Abstract

Volume data are widely used in many areas, especially in biomedical science and geology. Visualizing volume data is very important to enable intuitive understanding of 3D structures, making them easier to analyze. However, current visualization technologies for volume data cannot satisfy human requirements. In this study, we propose a holographic display method for volume data by volume rendering. Based on this holographic display method, we can generate holograms for transparent objects and multi-layer objects. In this paper, to increase the speed of calculation, we propose an approximate volume rendering based CGH calculation method with elemental holograms.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. A. Maintz and M. A. Viergever, “A survey of medical image registration,” Medical Images Analysis 2, 1–36 (1998).
    [Crossref]
  2. M. Levoy, “Display of surfaces from volume data,” IEEE Computer Graphics and Applications 8, 29–37 (1998).
    [Crossref]
  3. S. Napel, M. P. Marks, G. D. Rubin, M. D. Dake, C. H. McDonnell, S. M. Song, D. R. Enzmann, and J. R B Jeffrey, “CT angiography with spiral CT and maximum intensity projection,” Radiology 185, 607–610 (1992).
    [Crossref] [PubMed]
  4. R. A. Drebin, L. Carpenter, and P. Hanrahan, “Volume rendering,” ACM Siggraph Computer Graphics 22, 65–74 (1998).
    [Crossref]
  5. J. Kruger and R. Westermann, “Acceleration techniques for GPU-based volume rendering,” in Proceedings of the 14th IEEE Visualization 2003 (VIS’03), (IEEE Computer Society, Washington, DC, USA, 2003), VIS ’03, pp. 38–43.
  6. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am 53, 1123–1130 (1962).
    [Crossref]
  7. A. Wolfe and S. Hart, “Digital volumetric holograms for medical imaging,” http://spie.org/newsroom/digital-volumetric-holograms-for-medical-imaging .
  8. C. Honda, S. Takahashi, S. J. Hart, and R. J. Rankin, “The technology in the digital holography system,” Japan Science and Technology Information Aggregator, Electronic 15, 135–145 (1998).
  9. J. Khan, “Static digital holograms for medical images,” http://www.holoxica.com/digital-holograms/ .
  10. Y. Sakamoto, T. Aoyama, and Y. Aoki, “Volume rendering for computer generated holograms,” in Proceedings of 19th International Commission for Optics (ICO-19), (2002), pp. 555–556.
  11. Z. Lu and Y. Sakamoto, “Holographic display methods for volume data: polygon-based and MIP-based methods,” Appl Opt 57, A142–A149 (2018).
    [Crossref] [PubMed]
  12. T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray-tracing method,” Applied Optics 52, 201–209 (2013).
    [Crossref]
  13. W. E. Lorensen and H. E. Cline, “Marching cubes: A high resolution 3d surface construction algorithm,” ACM SIGGRAPH Computer Graphics 21, 163–169 (1987).
    [Crossref]
  14. J Image, “Image processing and analysis in Java,” https://imagej.net/Welcome .
  15. X Osiri, “The world famous DICOM viewer,” http://www.osirix-viewer.com/ .
  16. The Visualization Toolkit, http://www.vtk.org/ .
  17. Audtodesk 3ds Max, http://www.autodesk.com .
  18. T. Yoneyama, C. Yang, Y. Sakamoto, and F. Okuyama, “Eyepiece-type full-color electro-holographic binocular display with see-through vision,” in Proceedings of Digital Holography and Three-Dimensional Imaging, (Optical Society of America, 2013), p. DW2A.11.
    [Crossref]
  19. T. Shimobaba and T. Ito, “A color holographic reconstruction system by time division multiplexing with reference lights of laser,” Optical Review 10, 339–341 (2003).
    [Crossref]
  20. H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Applied optics 54, 10029–10034 (2015).
    [Crossref]
  21. DICOM Image Library, http://www.osirix-viewer.com/resources/dicom-image-library/ .

2018 (1)

Z. Lu and Y. Sakamoto, “Holographic display methods for volume data: polygon-based and MIP-based methods,” Appl Opt 57, A142–A149 (2018).
[Crossref] [PubMed]

2015 (1)

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Applied optics 54, 10029–10034 (2015).
[Crossref]

2013 (1)

T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray-tracing method,” Applied Optics 52, 201–209 (2013).
[Crossref]

2003 (1)

T. Shimobaba and T. Ito, “A color holographic reconstruction system by time division multiplexing with reference lights of laser,” Optical Review 10, 339–341 (2003).
[Crossref]

1998 (4)

C. Honda, S. Takahashi, S. J. Hart, and R. J. Rankin, “The technology in the digital holography system,” Japan Science and Technology Information Aggregator, Electronic 15, 135–145 (1998).

J. A. Maintz and M. A. Viergever, “A survey of medical image registration,” Medical Images Analysis 2, 1–36 (1998).
[Crossref]

M. Levoy, “Display of surfaces from volume data,” IEEE Computer Graphics and Applications 8, 29–37 (1998).
[Crossref]

R. A. Drebin, L. Carpenter, and P. Hanrahan, “Volume rendering,” ACM Siggraph Computer Graphics 22, 65–74 (1998).
[Crossref]

1992 (1)

S. Napel, M. P. Marks, G. D. Rubin, M. D. Dake, C. H. McDonnell, S. M. Song, D. R. Enzmann, and J. R B Jeffrey, “CT angiography with spiral CT and maximum intensity projection,” Radiology 185, 607–610 (1992).
[Crossref] [PubMed]

1987 (1)

W. E. Lorensen and H. E. Cline, “Marching cubes: A high resolution 3d surface construction algorithm,” ACM SIGGRAPH Computer Graphics 21, 163–169 (1987).
[Crossref]

1962 (1)

E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am 53, 1123–1130 (1962).
[Crossref]

Aoki, Y.

Y. Sakamoto, T. Aoyama, and Y. Aoki, “Volume rendering for computer generated holograms,” in Proceedings of 19th International Commission for Optics (ICO-19), (2002), pp. 555–556.

Aoyama, T.

Y. Sakamoto, T. Aoyama, and Y. Aoki, “Volume rendering for computer generated holograms,” in Proceedings of 19th International Commission for Optics (ICO-19), (2002), pp. 555–556.

Araki, H.

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Applied optics 54, 10029–10034 (2015).
[Crossref]

Carpenter, L.

R. A. Drebin, L. Carpenter, and P. Hanrahan, “Volume rendering,” ACM Siggraph Computer Graphics 22, 65–74 (1998).
[Crossref]

Cline, H. E.

W. E. Lorensen and H. E. Cline, “Marching cubes: A high resolution 3d surface construction algorithm,” ACM SIGGRAPH Computer Graphics 21, 163–169 (1987).
[Crossref]

Dake, M. D.

S. Napel, M. P. Marks, G. D. Rubin, M. D. Dake, C. H. McDonnell, S. M. Song, D. R. Enzmann, and J. R B Jeffrey, “CT angiography with spiral CT and maximum intensity projection,” Radiology 185, 607–610 (1992).
[Crossref] [PubMed]

Drebin, R. A.

R. A. Drebin, L. Carpenter, and P. Hanrahan, “Volume rendering,” ACM Siggraph Computer Graphics 22, 65–74 (1998).
[Crossref]

Enzmann, D. R.

S. Napel, M. P. Marks, G. D. Rubin, M. D. Dake, C. H. McDonnell, S. M. Song, D. R. Enzmann, and J. R B Jeffrey, “CT angiography with spiral CT and maximum intensity projection,” Radiology 185, 607–610 (1992).
[Crossref] [PubMed]

Fujiwara, M.

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Applied optics 54, 10029–10034 (2015).
[Crossref]

Hanrahan, P.

R. A. Drebin, L. Carpenter, and P. Hanrahan, “Volume rendering,” ACM Siggraph Computer Graphics 22, 65–74 (1998).
[Crossref]

Hart, S. J.

C. Honda, S. Takahashi, S. J. Hart, and R. J. Rankin, “The technology in the digital holography system,” Japan Science and Technology Information Aggregator, Electronic 15, 135–145 (1998).

Honda, C.

C. Honda, S. Takahashi, S. J. Hart, and R. J. Rankin, “The technology in the digital holography system,” Japan Science and Technology Information Aggregator, Electronic 15, 135–145 (1998).

Ichikawa, T.

T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray-tracing method,” Applied Optics 52, 201–209 (2013).
[Crossref]

Ikawa, S.

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Applied optics 54, 10029–10034 (2015).
[Crossref]

Ito, T.

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Applied optics 54, 10029–10034 (2015).
[Crossref]

T. Shimobaba and T. Ito, “A color holographic reconstruction system by time division multiplexing with reference lights of laser,” Optical Review 10, 339–341 (2003).
[Crossref]

Jeffrey, J. R B

S. Napel, M. P. Marks, G. D. Rubin, M. D. Dake, C. H. McDonnell, S. M. Song, D. R. Enzmann, and J. R B Jeffrey, “CT angiography with spiral CT and maximum intensity projection,” Radiology 185, 607–610 (1992).
[Crossref] [PubMed]

Kakue, T.

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Applied optics 54, 10029–10034 (2015).
[Crossref]

Kruger, J.

J. Kruger and R. Westermann, “Acceleration techniques for GPU-based volume rendering,” in Proceedings of the 14th IEEE Visualization 2003 (VIS’03), (IEEE Computer Society, Washington, DC, USA, 2003), VIS ’03, pp. 38–43.

Leith, E. N.

E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am 53, 1123–1130 (1962).
[Crossref]

Levoy, M.

M. Levoy, “Display of surfaces from volume data,” IEEE Computer Graphics and Applications 8, 29–37 (1998).
[Crossref]

Lorensen, W. E.

W. E. Lorensen and H. E. Cline, “Marching cubes: A high resolution 3d surface construction algorithm,” ACM SIGGRAPH Computer Graphics 21, 163–169 (1987).
[Crossref]

Lu, Z.

Z. Lu and Y. Sakamoto, “Holographic display methods for volume data: polygon-based and MIP-based methods,” Appl Opt 57, A142–A149 (2018).
[Crossref] [PubMed]

Maintz, J. A.

J. A. Maintz and M. A. Viergever, “A survey of medical image registration,” Medical Images Analysis 2, 1–36 (1998).
[Crossref]

Marks, M. P.

S. Napel, M. P. Marks, G. D. Rubin, M. D. Dake, C. H. McDonnell, S. M. Song, D. R. Enzmann, and J. R B Jeffrey, “CT angiography with spiral CT and maximum intensity projection,” Radiology 185, 607–610 (1992).
[Crossref] [PubMed]

McDonnell, C. H.

S. Napel, M. P. Marks, G. D. Rubin, M. D. Dake, C. H. McDonnell, S. M. Song, D. R. Enzmann, and J. R B Jeffrey, “CT angiography with spiral CT and maximum intensity projection,” Radiology 185, 607–610 (1992).
[Crossref] [PubMed]

Nakayama, H.

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Applied optics 54, 10029–10034 (2015).
[Crossref]

Napel, S.

S. Napel, M. P. Marks, G. D. Rubin, M. D. Dake, C. H. McDonnell, S. M. Song, D. R. Enzmann, and J. R B Jeffrey, “CT angiography with spiral CT and maximum intensity projection,” Radiology 185, 607–610 (1992).
[Crossref] [PubMed]

Niwase, H.

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Applied optics 54, 10029–10034 (2015).
[Crossref]

Okuyama, F.

T. Yoneyama, C. Yang, Y. Sakamoto, and F. Okuyama, “Eyepiece-type full-color electro-holographic binocular display with see-through vision,” in Proceedings of Digital Holography and Three-Dimensional Imaging, (Optical Society of America, 2013), p. DW2A.11.
[Crossref]

Rankin, R. J.

C. Honda, S. Takahashi, S. J. Hart, and R. J. Rankin, “The technology in the digital holography system,” Japan Science and Technology Information Aggregator, Electronic 15, 135–145 (1998).

Rubin, G. D.

S. Napel, M. P. Marks, G. D. Rubin, M. D. Dake, C. H. McDonnell, S. M. Song, D. R. Enzmann, and J. R B Jeffrey, “CT angiography with spiral CT and maximum intensity projection,” Radiology 185, 607–610 (1992).
[Crossref] [PubMed]

Sakamoto, Y.

Z. Lu and Y. Sakamoto, “Holographic display methods for volume data: polygon-based and MIP-based methods,” Appl Opt 57, A142–A149 (2018).
[Crossref] [PubMed]

T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray-tracing method,” Applied Optics 52, 201–209 (2013).
[Crossref]

T. Yoneyama, C. Yang, Y. Sakamoto, and F. Okuyama, “Eyepiece-type full-color electro-holographic binocular display with see-through vision,” in Proceedings of Digital Holography and Three-Dimensional Imaging, (Optical Society of America, 2013), p. DW2A.11.
[Crossref]

Y. Sakamoto, T. Aoyama, and Y. Aoki, “Volume rendering for computer generated holograms,” in Proceedings of 19th International Commission for Optics (ICO-19), (2002), pp. 555–556.

Shimobaba, T.

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Applied optics 54, 10029–10034 (2015).
[Crossref]

T. Shimobaba and T. Ito, “A color holographic reconstruction system by time division multiplexing with reference lights of laser,” Optical Review 10, 339–341 (2003).
[Crossref]

Song, S. M.

S. Napel, M. P. Marks, G. D. Rubin, M. D. Dake, C. H. McDonnell, S. M. Song, D. R. Enzmann, and J. R B Jeffrey, “CT angiography with spiral CT and maximum intensity projection,” Radiology 185, 607–610 (1992).
[Crossref] [PubMed]

Takada, N.

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Applied optics 54, 10029–10034 (2015).
[Crossref]

Takahashi, S.

C. Honda, S. Takahashi, S. J. Hart, and R. J. Rankin, “The technology in the digital holography system,” Japan Science and Technology Information Aggregator, Electronic 15, 135–145 (1998).

Upatnieks, J.

E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am 53, 1123–1130 (1962).
[Crossref]

Viergever, M. A.

J. A. Maintz and M. A. Viergever, “A survey of medical image registration,” Medical Images Analysis 2, 1–36 (1998).
[Crossref]

Westermann, R.

J. Kruger and R. Westermann, “Acceleration techniques for GPU-based volume rendering,” in Proceedings of the 14th IEEE Visualization 2003 (VIS’03), (IEEE Computer Society, Washington, DC, USA, 2003), VIS ’03, pp. 38–43.

Yamaguchi, K.

T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray-tracing method,” Applied Optics 52, 201–209 (2013).
[Crossref]

Yang, C.

T. Yoneyama, C. Yang, Y. Sakamoto, and F. Okuyama, “Eyepiece-type full-color electro-holographic binocular display with see-through vision,” in Proceedings of Digital Holography and Three-Dimensional Imaging, (Optical Society of America, 2013), p. DW2A.11.
[Crossref]

Yoneyama, T.

T. Yoneyama, C. Yang, Y. Sakamoto, and F. Okuyama, “Eyepiece-type full-color electro-holographic binocular display with see-through vision,” in Proceedings of Digital Holography and Three-Dimensional Imaging, (Optical Society of America, 2013), p. DW2A.11.
[Crossref]

ACM Siggraph Computer Graphics (1)

R. A. Drebin, L. Carpenter, and P. Hanrahan, “Volume rendering,” ACM Siggraph Computer Graphics 22, 65–74 (1998).
[Crossref]

W. E. Lorensen and H. E. Cline, “Marching cubes: A high resolution 3d surface construction algorithm,” ACM SIGGRAPH Computer Graphics 21, 163–169 (1987).
[Crossref]

Appl Opt (1)

Z. Lu and Y. Sakamoto, “Holographic display methods for volume data: polygon-based and MIP-based methods,” Appl Opt 57, A142–A149 (2018).
[Crossref] [PubMed]

Applied Optics (1)

T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray-tracing method,” Applied Optics 52, 201–209 (2013).
[Crossref]

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Applied optics 54, 10029–10034 (2015).
[Crossref]

IEEE Computer Graphics and Applications (1)

M. Levoy, “Display of surfaces from volume data,” IEEE Computer Graphics and Applications 8, 29–37 (1998).
[Crossref]

J. Opt. Soc. Am (1)

E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am 53, 1123–1130 (1962).
[Crossref]

Japan Science and Technology Information Aggregator, Electronic (1)

C. Honda, S. Takahashi, S. J. Hart, and R. J. Rankin, “The technology in the digital holography system,” Japan Science and Technology Information Aggregator, Electronic 15, 135–145 (1998).

Medical Images Analysis (1)

J. A. Maintz and M. A. Viergever, “A survey of medical image registration,” Medical Images Analysis 2, 1–36 (1998).
[Crossref]

Optical Review (1)

T. Shimobaba and T. Ito, “A color holographic reconstruction system by time division multiplexing with reference lights of laser,” Optical Review 10, 339–341 (2003).
[Crossref]

Radiology (1)

S. Napel, M. P. Marks, G. D. Rubin, M. D. Dake, C. H. McDonnell, S. M. Song, D. R. Enzmann, and J. R B Jeffrey, “CT angiography with spiral CT and maximum intensity projection,” Radiology 185, 607–610 (1992).
[Crossref] [PubMed]

Other (10)

J. Kruger and R. Westermann, “Acceleration techniques for GPU-based volume rendering,” in Proceedings of the 14th IEEE Visualization 2003 (VIS’03), (IEEE Computer Society, Washington, DC, USA, 2003), VIS ’03, pp. 38–43.

J. Khan, “Static digital holograms for medical images,” http://www.holoxica.com/digital-holograms/ .

Y. Sakamoto, T. Aoyama, and Y. Aoki, “Volume rendering for computer generated holograms,” in Proceedings of 19th International Commission for Optics (ICO-19), (2002), pp. 555–556.

A. Wolfe and S. Hart, “Digital volumetric holograms for medical imaging,” http://spie.org/newsroom/digital-volumetric-holograms-for-medical-imaging .

J Image, “Image processing and analysis in Java,” https://imagej.net/Welcome .

X Osiri, “The world famous DICOM viewer,” http://www.osirix-viewer.com/ .

The Visualization Toolkit, http://www.vtk.org/ .

Audtodesk 3ds Max, http://www.autodesk.com .

T. Yoneyama, C. Yang, Y. Sakamoto, and F. Okuyama, “Eyepiece-type full-color electro-holographic binocular display with see-through vision,” in Proceedings of Digital Holography and Three-Dimensional Imaging, (Optical Society of America, 2013), p. DW2A.11.
[Crossref]

DICOM Image Library, http://www.osirix-viewer.com/resources/dicom-image-library/ .

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Figures (15)

Fig. 1
Fig. 1 Volume data (set of voxels).
Fig. 2
Fig. 2 Schematic diagram of color map and α map.
Fig. 3
Fig. 3 Redefinition of R, G, B, and α based on color map and α map.
Fig. 4
Fig. 4 Volume rendering method for single view point.
Fig. 5
Fig. 5 Diffuse reflection model.
Fig. 6
Fig. 6 CGH calculation with volume rendering method.
Fig. 7
Fig. 7 CGH calculation with elemental holograms.
Fig. 8
Fig. 8 CGH reconstruction device.
Fig. 9
Fig. 9 Geometry for optical experiment.
Fig. 10
Fig. 10 Reconstructed images of optical experiment. (a) Focusing on data1. (b) Focusing on data2.
Fig. 11
Fig. 11 Reconstructed images of approximate experiment. (a) is CG result. (b) is reconstructed image of proposed method. (c) is reconstructed image of proposed method with EH.
Fig. 12
Fig. 12 Reconstructed images of different method. (a) is the CG rendering result. (b) is the reconstructed image of polygon-based method. (c) is the reconstructed image of MIP-based method. (d) is the reconstructed image of proposed method with EH.
Fig. 13
Fig. 13 Reconstructed images of VIX (transparent objects).
Fig. 14
Fig. 14 Reconstructed images of transparent object with different α maps. (a), (b), (c), and (d) are reconstructed images with different α maps.
Fig. 15
Fig. 15 Reconstructed images of multi-layer objects. (a) is CG rendering result of multi-layer objects. (b) is reconstructed image of multi-layer objects.

Tables (3)

Tables Icon

Table 1 Parameters of Reconstruction Device

Tables Icon

Table 2 Parameters of Computation Device

Tables Icon

Table 3 Calculation time of different methods

Equations (7)

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

C x = C 3 ( C 3 C 2 ) ( I 3 I x ) ( I 3 I 2 ) ,
I d i = I d 0 j = 0 i ( 1 α j ) ,
I r i = k d i I d i m a x ( 0 , L N ) ,
I r L i = I r i j = i + 1 n ( 1 α j ) ,
μ i ( ξ , η ) = I r L i n r i exp ( j ( k r i + ϕ i ) ) ,
μ ( ξ , η ) = i = 1 N μ i ( ξ , η ) ,
μ i ( ξ , η ) = I r L i e k r i exp ( j ( k r i + ϕ i ) ) ,

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