Abstract

Autostereoscopic printing is one of the most common ways for three-dimensional display, because it can present finer results by printing higher dots per inches (DPI). However, there are some problems for current methods. First, errors caused by dislocation between integer grids and non-customized lenticular lens result in severe vision quality. Second, the view-number and gray-level cannot be set arbitrarily. In this paper, an improved halftoning method for autostereoscopic printing based on float grid-division multiplexing (fGDM) is proposed. FGDM effectively addresses above two problems. GPU based program of fGDM is enabled to achieve the result very fast. Films with lenticular lens array are implemented in experiments to verify the effectiveness of proposed method which provides an improved three-dimensional performance, compared with the AM screening and random screening.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
High-performance autostereoscopic display based on the lenticular tracking method

Tianqi Huang, Boxuan Han, Xinran Zhang, and Hongen Liao
Opt. Express 27(15) 20421-20434 (2019)

Model-based error diffusion for high fidelity lenticular screening

Daniel L. Lau, Trebor Smith, and Trebor Smith
Opt. Express 14(8) 3214-3224 (2006)

High-fidelity lenticular screening by means of iterative tone correction

Daniel L. Lau and Trebor Smith
J. Opt. Soc. Am. A 23(11) 2714-2723 (2006)

References

  • View by:
  • |
  • |
  • |

  1. X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
    [Crossref]
  2. X. Yu, X. Sang, X. Gao, Z. Chen, D. Chen, W. Duan, B. Yan, C. Yu, and D. Xu, “Large viewing angle three-dimensional display with smooth motion parallax and accurate depth cues,” Opt. Express 23(20), 25950–25958 (2015).
    [Crossref] [PubMed]
  3. X. Gao, X. Sang, X. Yu, P. Wang, X. Cao, L. Sun, B. Yan, J. Yuan, K. Wang, C. Yu, and W. Dou, “Aberration analyses for improving the frontal projection three-dimensional display,” Opt. Express 22(19), 23496–23511 (2014).
    [Crossref] [PubMed]
  4. P. Wang, S. Xie, X. Sang, D. Chen, C. Li, X. Gao, X. Yu, C. Yu, B. Yan, W. Dou, and L. Xiao, “A large depth of field frontal multi-projection three-dimensional display with uniform light field distribution,” Opt. Commun. 354, 321–329 (2015).
    [Crossref]
  5. S. Xie, P. Wang, X. Sang, and C. Li, “Augmented reality three-dimensional display with light field fusion,” Opt. Express 24(11), 11483–11494 (2016).
    [Crossref] [PubMed]
  6. R. A. Ulichney, Digital Halftoning (MIT Press, 1987).
  7. D. L. Lau and G. R. Arce, Modern Digital Halftoning (CRC Press, 2001).
  8. R. L. Adler, B. P. Kitchens, M. Martens, C. P. Tresser, and C. W. Wu, “The mathematics of halftoning,” IBM J. Res. Develop. 47(1), 5–15 (2003).
    [Crossref]
  9. S. Sakamoto and Y. Takaki, “Three-dimensional print using a one-dimensional screen technique,” Jpn. J. Appl. Phys. 47(7), 5486–5492 (2008).
    [Crossref]
  10. H. Yamazaki and Y. Takaki, “Printing 3D Light Field with 1D Halftone Screening,” SIGGRAPH (2012).
  11. D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” Acm Trans. Graphic 30(6), 61–64 (2011).
    [Crossref]

2016 (1)

2015 (2)

X. Yu, X. Sang, X. Gao, Z. Chen, D. Chen, W. Duan, B. Yan, C. Yu, and D. Xu, “Large viewing angle three-dimensional display with smooth motion parallax and accurate depth cues,” Opt. Express 23(20), 25950–25958 (2015).
[Crossref] [PubMed]

P. Wang, S. Xie, X. Sang, D. Chen, C. Li, X. Gao, X. Yu, C. Yu, B. Yan, W. Dou, and L. Xiao, “A large depth of field frontal multi-projection three-dimensional display with uniform light field distribution,” Opt. Commun. 354, 321–329 (2015).
[Crossref]

2014 (2)

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

X. Gao, X. Sang, X. Yu, P. Wang, X. Cao, L. Sun, B. Yan, J. Yuan, K. Wang, C. Yu, and W. Dou, “Aberration analyses for improving the frontal projection three-dimensional display,” Opt. Express 22(19), 23496–23511 (2014).
[Crossref] [PubMed]

2011 (1)

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” Acm Trans. Graphic 30(6), 61–64 (2011).
[Crossref]

2008 (1)

S. Sakamoto and Y. Takaki, “Three-dimensional print using a one-dimensional screen technique,” Jpn. J. Appl. Phys. 47(7), 5486–5492 (2008).
[Crossref]

2003 (1)

R. L. Adler, B. P. Kitchens, M. Martens, C. P. Tresser, and C. W. Wu, “The mathematics of halftoning,” IBM J. Res. Develop. 47(1), 5–15 (2003).
[Crossref]

Adler, R. L.

R. L. Adler, B. P. Kitchens, M. Martens, C. P. Tresser, and C. W. Wu, “The mathematics of halftoning,” IBM J. Res. Develop. 47(1), 5–15 (2003).
[Crossref]

Cai, Y.

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

Cao, X.

Chen, D.

X. Yu, X. Sang, X. Gao, Z. Chen, D. Chen, W. Duan, B. Yan, C. Yu, and D. Xu, “Large viewing angle three-dimensional display with smooth motion parallax and accurate depth cues,” Opt. Express 23(20), 25950–25958 (2015).
[Crossref] [PubMed]

P. Wang, S. Xie, X. Sang, D. Chen, C. Li, X. Gao, X. Yu, C. Yu, B. Yan, W. Dou, and L. Xiao, “A large depth of field frontal multi-projection three-dimensional display with uniform light field distribution,” Opt. Commun. 354, 321–329 (2015).
[Crossref]

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

Chen, Z.

Dou, W.

P. Wang, S. Xie, X. Sang, D. Chen, C. Li, X. Gao, X. Yu, C. Yu, B. Yan, W. Dou, and L. Xiao, “A large depth of field frontal multi-projection three-dimensional display with uniform light field distribution,” Opt. Commun. 354, 321–329 (2015).
[Crossref]

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

X. Gao, X. Sang, X. Yu, P. Wang, X. Cao, L. Sun, B. Yan, J. Yuan, K. Wang, C. Yu, and W. Dou, “Aberration analyses for improving the frontal projection three-dimensional display,” Opt. Express 22(19), 23496–23511 (2014).
[Crossref] [PubMed]

Duan, W.

Gao, X.

Heidrich, W.

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” Acm Trans. Graphic 30(6), 61–64 (2011).
[Crossref]

Hirsch, M.

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” Acm Trans. Graphic 30(6), 61–64 (2011).
[Crossref]

Kitchens, B. P.

R. L. Adler, B. P. Kitchens, M. Martens, C. P. Tresser, and C. W. Wu, “The mathematics of halftoning,” IBM J. Res. Develop. 47(1), 5–15 (2003).
[Crossref]

Lanman, D.

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” Acm Trans. Graphic 30(6), 61–64 (2011).
[Crossref]

Li, C.

S. Xie, P. Wang, X. Sang, and C. Li, “Augmented reality three-dimensional display with light field fusion,” Opt. Express 24(11), 11483–11494 (2016).
[Crossref] [PubMed]

P. Wang, S. Xie, X. Sang, D. Chen, C. Li, X. Gao, X. Yu, C. Yu, B. Yan, W. Dou, and L. Xiao, “A large depth of field frontal multi-projection three-dimensional display with uniform light field distribution,” Opt. Commun. 354, 321–329 (2015).
[Crossref]

Martens, M.

R. L. Adler, B. P. Kitchens, M. Martens, C. P. Tresser, and C. W. Wu, “The mathematics of halftoning,” IBM J. Res. Develop. 47(1), 5–15 (2003).
[Crossref]

Raskar, R.

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” Acm Trans. Graphic 30(6), 61–64 (2011).
[Crossref]

Sakamoto, S.

S. Sakamoto and Y. Takaki, “Three-dimensional print using a one-dimensional screen technique,” Jpn. J. Appl. Phys. 47(7), 5486–5492 (2008).
[Crossref]

Sang, X.

S. Xie, P. Wang, X. Sang, and C. Li, “Augmented reality three-dimensional display with light field fusion,” Opt. Express 24(11), 11483–11494 (2016).
[Crossref] [PubMed]

X. Yu, X. Sang, X. Gao, Z. Chen, D. Chen, W. Duan, B. Yan, C. Yu, and D. Xu, “Large viewing angle three-dimensional display with smooth motion parallax and accurate depth cues,” Opt. Express 23(20), 25950–25958 (2015).
[Crossref] [PubMed]

P. Wang, S. Xie, X. Sang, D. Chen, C. Li, X. Gao, X. Yu, C. Yu, B. Yan, W. Dou, and L. Xiao, “A large depth of field frontal multi-projection three-dimensional display with uniform light field distribution,” Opt. Commun. 354, 321–329 (2015).
[Crossref]

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

X. Gao, X. Sang, X. Yu, P. Wang, X. Cao, L. Sun, B. Yan, J. Yuan, K. Wang, C. Yu, and W. Dou, “Aberration analyses for improving the frontal projection three-dimensional display,” Opt. Express 22(19), 23496–23511 (2014).
[Crossref] [PubMed]

Sun, L.

Takaki, Y.

S. Sakamoto and Y. Takaki, “Three-dimensional print using a one-dimensional screen technique,” Jpn. J. Appl. Phys. 47(7), 5486–5492 (2008).
[Crossref]

Tresser, C. P.

R. L. Adler, B. P. Kitchens, M. Martens, C. P. Tresser, and C. W. Wu, “The mathematics of halftoning,” IBM J. Res. Develop. 47(1), 5–15 (2003).
[Crossref]

Wang, K.

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

X. Gao, X. Sang, X. Yu, P. Wang, X. Cao, L. Sun, B. Yan, J. Yuan, K. Wang, C. Yu, and W. Dou, “Aberration analyses for improving the frontal projection three-dimensional display,” Opt. Express 22(19), 23496–23511 (2014).
[Crossref] [PubMed]

Wang, P.

Wetzstein, G.

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” Acm Trans. Graphic 30(6), 61–64 (2011).
[Crossref]

Wu, C. W.

R. L. Adler, B. P. Kitchens, M. Martens, C. P. Tresser, and C. W. Wu, “The mathematics of halftoning,” IBM J. Res. Develop. 47(1), 5–15 (2003).
[Crossref]

Xiao, L.

P. Wang, S. Xie, X. Sang, D. Chen, C. Li, X. Gao, X. Yu, C. Yu, B. Yan, W. Dou, and L. Xiao, “A large depth of field frontal multi-projection three-dimensional display with uniform light field distribution,” Opt. Commun. 354, 321–329 (2015).
[Crossref]

Xie, S.

S. Xie, P. Wang, X. Sang, and C. Li, “Augmented reality three-dimensional display with light field fusion,” Opt. Express 24(11), 11483–11494 (2016).
[Crossref] [PubMed]

P. Wang, S. Xie, X. Sang, D. Chen, C. Li, X. Gao, X. Yu, C. Yu, B. Yan, W. Dou, and L. Xiao, “A large depth of field frontal multi-projection three-dimensional display with uniform light field distribution,” Opt. Commun. 354, 321–329 (2015).
[Crossref]

Xing, S.

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

Xu, D.

Yan, B.

X. Yu, X. Sang, X. Gao, Z. Chen, D. Chen, W. Duan, B. Yan, C. Yu, and D. Xu, “Large viewing angle three-dimensional display with smooth motion parallax and accurate depth cues,” Opt. Express 23(20), 25950–25958 (2015).
[Crossref] [PubMed]

P. Wang, S. Xie, X. Sang, D. Chen, C. Li, X. Gao, X. Yu, C. Yu, B. Yan, W. Dou, and L. Xiao, “A large depth of field frontal multi-projection three-dimensional display with uniform light field distribution,” Opt. Commun. 354, 321–329 (2015).
[Crossref]

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

X. Gao, X. Sang, X. Yu, P. Wang, X. Cao, L. Sun, B. Yan, J. Yuan, K. Wang, C. Yu, and W. Dou, “Aberration analyses for improving the frontal projection three-dimensional display,” Opt. Express 22(19), 23496–23511 (2014).
[Crossref] [PubMed]

Yu, C.

X. Yu, X. Sang, X. Gao, Z. Chen, D. Chen, W. Duan, B. Yan, C. Yu, and D. Xu, “Large viewing angle three-dimensional display with smooth motion parallax and accurate depth cues,” Opt. Express 23(20), 25950–25958 (2015).
[Crossref] [PubMed]

P. Wang, S. Xie, X. Sang, D. Chen, C. Li, X. Gao, X. Yu, C. Yu, B. Yan, W. Dou, and L. Xiao, “A large depth of field frontal multi-projection three-dimensional display with uniform light field distribution,” Opt. Commun. 354, 321–329 (2015).
[Crossref]

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

X. Gao, X. Sang, X. Yu, P. Wang, X. Cao, L. Sun, B. Yan, J. Yuan, K. Wang, C. Yu, and W. Dou, “Aberration analyses for improving the frontal projection three-dimensional display,” Opt. Express 22(19), 23496–23511 (2014).
[Crossref] [PubMed]

Yu, X.

X. Yu, X. Sang, X. Gao, Z. Chen, D. Chen, W. Duan, B. Yan, C. Yu, and D. Xu, “Large viewing angle three-dimensional display with smooth motion parallax and accurate depth cues,” Opt. Express 23(20), 25950–25958 (2015).
[Crossref] [PubMed]

P. Wang, S. Xie, X. Sang, D. Chen, C. Li, X. Gao, X. Yu, C. Yu, B. Yan, W. Dou, and L. Xiao, “A large depth of field frontal multi-projection three-dimensional display with uniform light field distribution,” Opt. Commun. 354, 321–329 (2015).
[Crossref]

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

X. Gao, X. Sang, X. Yu, P. Wang, X. Cao, L. Sun, B. Yan, J. Yuan, K. Wang, C. Yu, and W. Dou, “Aberration analyses for improving the frontal projection three-dimensional display,” Opt. Express 22(19), 23496–23511 (2014).
[Crossref] [PubMed]

Yuan, J.

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

X. Gao, X. Sang, X. Yu, P. Wang, X. Cao, L. Sun, B. Yan, J. Yuan, K. Wang, C. Yu, and W. Dou, “Aberration analyses for improving the frontal projection three-dimensional display,” Opt. Express 22(19), 23496–23511 (2014).
[Crossref] [PubMed]

Zhao, T.

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

Acm Trans. Graphic (1)

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” Acm Trans. Graphic 30(6), 61–64 (2011).
[Crossref]

IBM J. Res. Develop. (1)

R. L. Adler, B. P. Kitchens, M. Martens, C. P. Tresser, and C. W. Wu, “The mathematics of halftoning,” IBM J. Res. Develop. 47(1), 5–15 (2003).
[Crossref]

Jpn. J. Appl. Phys. (1)

S. Sakamoto and Y. Takaki, “Three-dimensional print using a one-dimensional screen technique,” Jpn. J. Appl. Phys. 47(7), 5486–5492 (2008).
[Crossref]

Opt. Commun. (2)

X. Yu, X. Sang, S. Xing, T. Zhao, D. Chen, Y. Cai, B. Yan, K. Wang, J. Yuan, C. Yu, and W. Dou, “Natural three-dimensional display with smooth motion parallax using active partially pixelated masks,” Opt. Commun. 313(4), 146–151 (2014).
[Crossref]

P. Wang, S. Xie, X. Sang, D. Chen, C. Li, X. Gao, X. Yu, C. Yu, B. Yan, W. Dou, and L. Xiao, “A large depth of field frontal multi-projection three-dimensional display with uniform light field distribution,” Opt. Commun. 354, 321–329 (2015).
[Crossref]

Opt. Express (3)

Other (3)

H. Yamazaki and Y. Takaki, “Printing 3D Light Field with 1D Halftone Screening,” SIGGRAPH (2012).

R. A. Ulichney, Digital Halftoning (MIT Press, 1987).

D. L. Lau and G. R. Arce, Modern Digital Halftoning (CRC Press, 2001).

Supplementary Material (1)

NameDescription
» Visualization 1: MP4 (3405 KB)      This is the experiment on viewpoints multiplexing

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1 (a) A gray pixel is represented by an integer grid of binary dots. (b) The imaging process with different view-points. (c) The method of integer grids causes dislocation with the lenticular lens array.
Fig. 2
Fig. 2 (a) A gray pixel is represented by a float grid of partial binary dots. (b) The method of float grids keeps alignment with lenticular lens. (c) Views are printed on float grids. (d) All view-points are printed on float grids and kept alignment with the lenticular lens array.
Fig. 3
Fig. 3 The parameterization of the synthesized image, the simulated image, the halftoned image and the weight matrix.
Fig. 4
Fig. 4 (a) The example of the computation of f m , where w m n is the row float vector of W M × N at row m, mapping N dots to pixel m. (b) The example of the computation of x n , where w M n and e M n are the column float vectors mapping dot n to M pixels.
Fig. 5
Fig. 5 (a) The horizontal black dots are aggregated as lines in the halftone image. (b) The method of random solving sequence generates uniform dot distribution in the halftone image.
Fig. 6
Fig. 6 Three image groups. (a) is the image of “badge”. (b) is the image of “horse”. (c) is the image of “plane”.
Fig. 7
Fig. 7 Simulated and residual images generated by AM screening, random screening and fGDM. (a) The simulation and residues of “badge”. (b) The simulation and residues of “horse”. (c) The simulation and residues of “plane”.
Fig. 8
Fig. 8 PSNR of halftoning images when view-number ranges from 7 to 16 and gray-levels are set from 20 to 48. The white line identifies the area of n x t n x s . The other white line identifies 30dB.
Fig. 9
Fig. 9 Images of a single view generated by different methods. (a) Simulation image generated by AM screening and Random screening. (b) Photos of AM screening. (c) Photos of random screening. (d) The simulation image generated by fGDM. (e) Photos of fGDM. The magenta rectangle is the detail part of images above.
Fig. 10
Fig. 10 Photographs under different view-points at 36 gray levels. (a) The photo based on AM screening under 9 view-points. (b) The photo based on random screening under 9 view-points. (c) The photo based on fGDM under 12 view-points (see Visualization 1).
Fig. 11
Fig. 11 Photographs of fGDM under different view-points at different gray-levels.

Tables (1)

Tables Icon

Table 1 The configuration of each halftoning method.

Equations (9)

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

{ N v i e w = D P I L P I                 L g r a y = D P I L P I + 1
{ n x t = D P I N v i e w × L P I   n x s = L g r a y 1 n y
{ N v i e w = i n t ( D P I n x t × L P I )     L g r a y = i n t ( n x s × n y ) + 1
A ˜ m = i ( n x s × n y ) w i x i = n = 1 N w m n x n
f m = A m A ˜ m 2 = A m w m N x n 2
x n = a r g m i n ( M e m n f m ( x n ) )
X = a r g m i n ( e M n T F M × 1 ( X ) )
x n ' = e M n T ( A M × 1 ( W M × N X N × 1 ) ) w M n T e M n
x n = { 0 ( x n ' > 0 ) d o n o t h i n g ( x n ' = 0 ) 1 ( x n ' < 0 )

Metrics