Enhanced dynamic range x-ray imaging,” Comput. Biol. Med. 82(C), 40–48 (2017).
[Crossref]

Carlo, F. D.

X. G. Yang, V. D. Andrade, W. Scullin, E. L. Dyer, N. Kasthuri, F. D. Carlo, and D. Gürsoy, “Low-dose x-ray tomography through a deep convolutional neural network,” Sci. Rep. 8(1), 1–13 (2018).
[Crossref]

Carmignato, S.

D. L. Chiffre, S. Carmignato, J. P. Kruth, R. Schmitt, and A. Weckenmann, “Industrial applications of computed tomography,” CIRP Ann. 63(2), 655–677 (2014).
[Crossref]

Chan, J. L. H.

A. Macovski, R. E. Alvarez, J. L. H. Chan, J. P. Stonestrom, and L. M. Zatz, “Energy dependent reconstruction in X-ray computerized tomography,” Comput. Biol. Med. 6(4), 325–336 (1976).
[Crossref]

Chen, M. C.

M. C. Chen, H. M. Li, Z. Y. Chen, and J. Shen, “An examination of mass thickness measurements with X-ray sources,” Appl. Radiat. Isot. 66(10), 1387–1391 (2008).
[Crossref]

Chen, P.

Y. H. Li, Y. Han, and P. Chen, “X-ray energy self-adaption high dynamic range (HDR) imaging based on linear constraints with variable energy,” IEEE Photonics J. 10(1), 1 (2017).
[Crossref]

J. T. Wei, Y. Han, and P. Chen, “Improved contrast of materials based on multi-voltage images decomposition in X-ray CT,” Meas. Sci. Technol. 27(2), 025402 (2016).
[Crossref]

P. Chen and Y. Han, “Varying-energy CT imaging method based on EM-TV,” Meas. Sci. Technol. 27(11), 114004 (2016).
[Crossref]

P. Chen, Y. Han, and J. X. Pan, “High-Dynamic-Range CT Reconstruction Based on Varying Tube-Voltage Imaging,” PLoS One 10(11), e0141789 (2015).
[Crossref]

B. Liu, Y. Han, J. X. Pan, and P. Chen, “Multi-energy image sequence fusion based on variable energy X-ray imaging,” J. X-Ray Sci. Technol. 22(2), 241–251 (2014).
[Crossref]

J. T. Wei, P. Chen, and J. X. Pan, “Multi-voltage Digital Radiography images fusion based on gray consistency,” Nondestructive Evaluation testing: New Technology & Application IEEE. 209–211 (2013).

Chen, Z. Y.

M. C. Chen, H. M. Li, Z. Y. Chen, and J. Shen, “An examination of mass thickness measurements with X-ray sources,” Appl. Radiat. Isot. 66(10), 1387–1391 (2008).
[Crossref]

Chiffre, D. L.

D. L. Chiffre, S. Carmignato, J. P. Kruth, R. Schmitt, and A. Weckenmann, “Industrial applications of computed tomography,” CIRP Ann. 63(2), 655–677 (2014).
[Crossref]

Coan, P.

A. Mirone, E. Brun, and P. Coan, “A dictionary learning approach with overlap for the low dose computed tomography reconstruction and its vectorial application to differential phase tomography,” PLoS One 9(12), e114325 (2014).
[Crossref]

Davis, L. C.

L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. 1(6), 612–619 (1984).
[Crossref]

Dinten, J. M.

J. Rinkel, L. Gerfault, F. Estève, and J. M. Dinten, “A new method for x-ray scatter correction: first assessment on a cone-beam CT experimental setup,” Phys. Med. Biol. 52(15), 4633–4652 (2007).
[Crossref]

Dyer, E. L.

X. G. Yang, V. D. Andrade, W. Scullin, E. L. Dyer, N. Kasthuri, F. D. Carlo, and D. Gürsoy, “Low-dose x-ray tomography through a deep convolutional neural network,” Sci. Rep. 8(1), 1–13 (2018).
[Crossref]

Estève, F.

J. Rinkel, L. Gerfault, F. Estève, and J. M. Dinten, “A new method for x-ray scatter correction: first assessment on a cone-beam CT experimental setup,” Phys. Med. Biol. 52(15), 4633–4652 (2007).
[Crossref]

Feldkamp, L. A.

L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. 1(6), 612–619 (1984).
[Crossref]

Gerfault, L.

J. Rinkel, L. Gerfault, F. Estève, and J. M. Dinten, “A new method for x-ray scatter correction: first assessment on a cone-beam CT experimental setup,” Phys. Med. Biol. 52(15), 4633–4652 (2007).
[Crossref]

Gritzmann, P.

A. Alpers and P. Gritzmann, “On double-resolution imaging in discrete tomography,” SIAM J. Discrete Math. 32(2), 1369–1399 (2018).
[Crossref]

Guo, Y. M.

Y. M. Guo, L. Zeng, C. X. Wang, and L. L. Zhang, “Image reconstruction model for the exterior problem of computed tomography based on weighted directional total variation,” Appl Math Model. 52, 358–377 (2017).
[Crossref]

Gürsoy, D.

X. G. Yang, V. D. Andrade, W. Scullin, E. L. Dyer, N. Kasthuri, F. D. Carlo, and D. Gürsoy, “Low-dose x-ray tomography through a deep convolutional neural network,” Sci. Rep. 8(1), 1–13 (2018).
[Crossref]

Haidekker, M. A.

M. A. Haidekker, D. K. Morrison, A. Sharma, and E. Burke, “Enhanced dynamic range x-ray imaging,” Comput. Biol. Med. 82(C), 40–48 (2017).
[Crossref]

Han, Y.

Y. H. Li, Y. Han, and P. Chen, “X-ray energy self-adaption high dynamic range (HDR) imaging based on linear constraints with variable energy,” IEEE Photonics J. 10(1), 1 (2017).
[Crossref]

J. T. Wei, Y. Han, and P. Chen, “Improved contrast of materials based on multi-voltage images decomposition in X-ray CT,” Meas. Sci. Technol. 27(2), 025402 (2016).
[Crossref]

P. Chen and Y. Han, “Varying-energy CT imaging method based on EM-TV,” Meas. Sci. Technol. 27(11), 114004 (2016).
[Crossref]

P. Chen, Y. Han, and J. X. Pan, “High-Dynamic-Range CT Reconstruction Based on Varying Tube-Voltage Imaging,” PLoS One 10(11), e0141789 (2015).
[Crossref]

B. Liu, Y. Han, J. X. Pan, and P. Chen, “Multi-energy image sequence fusion based on variable energy X-ray imaging,” J. X-Ray Sci. Technol. 22(2), 241–251 (2014).
[Crossref]

Herman, G. T.

M. J. Schrapp and G. T. Herman, “Data fusion in X-ray computed tomography using a superiorization approach,” Rev. Sci. Instrum. 85(5), 053701 (2014).
[Crossref]

Inoue, K.

H. Kanamori, N. Nakamori, K. Inoue, and E. Takenaka, “Effects of scattered X-rays on CT images,” Phys. Med. Biol. 30(3), 239–249 (1985).
[Crossref]

Jiang, H.

H. Jiang, “Computed Tomography: Principles, Design, Artifacts, and Recent Advances,” SPIE-the International Society for Optical Engineering. 2003, 30–38 (2009).

Johns, P. C.

P. C. Johns and M. Yaffe, “Scattered radiation in fan beam imaging systems,” Med. Phys. 9(2), 231–239 (1982).
[Crossref]

Kanamori, H.

H. Kanamori, N. Nakamori, K. Inoue, and E. Takenaka, “Effects of scattered X-rays on CT images,” Phys. Med. Biol. 30(3), 239–249 (1985).
[Crossref]

Kasthuri, N.

X. G. Yang, V. D. Andrade, W. Scullin, E. L. Dyer, N. Kasthuri, F. D. Carlo, and D. Gürsoy, “Low-dose x-ray tomography through a deep convolutional neural network,” Sci. Rep. 8(1), 1–13 (2018).
[Crossref]

Kress, J. W.

L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. 1(6), 612–619 (1984).
[Crossref]

Kruth, J. P.

D. L. Chiffre, S. Carmignato, J. P. Kruth, R. Schmitt, and A. Weckenmann, “Industrial applications of computed tomography,” CIRP Ann. 63(2), 655–677 (2014).
[Crossref]

Li, H. M.

M. C. Chen, H. M. Li, Z. Y. Chen, and J. Shen, “An examination of mass thickness measurements with X-ray sources,” Appl. Radiat. Isot. 66(10), 1387–1391 (2008).
[Crossref]

Li, Y. H.

Y. H. Li, Y. Han, and P. Chen, “X-ray energy self-adaption high dynamic range (HDR) imaging based on linear constraints with variable energy,” IEEE Photonics J. 10(1), 1 (2017).
[Crossref]

Liu, B.

B. Liu, Y. Han, J. X. Pan, and P. Chen, “Multi-energy image sequence fusion based on variable energy X-ray imaging,” J. X-Ray Sci. Technol. 22(2), 241–251 (2014).
[Crossref]

Macovski, A.

A. Macovski, R. E. Alvarez, J. L. H. Chan, J. P. Stonestrom, and L. M. Zatz, “Energy dependent reconstruction in X-ray computerized tomography,” Comput. Biol. Med. 6(4), 325–336 (1976).
[Crossref]

Mirone, A.

A. Mirone, E. Brun, and P. Coan, “A dictionary learning approach with overlap for the low dose computed tomography reconstruction and its vectorial application to differential phase tomography,” PLoS One 9(12), e114325 (2014).
[Crossref]

Morrison, D. K.

M. A. Haidekker, D. K. Morrison, A. Sharma, and E. Burke, “Enhanced dynamic range x-ray imaging,” Comput. Biol. Med. 82(C), 40–48 (2017).
[Crossref]

Nakamori, N.

H. Kanamori, N. Nakamori, K. Inoue, and E. Takenaka, “Effects of scattered X-rays on CT images,” Phys. Med. Biol. 30(3), 239–249 (1985).
[Crossref]

Niu, T. Y.

T. Y. Niu and L. Zhu, “Overview of X-ray Scatter in Cone-beam Computed Tomography and Its Correction Methods,” Curr. Med. Imaging Rev. 6(2), 82–89 (2010).
[Crossref]

Pan, J. X.

P. Chen, Y. Han, and J. X. Pan, “High-Dynamic-Range CT Reconstruction Based on Varying Tube-Voltage Imaging,” PLoS One 10(11), e0141789 (2015).
[Crossref]

B. Liu, Y. Han, J. X. Pan, and P. Chen, “Multi-energy image sequence fusion based on variable energy X-ray imaging,” J. X-Ray Sci. Technol. 22(2), 241–251 (2014).
[Crossref]

J. T. Wei, P. Chen, and J. X. Pan, “Multi-voltage Digital Radiography images fusion based on gray consistency,” Nondestructive Evaluation testing: New Technology & Application IEEE. 209–211 (2013).

Rinkel, J.

J. Rinkel, L. Gerfault, F. Estève, and J. M. Dinten, “A new method for x-ray scatter correction: first assessment on a cone-beam CT experimental setup,” Phys. Med. Biol. 52(15), 4633–4652 (2007).
[Crossref]

Schmitt, R.

D. L. Chiffre, S. Carmignato, J. P. Kruth, R. Schmitt, and A. Weckenmann, “Industrial applications of computed tomography,” CIRP Ann. 63(2), 655–677 (2014).
[Crossref]

Schrapp, M. J.

M. J. Schrapp and G. T. Herman, “Data fusion in X-ray computed tomography using a superiorization approach,” Rev. Sci. Instrum. 85(5), 053701 (2014).
[Crossref]

Scullin, W.

X. G. Yang, V. D. Andrade, W. Scullin, E. L. Dyer, N. Kasthuri, F. D. Carlo, and D. Gürsoy, “Low-dose x-ray tomography through a deep convolutional neural network,” Sci. Rep. 8(1), 1–13 (2018).
[Crossref]

Sharma, A.

M. A. Haidekker, D. K. Morrison, A. Sharma, and E. Burke, “Enhanced dynamic range x-ray imaging,” Comput. Biol. Med. 82(C), 40–48 (2017).
[Crossref]

Shen, J.

M. C. Chen, H. M. Li, Z. Y. Chen, and J. Shen, “An examination of mass thickness measurements with X-ray sources,” Appl. Radiat. Isot. 66(10), 1387–1391 (2008).
[Crossref]

Stonestrom, J. P.

A. Macovski, R. E. Alvarez, J. L. H. Chan, J. P. Stonestrom, and L. M. Zatz, “Energy dependent reconstruction in X-ray computerized tomography,” Comput. Biol. Med. 6(4), 325–336 (1976).
[Crossref]

Takenaka, E.

H. Kanamori, N. Nakamori, K. Inoue, and E. Takenaka, “Effects of scattered X-rays on CT images,” Phys. Med. Biol. 30(3), 239–249 (1985).
[Crossref]

Wang, C. X.

Y. M. Guo, L. Zeng, C. X. Wang, and L. L. Zhang, “Image reconstruction model for the exterior problem of computed tomography based on weighted directional total variation,” Appl Math Model. 52, 358–377 (2017).
[Crossref]

Weckenmann, A.

D. L. Chiffre, S. Carmignato, J. P. Kruth, R. Schmitt, and A. Weckenmann, “Industrial applications of computed tomography,” CIRP Ann. 63(2), 655–677 (2014).
[Crossref]

Wei, J. T.

J. T. Wei, Y. Han, and P. Chen, “Improved contrast of materials based on multi-voltage images decomposition in X-ray CT,” Meas. Sci. Technol. 27(2), 025402 (2016).
[Crossref]

J. T. Wei, P. Chen, and J. X. Pan, “Multi-voltage Digital Radiography images fusion based on gray consistency,” Nondestructive Evaluation testing: New Technology & Application IEEE. 209–211 (2013).

Yaffe, M.

P. C. Johns and M. Yaffe, “Scattered radiation in fan beam imaging systems,” Med. Phys. 9(2), 231–239 (1982).
[Crossref]

Yang, X. G.

X. G. Yang, V. D. Andrade, W. Scullin, E. L. Dyer, N. Kasthuri, F. D. Carlo, and D. Gürsoy, “Low-dose x-ray tomography through a deep convolutional neural network,” Sci. Rep. 8(1), 1–13 (2018).
[Crossref]

Zatz, L. M.

A. Macovski, R. E. Alvarez, J. L. H. Chan, J. P. Stonestrom, and L. M. Zatz, “Energy dependent reconstruction in X-ray computerized tomography,” Comput. Biol. Med. 6(4), 325–336 (1976).
[Crossref]

Zeng, L.

Y. M. Guo, L. Zeng, C. X. Wang, and L. L. Zhang, “Image reconstruction model for the exterior problem of computed tomography based on weighted directional total variation,” Appl Math Model. 52, 358–377 (2017).
[Crossref]

Zhang, L. L.

Y. M. Guo, L. Zeng, C. X. Wang, and L. L. Zhang, “Image reconstruction model for the exterior problem of computed tomography based on weighted directional total variation,” Appl Math Model. 52, 358–377 (2017).
[Crossref]

Zhu, L.

T. Y. Niu and L. Zhu, “Overview of X-ray Scatter in Cone-beam Computed Tomography and Its Correction Methods,” Curr. Med. Imaging Rev. 6(2), 82–89 (2010).
[Crossref]

Appl Math Model. (1)

Y. M. Guo, L. Zeng, C. X. Wang, and L. L. Zhang, “Image reconstruction model for the exterior problem of computed tomography based on weighted directional total variation,” Appl Math Model. 52, 358–377 (2017).
[Crossref]

Appl. Radiat. Isot. (1)

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Figures (17)

Fig. 1.
Fig. 1. The actual titanium metal workpiece: (a) front view and (b) back view.
Fig. 2.
Fig. 2. The schematic diagram of the CT imaging system
Fig. 3.
Fig. 3. Partial projections at different angles with fixed-energy scanning.
Fig. 4.
Fig. 4. Minimum gray level curve at every rotation angle with fixed-energy scanning.
Fig. 5.
Fig. 5. Projection sequences with different voltages.
Fig. 6.
Fig. 6. Gray distribution histograms for different energies.
Fig. 7.
Fig. 7. The process of SAVE scanning.
Fig. 8.
Fig. 8. Gray curve contrast at all the scanning angles with fixed-scanning and SAVE scanning.
Fig. 9.
Fig. 9. The adaptive X-ray tube voltage at all the scanning angles.
Fig. 10.
Fig. 10. Projection and fusion projection at one angle: (a) SAVE scanning projection, (b) prescanning projection, (c) projection of the effective fusion area, (d) projection of the edge fusion area.
Fig. 11.
Fig. 11. CT reconstruction with fixed-energy CT scanning.
Fig. 12.
Fig. 12. CT reconstruction by different processes: (a) prescanning reconstruction, (b) SAVE scanning reconstruction, (c) projection reconstruction of the effective fusion area, and (d) projection reconstruction of the edge fusion area.
Fig. 13.
Fig. 13. Detailed comparisons of three areas labeled in Fig. 12. (a)-(c): prescanning, (d)-(f): SAVE scanning, (g)-(i): effective fusion area, (k)-(l): edge fusion area.
Fig. 14.
Fig. 14. Three-dimensional visualization result. Left: the whole structure, Middle: the sharp corner structure, Right: the hole structure.
Fig. 15.
Fig. 15. The pictures of the engine blade.
Fig. 16.
Fig. 16. CT reconstruction by different processes: (a) reconstruction with the fixed energy scanning, (b) reconstruction with SAVE scanning and projection fusion.
Fig. 17.
Fig. 17. Three-dimensional visualization result. Left: the front side, Right: the reverse side.

Equations (11)

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

Hmin=min(minIθ)(θ=0,1,2,,359)
I0=KZiV2
I=I0ES(E)eμ(E)ddE
α=ES(E)eμ(E)ddE
I=αKZiV2
Vn=Vn1/In1/In(n=2,3,4,,360)
I1=I01eμ(E1)d
I2=I02eμ(E2)d
I2=I02(I01)μ(E2)μ(E1)(I1)μ(E2)μ(E1)
H2=aH1b
H=H1+H22