Abstract

This study develops an interactive directional volumetric display that tracks a particular person and keeps displaying a directional image only to the person in real-time. Therefore, we construct a person-tracking system and combine it with the directional volumetric display to achieve interaction. There are limitations for real-time interaction due to the processing time of the algorithm. Thus, we accelerate the algorithm by utilizing a graphics processing unit (GPU). The GPU implementation processed images which comprise 64 × 64 pixels in 30 frames per second with image quality enough for practical applications.

© 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. B. G. Blundell, A. J. Schwarz, and D. K. Horrell, “Volumetric three-dimensional display systems: their past, present and future,” Eng. Sci. Educ. J. 2(5), 196–200 (1993).
    [Crossref]
  2. D. L. MacFarlane, “Volumetric three-dimensional display,” Appl. Opt. 33(31), 7453–7457 (1994).
    [Crossref]
  3. G. E. Favalora, “Volumetric 3D displays and application infrastructure,” Computer 38(8), 37–44 (2005).
    [Crossref]
  4. M. Parker, “Lumarca,” in ACM SIGGRAPH ASIA 2009 Art Gallery &; Emerging Technologies: Adaptation, (ACM, New York, NY, USA, 2009), SIGGRAPH ASIA ’09, p. 77.
  5. A. Shiraki, M. Ikeda, H. Nakayama, R. Hirayama, T. Kakue, T. Shimobaba, and T. Ito, “Efficient method for fabricating a directional volumetric display using strings displaying multiple images,” Appl. Opt. 57(1), A33–A38 (2018).
    [Crossref]
  6. R. Hirayama, A. Shiraki, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Operating scheme for the light-emitting diode array of a volumetric display that exhibits multiple full-color dynamic images,” Opt. Eng. 56(7), 073108 (2017).
    [Crossref]
  7. R. Hirayama, M. Naruse, H. Nakayama, N. Tate, A. Shiraki, T. Kakue, T. Shimobaba, M. Ohtsu, and T. Ito, “Design, implementation and characterization of a quantum-dot-based volumetric display,” Sci. Rep. 5(1), 8472 (2015).
    [Crossref]
  8. S. K. Nayar and V. N. Anand, “3D Display Using Passive Optical Scatterers,” Computer 40(7), 54–63 (2007).
    [Crossref]
  9. K. Kumagai, S. Hasegawa, and Y. Hayasaki, “Volumetric bubble display,” Optica 4(3), 298–302 (2017).
    [Crossref]
  10. M. Gately, Y. Zhai, M. Yeary, E. Petrich, and L. Sawalha, “A Three-Dimensional Swept Volume Display Based on LED Arrays,” J. Display Technol. 7(9), 503–514 (2011).
    [Crossref]
  11. G. Sela and G. Elber, “Generation of view dependent models using free form deformation,” Visual Comput. 23(3), 219–229 (2007).
    [Crossref]
  12. N. J. Mitra and M. Pauly, “Shadow Art,” ACM Trans. Graph. 28(5), 1–7 (2009).
    [Crossref]
  13. D. R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (Basic Books, Inc., New York, NY, USA, 1979).
  14. H. Nakayama, A. Shiraki, R. Hirayama, N. Masuda, T. Shimobaba, and T. Ito, “Three-dimensional volume containing multiple two-dimensional information patterns,” Sci. Rep. 3(1), 1931 (2013).
    [Crossref]
  15. R. Hirayama, H. Nakayama, A. Shiraki, T. Kakue, T. Shimobaba, and T. Ito, “Image quality improvement for a 3D structure exhibiting multiple 2D patterns and its implementation,” Opt. Express 24(7), 7319–7327 (2016).
    [Crossref]
  16. A. Shiraki, D. Matsumoto, R. Hirayama, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Improvement of an algorithm for displaying multiple images in one space,” Appl. Opt. 58(5), A1–A6 (2019).
    [Crossref]
  17. J. Müller, F. Alt, D. Michelis, and A. Schmidt, “Requirements and Design Space for Interactive Public Displays,” in Proceedings of the 18th ACM International Conference on Multimedia, (ACM, New York, NY, USA, 2010), MM ’10, pp. 1285–1294.
  18. T. Ojala, V. Kostakos, H. Kukka, T. Heikkinen, T. Linden, M. Jurmu, S. Hosio, F. Kruger, and D. Zanni, “Multipurpose Interactive Public Displays in the Wild: Three Years Later,” Computer 45(5), 42–49 (2012).
    [Crossref]
  19. K. Kuikkaniemi, G. Jacucci, M. Turpeinen, E. Hoggan, and J. Müller, “From Space to Stage: How Interactive Screens Will Change Urban Life,” Computer 44(6), 40–47 (2011).
    [Crossref]
  20. A. D. Wilson, “TouchLight: An Imaging Touch Screen and Display for Gesture-Based Interaction,” in Proceedings of the 6th International Conference on Multimodal Interfaces, (ACM, New York, NY, USA, 2004), ICMI ’04, pp. 69–76.
  21. H. Benko, A. D. Wilson, and R. Balakrishnan, “Sphere: Multi-touch Interactions on a Spherical Display,” in Proceedings of the 21st Annual ACM Symposium on User Interface Software and Technology, (ACM, New York, NY, USA, 2008), UIST ’08, pp. 77–86.
  22. I. Stavness, B. Lam, and S. Fels, “pCubee: A Perspective-Corrected Handheld Cubic Display,” in Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, (ACM, New York, NY, USA, 2010), CHI ’10, pp. 1381–1390.
  23. R. Balakrishnan, G. W. Fitzmaurice, and G. Kurtenbach, “User interfaces for volumetric displays,” Computer 34(3), 37–45 (2001).
    [Crossref]
  24. T. Grossman, D. Wigdor, and R. Balakrishnan, “Multi-finger Gestural Interaction with 3D Volumetric Displays,” in Proceedings of the 17th Annual ACM Symposium on User Interface Software and Technology, (ACM, New York, NY, USA, 2004), UIST ’04, pp. 61–70.
  25. J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
    [Crossref]
  26. D. Luebke and G. Humphreys, “How GPUs Work,” Computer 40(2), 96–100 (2007).
    [Crossref]
  27. F. Xu and K. Mueller, “Accelerating Popular Tomographic Reconstruction Algorithms on Commodity PC Graphics Hardware,” IEEE Trans. Nucl. Sci. 52(3), 654–663 (2005).
    [Crossref]
  28. M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. Medica 28(2), 94–108 (2012).
    [Crossref]
  29. A. Eklund, P. Dufort, D. Forsberg, and S. M. LaConte, “Medical image processing on the GPU - Past, present and future,” Med. Image Anal. 17(8), 1073–1094 (2013).
    [Crossref]
  30. “Depth Camera D415 - Intel® RealSense™ Depth and Tracking Cameras,” https://www.intelrealsense.com/depth-camera-d415/ .
  31. “Release Intel® RealSense™ SDK 2.0 (build 2.16.5),” https://github.com/IntelRealSense/librealsense/releases/tag/v2.16.5 .
  32. “Nuitrack Full Body Skeletal Tracking Software - Kinect replacement for Android, Windows, Linux, iOS, Intel RealSense, Orbbec,” https://nuitrack.com/ .
  33. “OpenGL - The Industry Standard for High Performance Graphics,” https://www.opengl.org/ .
  34. “CUDA Toolkit 9.0 Downloads | NVIDIA Developer,” https://developer.nvidia.com/cuda-90-download-archive .
  35. J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable Parallel Programming with CUDA,” Queue 6(2), 40–53 (2008).
    [Crossref]
  36. A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
    [Crossref]

2019 (1)

2018 (1)

2017 (2)

K. Kumagai, S. Hasegawa, and Y. Hayasaki, “Volumetric bubble display,” Optica 4(3), 298–302 (2017).
[Crossref]

R. Hirayama, A. Shiraki, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Operating scheme for the light-emitting diode array of a volumetric display that exhibits multiple full-color dynamic images,” Opt. Eng. 56(7), 073108 (2017).
[Crossref]

2016 (1)

2015 (1)

R. Hirayama, M. Naruse, H. Nakayama, N. Tate, A. Shiraki, T. Kakue, T. Shimobaba, M. Ohtsu, and T. Ito, “Design, implementation and characterization of a quantum-dot-based volumetric display,” Sci. Rep. 5(1), 8472 (2015).
[Crossref]

2013 (2)

A. Eklund, P. Dufort, D. Forsberg, and S. M. LaConte, “Medical image processing on the GPU - Past, present and future,” Med. Image Anal. 17(8), 1073–1094 (2013).
[Crossref]

H. Nakayama, A. Shiraki, R. Hirayama, N. Masuda, T. Shimobaba, and T. Ito, “Three-dimensional volume containing multiple two-dimensional information patterns,” Sci. Rep. 3(1), 1931 (2013).
[Crossref]

2012 (2)

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. Medica 28(2), 94–108 (2012).
[Crossref]

T. Ojala, V. Kostakos, H. Kukka, T. Heikkinen, T. Linden, M. Jurmu, S. Hosio, F. Kruger, and D. Zanni, “Multipurpose Interactive Public Displays in the Wild: Three Years Later,” Computer 45(5), 42–49 (2012).
[Crossref]

2011 (2)

K. Kuikkaniemi, G. Jacucci, M. Turpeinen, E. Hoggan, and J. Müller, “From Space to Stage: How Interactive Screens Will Change Urban Life,” Computer 44(6), 40–47 (2011).
[Crossref]

M. Gately, Y. Zhai, M. Yeary, E. Petrich, and L. Sawalha, “A Three-Dimensional Swept Volume Display Based on LED Arrays,” J. Display Technol. 7(9), 503–514 (2011).
[Crossref]

2009 (1)

N. J. Mitra and M. Pauly, “Shadow Art,” ACM Trans. Graph. 28(5), 1–7 (2009).
[Crossref]

2008 (2)

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable Parallel Programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

2007 (3)

D. Luebke and G. Humphreys, “How GPUs Work,” Computer 40(2), 96–100 (2007).
[Crossref]

S. K. Nayar and V. N. Anand, “3D Display Using Passive Optical Scatterers,” Computer 40(7), 54–63 (2007).
[Crossref]

G. Sela and G. Elber, “Generation of view dependent models using free form deformation,” Visual Comput. 23(3), 219–229 (2007).
[Crossref]

2005 (2)

G. E. Favalora, “Volumetric 3D displays and application infrastructure,” Computer 38(8), 37–44 (2005).
[Crossref]

F. Xu and K. Mueller, “Accelerating Popular Tomographic Reconstruction Algorithms on Commodity PC Graphics Hardware,” IEEE Trans. Nucl. Sci. 52(3), 654–663 (2005).
[Crossref]

2004 (1)

A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

2001 (1)

R. Balakrishnan, G. W. Fitzmaurice, and G. Kurtenbach, “User interfaces for volumetric displays,” Computer 34(3), 37–45 (2001).
[Crossref]

1994 (1)

1993 (1)

B. G. Blundell, A. J. Schwarz, and D. K. Horrell, “Volumetric three-dimensional display systems: their past, present and future,” Eng. Sci. Educ. J. 2(5), 196–200 (1993).
[Crossref]

Alt, F.

J. Müller, F. Alt, D. Michelis, and A. Schmidt, “Requirements and Design Space for Interactive Public Displays,” in Proceedings of the 18th ACM International Conference on Multimedia, (ACM, New York, NY, USA, 2010), MM ’10, pp. 1285–1294.

Anand, V. N.

S. K. Nayar and V. N. Anand, “3D Display Using Passive Optical Scatterers,” Computer 40(7), 54–63 (2007).
[Crossref]

Balakrishnan, R.

R. Balakrishnan, G. W. Fitzmaurice, and G. Kurtenbach, “User interfaces for volumetric displays,” Computer 34(3), 37–45 (2001).
[Crossref]

H. Benko, A. D. Wilson, and R. Balakrishnan, “Sphere: Multi-touch Interactions on a Spherical Display,” in Proceedings of the 21st Annual ACM Symposium on User Interface Software and Technology, (ACM, New York, NY, USA, 2008), UIST ’08, pp. 77–86.

T. Grossman, D. Wigdor, and R. Balakrishnan, “Multi-finger Gestural Interaction with 3D Volumetric Displays,” in Proceedings of the 17th Annual ACM Symposium on User Interface Software and Technology, (ACM, New York, NY, USA, 2004), UIST ’04, pp. 61–70.

Beister, M.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. Medica 28(2), 94–108 (2012).
[Crossref]

Benko, H.

H. Benko, A. D. Wilson, and R. Balakrishnan, “Sphere: Multi-touch Interactions on a Spherical Display,” in Proceedings of the 21st Annual ACM Symposium on User Interface Software and Technology, (ACM, New York, NY, USA, 2008), UIST ’08, pp. 77–86.

Blundell, B. G.

B. G. Blundell, A. J. Schwarz, and D. K. Horrell, “Volumetric three-dimensional display systems: their past, present and future,” Eng. Sci. Educ. J. 2(5), 196–200 (1993).
[Crossref]

Bovik, A. C.

A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

Buck, I.

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable Parallel Programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Dufort, P.

A. Eklund, P. Dufort, D. Forsberg, and S. M. LaConte, “Medical image processing on the GPU - Past, present and future,” Med. Image Anal. 17(8), 1073–1094 (2013).
[Crossref]

Eklund, A.

A. Eklund, P. Dufort, D. Forsberg, and S. M. LaConte, “Medical image processing on the GPU - Past, present and future,” Med. Image Anal. 17(8), 1073–1094 (2013).
[Crossref]

Elber, G.

G. Sela and G. Elber, “Generation of view dependent models using free form deformation,” Visual Comput. 23(3), 219–229 (2007).
[Crossref]

Favalora, G. E.

G. E. Favalora, “Volumetric 3D displays and application infrastructure,” Computer 38(8), 37–44 (2005).
[Crossref]

Fels, S.

I. Stavness, B. Lam, and S. Fels, “pCubee: A Perspective-Corrected Handheld Cubic Display,” in Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, (ACM, New York, NY, USA, 2010), CHI ’10, pp. 1381–1390.

Fitzmaurice, G. W.

R. Balakrishnan, G. W. Fitzmaurice, and G. Kurtenbach, “User interfaces for volumetric displays,” Computer 34(3), 37–45 (2001).
[Crossref]

Forsberg, D.

A. Eklund, P. Dufort, D. Forsberg, and S. M. LaConte, “Medical image processing on the GPU - Past, present and future,” Med. Image Anal. 17(8), 1073–1094 (2013).
[Crossref]

Garland, M.

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable Parallel Programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Gately, M.

Green, S.

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Grossman, T.

T. Grossman, D. Wigdor, and R. Balakrishnan, “Multi-finger Gestural Interaction with 3D Volumetric Displays,” in Proceedings of the 17th Annual ACM Symposium on User Interface Software and Technology, (ACM, New York, NY, USA, 2004), UIST ’04, pp. 61–70.

Hasegawa, S.

Hayasaki, Y.

Heikkinen, T.

T. Ojala, V. Kostakos, H. Kukka, T. Heikkinen, T. Linden, M. Jurmu, S. Hosio, F. Kruger, and D. Zanni, “Multipurpose Interactive Public Displays in the Wild: Three Years Later,” Computer 45(5), 42–49 (2012).
[Crossref]

Hirayama, R.

A. Shiraki, D. Matsumoto, R. Hirayama, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Improvement of an algorithm for displaying multiple images in one space,” Appl. Opt. 58(5), A1–A6 (2019).
[Crossref]

A. Shiraki, M. Ikeda, H. Nakayama, R. Hirayama, T. Kakue, T. Shimobaba, and T. Ito, “Efficient method for fabricating a directional volumetric display using strings displaying multiple images,” Appl. Opt. 57(1), A33–A38 (2018).
[Crossref]

R. Hirayama, A. Shiraki, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Operating scheme for the light-emitting diode array of a volumetric display that exhibits multiple full-color dynamic images,” Opt. Eng. 56(7), 073108 (2017).
[Crossref]

R. Hirayama, H. Nakayama, A. Shiraki, T. Kakue, T. Shimobaba, and T. Ito, “Image quality improvement for a 3D structure exhibiting multiple 2D patterns and its implementation,” Opt. Express 24(7), 7319–7327 (2016).
[Crossref]

R. Hirayama, M. Naruse, H. Nakayama, N. Tate, A. Shiraki, T. Kakue, T. Shimobaba, M. Ohtsu, and T. Ito, “Design, implementation and characterization of a quantum-dot-based volumetric display,” Sci. Rep. 5(1), 8472 (2015).
[Crossref]

H. Nakayama, A. Shiraki, R. Hirayama, N. Masuda, T. Shimobaba, and T. Ito, “Three-dimensional volume containing multiple two-dimensional information patterns,” Sci. Rep. 3(1), 1931 (2013).
[Crossref]

Hofstadter, D. R.

D. R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (Basic Books, Inc., New York, NY, USA, 1979).

Hoggan, E.

K. Kuikkaniemi, G. Jacucci, M. Turpeinen, E. Hoggan, and J. Müller, “From Space to Stage: How Interactive Screens Will Change Urban Life,” Computer 44(6), 40–47 (2011).
[Crossref]

Horrell, D. K.

B. G. Blundell, A. J. Schwarz, and D. K. Horrell, “Volumetric three-dimensional display systems: their past, present and future,” Eng. Sci. Educ. J. 2(5), 196–200 (1993).
[Crossref]

Hosio, S.

T. Ojala, V. Kostakos, H. Kukka, T. Heikkinen, T. Linden, M. Jurmu, S. Hosio, F. Kruger, and D. Zanni, “Multipurpose Interactive Public Displays in the Wild: Three Years Later,” Computer 45(5), 42–49 (2012).
[Crossref]

Houston, M.

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Humphreys, G.

D. Luebke and G. Humphreys, “How GPUs Work,” Computer 40(2), 96–100 (2007).
[Crossref]

Ikeda, M.

Ito, T.

A. Shiraki, D. Matsumoto, R. Hirayama, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Improvement of an algorithm for displaying multiple images in one space,” Appl. Opt. 58(5), A1–A6 (2019).
[Crossref]

A. Shiraki, M. Ikeda, H. Nakayama, R. Hirayama, T. Kakue, T. Shimobaba, and T. Ito, “Efficient method for fabricating a directional volumetric display using strings displaying multiple images,” Appl. Opt. 57(1), A33–A38 (2018).
[Crossref]

R. Hirayama, A. Shiraki, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Operating scheme for the light-emitting diode array of a volumetric display that exhibits multiple full-color dynamic images,” Opt. Eng. 56(7), 073108 (2017).
[Crossref]

R. Hirayama, H. Nakayama, A. Shiraki, T. Kakue, T. Shimobaba, and T. Ito, “Image quality improvement for a 3D structure exhibiting multiple 2D patterns and its implementation,” Opt. Express 24(7), 7319–7327 (2016).
[Crossref]

R. Hirayama, M. Naruse, H. Nakayama, N. Tate, A. Shiraki, T. Kakue, T. Shimobaba, M. Ohtsu, and T. Ito, “Design, implementation and characterization of a quantum-dot-based volumetric display,” Sci. Rep. 5(1), 8472 (2015).
[Crossref]

H. Nakayama, A. Shiraki, R. Hirayama, N. Masuda, T. Shimobaba, and T. Ito, “Three-dimensional volume containing multiple two-dimensional information patterns,” Sci. Rep. 3(1), 1931 (2013).
[Crossref]

Jacucci, G.

K. Kuikkaniemi, G. Jacucci, M. Turpeinen, E. Hoggan, and J. Müller, “From Space to Stage: How Interactive Screens Will Change Urban Life,” Computer 44(6), 40–47 (2011).
[Crossref]

Jurmu, M.

T. Ojala, V. Kostakos, H. Kukka, T. Heikkinen, T. Linden, M. Jurmu, S. Hosio, F. Kruger, and D. Zanni, “Multipurpose Interactive Public Displays in the Wild: Three Years Later,” Computer 45(5), 42–49 (2012).
[Crossref]

Kakue, T.

Kalender, W. A.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. Medica 28(2), 94–108 (2012).
[Crossref]

Kolditz, D.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. Medica 28(2), 94–108 (2012).
[Crossref]

Kostakos, V.

T. Ojala, V. Kostakos, H. Kukka, T. Heikkinen, T. Linden, M. Jurmu, S. Hosio, F. Kruger, and D. Zanni, “Multipurpose Interactive Public Displays in the Wild: Three Years Later,” Computer 45(5), 42–49 (2012).
[Crossref]

Kruger, F.

T. Ojala, V. Kostakos, H. Kukka, T. Heikkinen, T. Linden, M. Jurmu, S. Hosio, F. Kruger, and D. Zanni, “Multipurpose Interactive Public Displays in the Wild: Three Years Later,” Computer 45(5), 42–49 (2012).
[Crossref]

Kuikkaniemi, K.

K. Kuikkaniemi, G. Jacucci, M. Turpeinen, E. Hoggan, and J. Müller, “From Space to Stage: How Interactive Screens Will Change Urban Life,” Computer 44(6), 40–47 (2011).
[Crossref]

Kukka, H.

T. Ojala, V. Kostakos, H. Kukka, T. Heikkinen, T. Linden, M. Jurmu, S. Hosio, F. Kruger, and D. Zanni, “Multipurpose Interactive Public Displays in the Wild: Three Years Later,” Computer 45(5), 42–49 (2012).
[Crossref]

Kumagai, K.

Kurtenbach, G.

R. Balakrishnan, G. W. Fitzmaurice, and G. Kurtenbach, “User interfaces for volumetric displays,” Computer 34(3), 37–45 (2001).
[Crossref]

LaConte, S. M.

A. Eklund, P. Dufort, D. Forsberg, and S. M. LaConte, “Medical image processing on the GPU - Past, present and future,” Med. Image Anal. 17(8), 1073–1094 (2013).
[Crossref]

Lam, B.

I. Stavness, B. Lam, and S. Fels, “pCubee: A Perspective-Corrected Handheld Cubic Display,” in Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, (ACM, New York, NY, USA, 2010), CHI ’10, pp. 1381–1390.

Linden, T.

T. Ojala, V. Kostakos, H. Kukka, T. Heikkinen, T. Linden, M. Jurmu, S. Hosio, F. Kruger, and D. Zanni, “Multipurpose Interactive Public Displays in the Wild: Three Years Later,” Computer 45(5), 42–49 (2012).
[Crossref]

Luebke, D.

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

D. Luebke and G. Humphreys, “How GPUs Work,” Computer 40(2), 96–100 (2007).
[Crossref]

MacFarlane, D. L.

Masuda, N.

H. Nakayama, A. Shiraki, R. Hirayama, N. Masuda, T. Shimobaba, and T. Ito, “Three-dimensional volume containing multiple two-dimensional information patterns,” Sci. Rep. 3(1), 1931 (2013).
[Crossref]

Matsumoto, D.

Michelis, D.

J. Müller, F. Alt, D. Michelis, and A. Schmidt, “Requirements and Design Space for Interactive Public Displays,” in Proceedings of the 18th ACM International Conference on Multimedia, (ACM, New York, NY, USA, 2010), MM ’10, pp. 1285–1294.

Mitra, N. J.

N. J. Mitra and M. Pauly, “Shadow Art,” ACM Trans. Graph. 28(5), 1–7 (2009).
[Crossref]

Mueller, K.

F. Xu and K. Mueller, “Accelerating Popular Tomographic Reconstruction Algorithms on Commodity PC Graphics Hardware,” IEEE Trans. Nucl. Sci. 52(3), 654–663 (2005).
[Crossref]

Müller, J.

K. Kuikkaniemi, G. Jacucci, M. Turpeinen, E. Hoggan, and J. Müller, “From Space to Stage: How Interactive Screens Will Change Urban Life,” Computer 44(6), 40–47 (2011).
[Crossref]

J. Müller, F. Alt, D. Michelis, and A. Schmidt, “Requirements and Design Space for Interactive Public Displays,” in Proceedings of the 18th ACM International Conference on Multimedia, (ACM, New York, NY, USA, 2010), MM ’10, pp. 1285–1294.

Nakayama, H.

A. Shiraki, D. Matsumoto, R. Hirayama, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Improvement of an algorithm for displaying multiple images in one space,” Appl. Opt. 58(5), A1–A6 (2019).
[Crossref]

A. Shiraki, M. Ikeda, H. Nakayama, R. Hirayama, T. Kakue, T. Shimobaba, and T. Ito, “Efficient method for fabricating a directional volumetric display using strings displaying multiple images,” Appl. Opt. 57(1), A33–A38 (2018).
[Crossref]

R. Hirayama, A. Shiraki, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Operating scheme for the light-emitting diode array of a volumetric display that exhibits multiple full-color dynamic images,” Opt. Eng. 56(7), 073108 (2017).
[Crossref]

R. Hirayama, H. Nakayama, A. Shiraki, T. Kakue, T. Shimobaba, and T. Ito, “Image quality improvement for a 3D structure exhibiting multiple 2D patterns and its implementation,” Opt. Express 24(7), 7319–7327 (2016).
[Crossref]

R. Hirayama, M. Naruse, H. Nakayama, N. Tate, A. Shiraki, T. Kakue, T. Shimobaba, M. Ohtsu, and T. Ito, “Design, implementation and characterization of a quantum-dot-based volumetric display,” Sci. Rep. 5(1), 8472 (2015).
[Crossref]

H. Nakayama, A. Shiraki, R. Hirayama, N. Masuda, T. Shimobaba, and T. Ito, “Three-dimensional volume containing multiple two-dimensional information patterns,” Sci. Rep. 3(1), 1931 (2013).
[Crossref]

Naruse, M.

R. Hirayama, M. Naruse, H. Nakayama, N. Tate, A. Shiraki, T. Kakue, T. Shimobaba, M. Ohtsu, and T. Ito, “Design, implementation and characterization of a quantum-dot-based volumetric display,” Sci. Rep. 5(1), 8472 (2015).
[Crossref]

Nayar, S. K.

S. K. Nayar and V. N. Anand, “3D Display Using Passive Optical Scatterers,” Computer 40(7), 54–63 (2007).
[Crossref]

Nickolls, J.

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable Parallel Programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Ohtsu, M.

R. Hirayama, M. Naruse, H. Nakayama, N. Tate, A. Shiraki, T. Kakue, T. Shimobaba, M. Ohtsu, and T. Ito, “Design, implementation and characterization of a quantum-dot-based volumetric display,” Sci. Rep. 5(1), 8472 (2015).
[Crossref]

Ojala, T.

T. Ojala, V. Kostakos, H. Kukka, T. Heikkinen, T. Linden, M. Jurmu, S. Hosio, F. Kruger, and D. Zanni, “Multipurpose Interactive Public Displays in the Wild: Three Years Later,” Computer 45(5), 42–49 (2012).
[Crossref]

Owens, J. D.

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Parker, M.

M. Parker, “Lumarca,” in ACM SIGGRAPH ASIA 2009 Art Gallery &; Emerging Technologies: Adaptation, (ACM, New York, NY, USA, 2009), SIGGRAPH ASIA ’09, p. 77.

Pauly, M.

N. J. Mitra and M. Pauly, “Shadow Art,” ACM Trans. Graph. 28(5), 1–7 (2009).
[Crossref]

Petrich, E.

Phillips, J. C.

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Sawalha, L.

Schmidt, A.

J. Müller, F. Alt, D. Michelis, and A. Schmidt, “Requirements and Design Space for Interactive Public Displays,” in Proceedings of the 18th ACM International Conference on Multimedia, (ACM, New York, NY, USA, 2010), MM ’10, pp. 1285–1294.

Schwarz, A. J.

B. G. Blundell, A. J. Schwarz, and D. K. Horrell, “Volumetric three-dimensional display systems: their past, present and future,” Eng. Sci. Educ. J. 2(5), 196–200 (1993).
[Crossref]

Sela, G.

G. Sela and G. Elber, “Generation of view dependent models using free form deformation,” Visual Comput. 23(3), 219–229 (2007).
[Crossref]

Sheikh, H. R.

A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

Shimobaba, T.

A. Shiraki, D. Matsumoto, R. Hirayama, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Improvement of an algorithm for displaying multiple images in one space,” Appl. Opt. 58(5), A1–A6 (2019).
[Crossref]

A. Shiraki, M. Ikeda, H. Nakayama, R. Hirayama, T. Kakue, T. Shimobaba, and T. Ito, “Efficient method for fabricating a directional volumetric display using strings displaying multiple images,” Appl. Opt. 57(1), A33–A38 (2018).
[Crossref]

R. Hirayama, A. Shiraki, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Operating scheme for the light-emitting diode array of a volumetric display that exhibits multiple full-color dynamic images,” Opt. Eng. 56(7), 073108 (2017).
[Crossref]

R. Hirayama, H. Nakayama, A. Shiraki, T. Kakue, T. Shimobaba, and T. Ito, “Image quality improvement for a 3D structure exhibiting multiple 2D patterns and its implementation,” Opt. Express 24(7), 7319–7327 (2016).
[Crossref]

R. Hirayama, M. Naruse, H. Nakayama, N. Tate, A. Shiraki, T. Kakue, T. Shimobaba, M. Ohtsu, and T. Ito, “Design, implementation and characterization of a quantum-dot-based volumetric display,” Sci. Rep. 5(1), 8472 (2015).
[Crossref]

H. Nakayama, A. Shiraki, R. Hirayama, N. Masuda, T. Shimobaba, and T. Ito, “Three-dimensional volume containing multiple two-dimensional information patterns,” Sci. Rep. 3(1), 1931 (2013).
[Crossref]

Shiraki, A.

A. Shiraki, D. Matsumoto, R. Hirayama, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Improvement of an algorithm for displaying multiple images in one space,” Appl. Opt. 58(5), A1–A6 (2019).
[Crossref]

A. Shiraki, M. Ikeda, H. Nakayama, R. Hirayama, T. Kakue, T. Shimobaba, and T. Ito, “Efficient method for fabricating a directional volumetric display using strings displaying multiple images,” Appl. Opt. 57(1), A33–A38 (2018).
[Crossref]

R. Hirayama, A. Shiraki, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Operating scheme for the light-emitting diode array of a volumetric display that exhibits multiple full-color dynamic images,” Opt. Eng. 56(7), 073108 (2017).
[Crossref]

R. Hirayama, H. Nakayama, A. Shiraki, T. Kakue, T. Shimobaba, and T. Ito, “Image quality improvement for a 3D structure exhibiting multiple 2D patterns and its implementation,” Opt. Express 24(7), 7319–7327 (2016).
[Crossref]

R. Hirayama, M. Naruse, H. Nakayama, N. Tate, A. Shiraki, T. Kakue, T. Shimobaba, M. Ohtsu, and T. Ito, “Design, implementation and characterization of a quantum-dot-based volumetric display,” Sci. Rep. 5(1), 8472 (2015).
[Crossref]

H. Nakayama, A. Shiraki, R. Hirayama, N. Masuda, T. Shimobaba, and T. Ito, “Three-dimensional volume containing multiple two-dimensional information patterns,” Sci. Rep. 3(1), 1931 (2013).
[Crossref]

Simoncelli, E. P.

A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

Skadron, K.

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable Parallel Programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Stavness, I.

I. Stavness, B. Lam, and S. Fels, “pCubee: A Perspective-Corrected Handheld Cubic Display,” in Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, (ACM, New York, NY, USA, 2010), CHI ’10, pp. 1381–1390.

Stone, J. E.

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Tate, N.

R. Hirayama, M. Naruse, H. Nakayama, N. Tate, A. Shiraki, T. Kakue, T. Shimobaba, M. Ohtsu, and T. Ito, “Design, implementation and characterization of a quantum-dot-based volumetric display,” Sci. Rep. 5(1), 8472 (2015).
[Crossref]

Turpeinen, M.

K. Kuikkaniemi, G. Jacucci, M. Turpeinen, E. Hoggan, and J. Müller, “From Space to Stage: How Interactive Screens Will Change Urban Life,” Computer 44(6), 40–47 (2011).
[Crossref]

Wigdor, D.

T. Grossman, D. Wigdor, and R. Balakrishnan, “Multi-finger Gestural Interaction with 3D Volumetric Displays,” in Proceedings of the 17th Annual ACM Symposium on User Interface Software and Technology, (ACM, New York, NY, USA, 2004), UIST ’04, pp. 61–70.

Wilson, A. D.

A. D. Wilson, “TouchLight: An Imaging Touch Screen and Display for Gesture-Based Interaction,” in Proceedings of the 6th International Conference on Multimodal Interfaces, (ACM, New York, NY, USA, 2004), ICMI ’04, pp. 69–76.

H. Benko, A. D. Wilson, and R. Balakrishnan, “Sphere: Multi-touch Interactions on a Spherical Display,” in Proceedings of the 21st Annual ACM Symposium on User Interface Software and Technology, (ACM, New York, NY, USA, 2008), UIST ’08, pp. 77–86.

Xu, F.

F. Xu and K. Mueller, “Accelerating Popular Tomographic Reconstruction Algorithms on Commodity PC Graphics Hardware,” IEEE Trans. Nucl. Sci. 52(3), 654–663 (2005).
[Crossref]

Yeary, M.

Zanni, D.

T. Ojala, V. Kostakos, H. Kukka, T. Heikkinen, T. Linden, M. Jurmu, S. Hosio, F. Kruger, and D. Zanni, “Multipurpose Interactive Public Displays in the Wild: Three Years Later,” Computer 45(5), 42–49 (2012).
[Crossref]

Zhai, Y.

ACM Trans. Graph. (1)

N. J. Mitra and M. Pauly, “Shadow Art,” ACM Trans. Graph. 28(5), 1–7 (2009).
[Crossref]

Appl. Opt. (3)

Computer (6)

R. Balakrishnan, G. W. Fitzmaurice, and G. Kurtenbach, “User interfaces for volumetric displays,” Computer 34(3), 37–45 (2001).
[Crossref]

D. Luebke and G. Humphreys, “How GPUs Work,” Computer 40(2), 96–100 (2007).
[Crossref]

G. E. Favalora, “Volumetric 3D displays and application infrastructure,” Computer 38(8), 37–44 (2005).
[Crossref]

S. K. Nayar and V. N. Anand, “3D Display Using Passive Optical Scatterers,” Computer 40(7), 54–63 (2007).
[Crossref]

T. Ojala, V. Kostakos, H. Kukka, T. Heikkinen, T. Linden, M. Jurmu, S. Hosio, F. Kruger, and D. Zanni, “Multipurpose Interactive Public Displays in the Wild: Three Years Later,” Computer 45(5), 42–49 (2012).
[Crossref]

K. Kuikkaniemi, G. Jacucci, M. Turpeinen, E. Hoggan, and J. Müller, “From Space to Stage: How Interactive Screens Will Change Urban Life,” Computer 44(6), 40–47 (2011).
[Crossref]

Eng. Sci. Educ. J. (1)

B. G. Blundell, A. J. Schwarz, and D. K. Horrell, “Volumetric three-dimensional display systems: their past, present and future,” Eng. Sci. Educ. J. 2(5), 196–200 (1993).
[Crossref]

IEEE Trans. Nucl. Sci. (1)

F. Xu and K. Mueller, “Accelerating Popular Tomographic Reconstruction Algorithms on Commodity PC Graphics Hardware,” IEEE Trans. Nucl. Sci. 52(3), 654–663 (2005).
[Crossref]

IEEE Trans. on Image Process. (1)

A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

J. Display Technol. (1)

Med. Image Anal. (1)

A. Eklund, P. Dufort, D. Forsberg, and S. M. LaConte, “Medical image processing on the GPU - Past, present and future,” Med. Image Anal. 17(8), 1073–1094 (2013).
[Crossref]

Opt. Eng. (1)

R. Hirayama, A. Shiraki, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Operating scheme for the light-emitting diode array of a volumetric display that exhibits multiple full-color dynamic images,” Opt. Eng. 56(7), 073108 (2017).
[Crossref]

Opt. Express (1)

Optica (1)

Phys. Medica (1)

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Phys. Medica 28(2), 94–108 (2012).
[Crossref]

Proc. IEEE (1)

J. D. Owens, M. Houston, D. Luebke, S. Green, J. E. Stone, and J. C. Phillips, “GPU Computing,” Proc. IEEE 96(5), 879–899 (2008).
[Crossref]

Queue (1)

J. Nickolls, I. Buck, M. Garland, and K. Skadron, “Scalable Parallel Programming with CUDA,” Queue 6(2), 40–53 (2008).
[Crossref]

Sci. Rep. (2)

R. Hirayama, M. Naruse, H. Nakayama, N. Tate, A. Shiraki, T. Kakue, T. Shimobaba, M. Ohtsu, and T. Ito, “Design, implementation and characterization of a quantum-dot-based volumetric display,” Sci. Rep. 5(1), 8472 (2015).
[Crossref]

H. Nakayama, A. Shiraki, R. Hirayama, N. Masuda, T. Shimobaba, and T. Ito, “Three-dimensional volume containing multiple two-dimensional information patterns,” Sci. Rep. 3(1), 1931 (2013).
[Crossref]

Visual Comput. (1)

G. Sela and G. Elber, “Generation of view dependent models using free form deformation,” Visual Comput. 23(3), 219–229 (2007).
[Crossref]

Other (12)

M. Parker, “Lumarca,” in ACM SIGGRAPH ASIA 2009 Art Gallery &; Emerging Technologies: Adaptation, (ACM, New York, NY, USA, 2009), SIGGRAPH ASIA ’09, p. 77.

D. R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (Basic Books, Inc., New York, NY, USA, 1979).

A. D. Wilson, “TouchLight: An Imaging Touch Screen and Display for Gesture-Based Interaction,” in Proceedings of the 6th International Conference on Multimodal Interfaces, (ACM, New York, NY, USA, 2004), ICMI ’04, pp. 69–76.

H. Benko, A. D. Wilson, and R. Balakrishnan, “Sphere: Multi-touch Interactions on a Spherical Display,” in Proceedings of the 21st Annual ACM Symposium on User Interface Software and Technology, (ACM, New York, NY, USA, 2008), UIST ’08, pp. 77–86.

I. Stavness, B. Lam, and S. Fels, “pCubee: A Perspective-Corrected Handheld Cubic Display,” in Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, (ACM, New York, NY, USA, 2010), CHI ’10, pp. 1381–1390.

“Depth Camera D415 - Intel® RealSense™ Depth and Tracking Cameras,” https://www.intelrealsense.com/depth-camera-d415/ .

“Release Intel® RealSense™ SDK 2.0 (build 2.16.5),” https://github.com/IntelRealSense/librealsense/releases/tag/v2.16.5 .

“Nuitrack Full Body Skeletal Tracking Software - Kinect replacement for Android, Windows, Linux, iOS, Intel RealSense, Orbbec,” https://nuitrack.com/ .

“OpenGL - The Industry Standard for High Performance Graphics,” https://www.opengl.org/ .

“CUDA Toolkit 9.0 Downloads | NVIDIA Developer,” https://developer.nvidia.com/cuda-90-download-archive .

T. Grossman, D. Wigdor, and R. Balakrishnan, “Multi-finger Gestural Interaction with 3D Volumetric Displays,” in Proceedings of the 17th Annual ACM Symposium on User Interface Software and Technology, (ACM, New York, NY, USA, 2004), UIST ’04, pp. 61–70.

J. Müller, F. Alt, D. Michelis, and A. Schmidt, “Requirements and Design Space for Interactive Public Displays,” in Proceedings of the 18th ACM International Conference on Multimedia, (ACM, New York, NY, USA, 2010), MM ’10, pp. 1285–1294.

Supplementary Material (2)

NameDescription
» Visualization 1       The observation results of the interactive directional volumetric display
» Visualization 2       The comparison of the observation results in the directional volumetric display with and without the person-tracking system.

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Figures (12)

Fig. 1.
Fig. 1. Directional volumetric display: (a) composed of LEDs and (b) composed of threads and a projector [14].
Fig. 2.
Fig. 2. The setting of the coordinate system and vectors of the person-tracking system.
Fig. 3.
Fig. 3. Scheme of the voxel calculation algorithm: (a) the relationship between the voxels and the input images and (b) the iteration process for improving the image quality of the displayed images.
Fig. 4.
Fig. 4. The flow of a projection system to fabricate the interactive directional volumetric display using GPU implementation.
Fig. 5.
Fig. 5. The convergence of the SSIM values of the displayed images consisting of 64 $\times$ 64 pixels.
Fig. 6.
Fig. 6. The processing time of the GPU implementation using the original images, which consist of 64 $\times$ 64 pixels.
Fig. 7.
Fig. 7. Simulation result of the displayed image using two original images which comprise 20 $\times$ 20 pixels: (a) original image 1, (b) original image 2, (c) displayed image 1, (d) displayed image 2, and (e) an image observed from an undesignated viewing position.
Fig. 8.
Fig. 8. Simulation result of the displayed image using two original images which comprise 64 $\times$ 64 pixels: (a) original image 1, (b) original image 2, (c) displayed image 1, (d) displayed image 2, and (e) an image observed from an undesignated viewing position.
Fig. 9.
Fig. 9. The system overview of the interactive directional volumetric display.
Fig. 10.
Fig. 10. The relationship between the directional volumetric display, the projector, RealSense, and the angles of observer.
Fig. 11.
Fig. 11. The observation results of the interactive directional volumetric display: (a) original image 1, (b) original image 2, (c) displayed image 1, (d) displayed image 2 observed from an angle of $45^\circ$, (e) displayed images 2 observed from an angle of $90^\circ$, (f) displayed image 2 observed from an angle of $135^\circ$, and (g) an image observed from an undesignated viewing position (see Visualization 1).
Fig. 12.
Fig. 12. The placement of the interactive volumetric display and RealSense(s) to achieve a person tracking for the whole circumference: (a) with multiple RealSenses and (b) with one RealSense.

Tables (6)

Tables Icon

Table 1. The specifications of Intel RealSense D415

Tables Icon

Table 2. The specifications of NVIDIA GeForce GTX 1050

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Table 3. The specifications of Intel Core i5-6500

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Table 4. Comparison of the processing time between the CPU implementation and the GPU implementation with two original images

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Table 5. Comparison of the processing time between the CPU implementation and the GPU implementation with three original images

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Table 6. The specifications of MH550 projector

Equations (6)

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

θ = cos 1 H y 2 + C H z H y 2 + C 2 H x 2 + H y 2 + H z 2 .
V ( x , y , z ) = i = 1 N I i ( u i , v i ) ( k ) N ,
D i ( u i , v i ) ( k ) = W i V i ( x , y , z ) ( k ) ,
I i ( u i , v i ) ( k + 1 ) = I i ( u i , v i ) ( k ) O i ( u i , v i ) D i ( u i , v i ) ( k ) ,
V ( x , y , z ) = i = 1 N I i ( u i , v i ) ( k ) ,
I i ( u i , v i ) ( k + 1 ) = I i ( u i , v i ) ( k ) + ( O i ( u i , v i ) D i ( u i , v i ) ( k ) ) .

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