Abstract

Optical display quality can be degraded by the appearance of moiré pattern occurring in a display system consisting of a basic matrix superimposed with a functional structured optical layer. We propose in this paper a novel pseudo-random arranged color filter array with the table number arranged with an optimal design scenario. We show that the moiré pattern can be significantly reduced with the introduction of the special color filter array. The idea is tested with an experiment that gives rise to a substantially reduced moiré pattern in a display system. It is believed that the novel functional optical structures have significant impact to complex structured display system in general and to the autostereoscopic and integrated display systems in particular.

© 2015 Optical Society of America

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References

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  1. I. Amidror, The Theory of the Moiré Phenomenon, (Springer, 2000), Ch. 1.
  2. W. X. Zhao, Q. H. Wang, A. H. Wang, and D. H. Li, “Autostereoscopic display based on two-layer lenticular lenses,” Opt. Lett. 35(24), 4127–4129 (2010).
    [Crossref] [PubMed]
  3. N. A. Dodgson, “Autostereoscopic 3D displays,” Computer 38(8), 31–36 (2005).
    [Crossref]
  4. V. Saveljev and S. K. Kim, “Simulation and measurement of moiré patterns at finite distance,” Opt. Express 20(3), 2163–2177 (2012).
    [Crossref] [PubMed]
  5. S. J. Byun, S. Y. Byun, J. Lee, W. M. Kim, H. P. Kim, M. Y. Jeon, and T. S. Lee, “An efficient simulation and analysis method of moiré patterns in display systems,” Opt. Express 22(3), 3128–3136 (2014).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  9. V. V. Saveljev, J. Y. Son, J. H. Chun, and K. D. Kwack, “About a Moiré-less condition for non-square grids,” J. Disp. Technol. 4(3), 332–339 (2008).
    [Crossref]
  10. Y. Kim, G. Park, J.-H. Jung, J. Kim, and B. Lee, “Color moiré pattern simulation and analysis in three-dimensional integral imaging for finding the moiré-reduced tilted angle of a lens array,” Appl. Opt. 48(11), 2178–2187 (2009).
    [Crossref] [PubMed]
  11. L. Kong, G. Jin, T. Wang, S. Cai, X. Zhong, and K. Xu, “Parameter design of a parallax barrier based on the color moiré patterns in autostereoscopic display,” Appl. Opt. 50(34), H153–H158 (2011).
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  13. S. T. Hodgkin, M. J. Irwin, P. C. Hewett, and S. J. Warren, “The UKIRT wide field camera ZYJHK photometric system: calibration from 2MASS,” Mon. Not. R. Astron. Soc. 394(2), 675–692 (2009).
    [Crossref]
  14. J. Wang, H. Liang, H. Fan, Y. Zhou, P. Krebs, J. Su, Y. Deng, and J. Zhou, “High-quality autostereoscopic display with spatial and sequential hybrid control,” Appl. Opt. 52(35), 8549–8553 (2013).
    [Crossref] [PubMed]
  15. L. Condat, “Color filter array design using random patterns with blue noise chromatic spectra,” Image Vis. Comput. 28(8), 1196–1202 (2010).
    [Crossref]
  16. Y. Zhou, P. Krebs, H. Fan, H. Liang, J. Su, J. Wang, and J. Zhou, “Quantitative measurement and control of optical Moiré pattern in an autostereoscopic liquid crystal display system,” Appl. Opt. 54(6), 1521–1527 (2015).
    [Crossref] [PubMed]
  17. H. Liang, S. An, J. Wang, Y. Zhou, H. Fan, P. Krebs, and J. Zhou, “Optimizing time-multiplexing auto-stereoscopic displays with a genetic algorithm,” J. Disp. Technol. 10(8), 695–699 (2014).
    [Crossref]
  18. H. Fan, Y. Zhou, J. Wang, H. Liang, P. Krebs, J. Su, D. Lin, K. Li, and J. Zhou, “Full resolution, low crosstalk and wide viewing angle auto-stereoscopic display with a backlight unit on free-form surface,” J. Disp. Technol. 11(7), 620–624 (2015).
    [Crossref]
  19. B. O’Brien, “Vision and resolution in the central retina,” J. Opt. Soc. Am. 41(12), 882–894 (1951).
    [Crossref] [PubMed]
  20. P. G. J. Barten, Contrast Sensitivity of the Human Eye and its Effects on Image Quality (SPIE press, 1999), Ch. 3.

2015 (2)

H. Fan, Y. Zhou, J. Wang, H. Liang, P. Krebs, J. Su, D. Lin, K. Li, and J. Zhou, “Full resolution, low crosstalk and wide viewing angle auto-stereoscopic display with a backlight unit on free-form surface,” J. Disp. Technol. 11(7), 620–624 (2015).
[Crossref]

Y. Zhou, P. Krebs, H. Fan, H. Liang, J. Su, J. Wang, and J. Zhou, “Quantitative measurement and control of optical Moiré pattern in an autostereoscopic liquid crystal display system,” Appl. Opt. 54(6), 1521–1527 (2015).
[Crossref] [PubMed]

2014 (2)

S. J. Byun, S. Y. Byun, J. Lee, W. M. Kim, H. P. Kim, M. Y. Jeon, and T. S. Lee, “An efficient simulation and analysis method of moiré patterns in display systems,” Opt. Express 22(3), 3128–3136 (2014).
[Crossref] [PubMed]

H. Liang, S. An, J. Wang, Y. Zhou, H. Fan, P. Krebs, and J. Zhou, “Optimizing time-multiplexing auto-stereoscopic displays with a genetic algorithm,” J. Disp. Technol. 10(8), 695–699 (2014).
[Crossref]

2013 (2)

2012 (2)

2011 (1)

2010 (2)

W. X. Zhao, Q. H. Wang, A. H. Wang, and D. H. Li, “Autostereoscopic display based on two-layer lenticular lenses,” Opt. Lett. 35(24), 4127–4129 (2010).
[Crossref] [PubMed]

L. Condat, “Color filter array design using random patterns with blue noise chromatic spectra,” Image Vis. Comput. 28(8), 1196–1202 (2010).
[Crossref]

2009 (2)

S. T. Hodgkin, M. J. Irwin, P. C. Hewett, and S. J. Warren, “The UKIRT wide field camera ZYJHK photometric system: calibration from 2MASS,” Mon. Not. R. Astron. Soc. 394(2), 675–692 (2009).
[Crossref]

Y. Kim, G. Park, J.-H. Jung, J. Kim, and B. Lee, “Color moiré pattern simulation and analysis in three-dimensional integral imaging for finding the moiré-reduced tilted angle of a lens array,” Appl. Opt. 48(11), 2178–2187 (2009).
[Crossref] [PubMed]

2008 (1)

V. V. Saveljev, J. Y. Son, J. H. Chun, and K. D. Kwack, “About a Moiré-less condition for non-square grids,” J. Disp. Technol. 4(3), 332–339 (2008).
[Crossref]

2005 (2)

N. A. Dodgson, “Autostereoscopic 3D displays,” Computer 38(8), 31–36 (2005).
[Crossref]

V. V. Saveljev, J. Y. Son, B. Javidi, S. K. Kim, and D. S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Disp. Technol. 1(2), 347–353 (2005).
[Crossref]

1998 (1)

1951 (1)

An, S.

H. Liang, S. An, J. Wang, Y. Zhou, H. Fan, P. Krebs, and J. Zhou, “Optimizing time-multiplexing auto-stereoscopic displays with a genetic algorithm,” J. Disp. Technol. 10(8), 695–699 (2014).
[Crossref]

Byun, S. J.

Byun, S. Y.

Cai, S.

Chun, J. H.

V. V. Saveljev, J. Y. Son, J. H. Chun, and K. D. Kwack, “About a Moiré-less condition for non-square grids,” J. Disp. Technol. 4(3), 332–339 (2008).
[Crossref]

Condat, L.

L. Condat, “Color filter array design using random patterns with blue noise chromatic spectra,” Image Vis. Comput. 28(8), 1196–1202 (2010).
[Crossref]

Deng, Y.

Dodgson, N. A.

N. A. Dodgson, “Autostereoscopic 3D displays,” Computer 38(8), 31–36 (2005).
[Crossref]

Fan, H.

H. Fan, Y. Zhou, J. Wang, H. Liang, P. Krebs, J. Su, D. Lin, K. Li, and J. Zhou, “Full resolution, low crosstalk and wide viewing angle auto-stereoscopic display with a backlight unit on free-form surface,” J. Disp. Technol. 11(7), 620–624 (2015).
[Crossref]

Y. Zhou, P. Krebs, H. Fan, H. Liang, J. Su, J. Wang, and J. Zhou, “Quantitative measurement and control of optical Moiré pattern in an autostereoscopic liquid crystal display system,” Appl. Opt. 54(6), 1521–1527 (2015).
[Crossref] [PubMed]

H. Liang, S. An, J. Wang, Y. Zhou, H. Fan, P. Krebs, and J. Zhou, “Optimizing time-multiplexing auto-stereoscopic displays with a genetic algorithm,” J. Disp. Technol. 10(8), 695–699 (2014).
[Crossref]

J. Wang, H. Liang, H. Fan, Y. Zhou, P. Krebs, J. Su, Y. Deng, and J. Zhou, “High-quality autostereoscopic display with spatial and sequential hybrid control,” Appl. Opt. 52(35), 8549–8553 (2013).
[Crossref] [PubMed]

Hewett, P. C.

S. T. Hodgkin, M. J. Irwin, P. C. Hewett, and S. J. Warren, “The UKIRT wide field camera ZYJHK photometric system: calibration from 2MASS,” Mon. Not. R. Astron. Soc. 394(2), 675–692 (2009).
[Crossref]

Hodgkin, S. T.

S. T. Hodgkin, M. J. Irwin, P. C. Hewett, and S. J. Warren, “The UKIRT wide field camera ZYJHK photometric system: calibration from 2MASS,” Mon. Not. R. Astron. Soc. 394(2), 675–692 (2009).
[Crossref]

Irwin, M. J.

S. T. Hodgkin, M. J. Irwin, P. C. Hewett, and S. J. Warren, “The UKIRT wide field camera ZYJHK photometric system: calibration from 2MASS,” Mon. Not. R. Astron. Soc. 394(2), 675–692 (2009).
[Crossref]

Javidi, B.

V. V. Saveljev, J. Y. Son, B. Javidi, S. K. Kim, and D. S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Disp. Technol. 1(2), 347–353 (2005).
[Crossref]

Jeon, M. Y.

Jin, G.

Jung, J.-H.

Kakeya, H.

Kim, D. S.

V. V. Saveljev, J. Y. Son, B. Javidi, S. K. Kim, and D. S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Disp. Technol. 1(2), 347–353 (2005).
[Crossref]

Kim, H. P.

Kim, J.

Kim, S. K.

V. Saveljev and S. K. Kim, “Simulation and measurement of moiré patterns at finite distance,” Opt. Express 20(3), 2163–2177 (2012).
[Crossref] [PubMed]

V. V. Saveljev, J. Y. Son, B. Javidi, S. K. Kim, and D. S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Disp. Technol. 1(2), 347–353 (2005).
[Crossref]

Kim, W. M.

Kim, Y.

Kong, L.

Krebs, P.

H. Fan, Y. Zhou, J. Wang, H. Liang, P. Krebs, J. Su, D. Lin, K. Li, and J. Zhou, “Full resolution, low crosstalk and wide viewing angle auto-stereoscopic display with a backlight unit on free-form surface,” J. Disp. Technol. 11(7), 620–624 (2015).
[Crossref]

Y. Zhou, P. Krebs, H. Fan, H. Liang, J. Su, J. Wang, and J. Zhou, “Quantitative measurement and control of optical Moiré pattern in an autostereoscopic liquid crystal display system,” Appl. Opt. 54(6), 1521–1527 (2015).
[Crossref] [PubMed]

H. Liang, S. An, J. Wang, Y. Zhou, H. Fan, P. Krebs, and J. Zhou, “Optimizing time-multiplexing auto-stereoscopic displays with a genetic algorithm,” J. Disp. Technol. 10(8), 695–699 (2014).
[Crossref]

J. Wang, H. Liang, H. Fan, Y. Zhou, P. Krebs, J. Su, Y. Deng, and J. Zhou, “High-quality autostereoscopic display with spatial and sequential hybrid control,” Appl. Opt. 52(35), 8549–8553 (2013).
[Crossref] [PubMed]

Kurokawa, T.

Kwack, K. D.

V. V. Saveljev, J. Y. Son, J. H. Chun, and K. D. Kwack, “About a Moiré-less condition for non-square grids,” J. Disp. Technol. 4(3), 332–339 (2008).
[Crossref]

Lee, B.

Lee, J.

Lee, T. S.

Li, D. H.

Li, K.

H. Fan, Y. Zhou, J. Wang, H. Liang, P. Krebs, J. Su, D. Lin, K. Li, and J. Zhou, “Full resolution, low crosstalk and wide viewing angle auto-stereoscopic display with a backlight unit on free-form surface,” J. Disp. Technol. 11(7), 620–624 (2015).
[Crossref]

Liang, H.

H. Fan, Y. Zhou, J. Wang, H. Liang, P. Krebs, J. Su, D. Lin, K. Li, and J. Zhou, “Full resolution, low crosstalk and wide viewing angle auto-stereoscopic display with a backlight unit on free-form surface,” J. Disp. Technol. 11(7), 620–624 (2015).
[Crossref]

Y. Zhou, P. Krebs, H. Fan, H. Liang, J. Su, J. Wang, and J. Zhou, “Quantitative measurement and control of optical Moiré pattern in an autostereoscopic liquid crystal display system,” Appl. Opt. 54(6), 1521–1527 (2015).
[Crossref] [PubMed]

H. Liang, S. An, J. Wang, Y. Zhou, H. Fan, P. Krebs, and J. Zhou, “Optimizing time-multiplexing auto-stereoscopic displays with a genetic algorithm,” J. Disp. Technol. 10(8), 695–699 (2014).
[Crossref]

J. Wang, H. Liang, H. Fan, Y. Zhou, P. Krebs, J. Su, Y. Deng, and J. Zhou, “High-quality autostereoscopic display with spatial and sequential hybrid control,” Appl. Opt. 52(35), 8549–8553 (2013).
[Crossref] [PubMed]

Lin, D.

H. Fan, Y. Zhou, J. Wang, H. Liang, P. Krebs, J. Su, D. Lin, K. Li, and J. Zhou, “Full resolution, low crosstalk and wide viewing angle auto-stereoscopic display with a backlight unit on free-form surface,” J. Disp. Technol. 11(7), 620–624 (2015).
[Crossref]

Masters, B.

O’Brien, B.

Park, G.

Saveljev, V.

Saveljev, V. V.

V. V. Saveljev, J. Y. Son, J. H. Chun, and K. D. Kwack, “About a Moiré-less condition for non-square grids,” J. Disp. Technol. 4(3), 332–339 (2008).
[Crossref]

V. V. Saveljev, J. Y. Son, B. Javidi, S. K. Kim, and D. S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Disp. Technol. 1(2), 347–353 (2005).
[Crossref]

Sawada, S.

Son, J. Y.

V. V. Saveljev, J. Y. Son, J. H. Chun, and K. D. Kwack, “About a Moiré-less condition for non-square grids,” J. Disp. Technol. 4(3), 332–339 (2008).
[Crossref]

V. V. Saveljev, J. Y. Son, B. Javidi, S. K. Kim, and D. S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Disp. Technol. 1(2), 347–353 (2005).
[Crossref]

Su, J.

Ueda, Y.

Wang, A. H.

Wang, J.

H. Fan, Y. Zhou, J. Wang, H. Liang, P. Krebs, J. Su, D. Lin, K. Li, and J. Zhou, “Full resolution, low crosstalk and wide viewing angle auto-stereoscopic display with a backlight unit on free-form surface,” J. Disp. Technol. 11(7), 620–624 (2015).
[Crossref]

Y. Zhou, P. Krebs, H. Fan, H. Liang, J. Su, J. Wang, and J. Zhou, “Quantitative measurement and control of optical Moiré pattern in an autostereoscopic liquid crystal display system,” Appl. Opt. 54(6), 1521–1527 (2015).
[Crossref] [PubMed]

H. Liang, S. An, J. Wang, Y. Zhou, H. Fan, P. Krebs, and J. Zhou, “Optimizing time-multiplexing auto-stereoscopic displays with a genetic algorithm,” J. Disp. Technol. 10(8), 695–699 (2014).
[Crossref]

J. Wang, H. Liang, H. Fan, Y. Zhou, P. Krebs, J. Su, Y. Deng, and J. Zhou, “High-quality autostereoscopic display with spatial and sequential hybrid control,” Appl. Opt. 52(35), 8549–8553 (2013).
[Crossref] [PubMed]

Wang, Q. H.

Wang, T.

Warren, S. J.

S. T. Hodgkin, M. J. Irwin, P. C. Hewett, and S. J. Warren, “The UKIRT wide field camera ZYJHK photometric system: calibration from 2MASS,” Mon. Not. R. Astron. Soc. 394(2), 675–692 (2009).
[Crossref]

Xu, K.

Zhao, W. X.

Zhong, X.

Zhou, J.

H. Fan, Y. Zhou, J. Wang, H. Liang, P. Krebs, J. Su, D. Lin, K. Li, and J. Zhou, “Full resolution, low crosstalk and wide viewing angle auto-stereoscopic display with a backlight unit on free-form surface,” J. Disp. Technol. 11(7), 620–624 (2015).
[Crossref]

Y. Zhou, P. Krebs, H. Fan, H. Liang, J. Su, J. Wang, and J. Zhou, “Quantitative measurement and control of optical Moiré pattern in an autostereoscopic liquid crystal display system,” Appl. Opt. 54(6), 1521–1527 (2015).
[Crossref] [PubMed]

H. Liang, S. An, J. Wang, Y. Zhou, H. Fan, P. Krebs, and J. Zhou, “Optimizing time-multiplexing auto-stereoscopic displays with a genetic algorithm,” J. Disp. Technol. 10(8), 695–699 (2014).
[Crossref]

J. Wang, H. Liang, H. Fan, Y. Zhou, P. Krebs, J. Su, Y. Deng, and J. Zhou, “High-quality autostereoscopic display with spatial and sequential hybrid control,” Appl. Opt. 52(35), 8549–8553 (2013).
[Crossref] [PubMed]

Zhou, Y.

H. Fan, Y. Zhou, J. Wang, H. Liang, P. Krebs, J. Su, D. Lin, K. Li, and J. Zhou, “Full resolution, low crosstalk and wide viewing angle auto-stereoscopic display with a backlight unit on free-form surface,” J. Disp. Technol. 11(7), 620–624 (2015).
[Crossref]

Y. Zhou, P. Krebs, H. Fan, H. Liang, J. Su, J. Wang, and J. Zhou, “Quantitative measurement and control of optical Moiré pattern in an autostereoscopic liquid crystal display system,” Appl. Opt. 54(6), 1521–1527 (2015).
[Crossref] [PubMed]

H. Liang, S. An, J. Wang, Y. Zhou, H. Fan, P. Krebs, and J. Zhou, “Optimizing time-multiplexing auto-stereoscopic displays with a genetic algorithm,” J. Disp. Technol. 10(8), 695–699 (2014).
[Crossref]

J. Wang, H. Liang, H. Fan, Y. Zhou, P. Krebs, J. Su, Y. Deng, and J. Zhou, “High-quality autostereoscopic display with spatial and sequential hybrid control,” Appl. Opt. 52(35), 8549–8553 (2013).
[Crossref] [PubMed]

Appl. Opt. (4)

Computer (1)

N. A. Dodgson, “Autostereoscopic 3D displays,” Computer 38(8), 31–36 (2005).
[Crossref]

Image Vis. Comput. (1)

L. Condat, “Color filter array design using random patterns with blue noise chromatic spectra,” Image Vis. Comput. 28(8), 1196–1202 (2010).
[Crossref]

J. Disp. Technol. (4)

H. Liang, S. An, J. Wang, Y. Zhou, H. Fan, P. Krebs, and J. Zhou, “Optimizing time-multiplexing auto-stereoscopic displays with a genetic algorithm,” J. Disp. Technol. 10(8), 695–699 (2014).
[Crossref]

H. Fan, Y. Zhou, J. Wang, H. Liang, P. Krebs, J. Su, D. Lin, K. Li, and J. Zhou, “Full resolution, low crosstalk and wide viewing angle auto-stereoscopic display with a backlight unit on free-form surface,” J. Disp. Technol. 11(7), 620–624 (2015).
[Crossref]

V. V. Saveljev, J. Y. Son, B. Javidi, S. K. Kim, and D. S. Kim, “Moiré minimization condition in three-dimensional image displays,” J. Disp. Technol. 1(2), 347–353 (2005).
[Crossref]

V. V. Saveljev, J. Y. Son, J. H. Chun, and K. D. Kwack, “About a Moiré-less condition for non-square grids,” J. Disp. Technol. 4(3), 332–339 (2008).
[Crossref]

J. Opt. Soc. Am. (1)

Mon. Not. R. Astron. Soc. (1)

S. T. Hodgkin, M. J. Irwin, P. C. Hewett, and S. J. Warren, “The UKIRT wide field camera ZYJHK photometric system: calibration from 2MASS,” Mon. Not. R. Astron. Soc. 394(2), 675–692 (2009).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Other (2)

P. G. J. Barten, Contrast Sensitivity of the Human Eye and its Effects on Image Quality (SPIE press, 1999), Ch. 3.

I. Amidror, The Theory of the Moiré Phenomenon, (Springer, 2000), Ch. 1.

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Figures (8)

Fig. 1
Fig. 1 (a) The structure of LCD panel consists of a trio-color filter; (b) The scheme of 1-fold binary grating that represent red color in (a); (c) equivalent binary grating corresponding to one color shown in (b) is shown as a transmittance function (c).
Fig. 2
Fig. 2 Notation of position of the sub pixel.
Fig. 3
Fig. 3 Optimum arrangement of sub pixel (here first 20 pixels is schemed).
Fig. 4
Fig. 4 Average intensity si of the ordinary RGB arranged color filter and the optimum arranged one. The dot line represents the ordinary one, and the solid line shows the optimum one. (a) the average intensity of red sub pixel; (b) the Fourier transform of red moiré; (c) si of green sub pixel; (d) Fourier spectrum of green one; (e) si of blue sub pixel; (f) Fourier spectrum of blue one.
Fig. 5
Fig. 5 Visual effect of the superposing of the binary grating and the CF (100 pixels). (a) Ordinary RGB arrangement; (b) re-arranged sub pixels.
Fig. 6
Fig. 6 Experiment gratings. (a) printed binary grating, T2 = 5 mm, opening ratio = 0.8; (b) ordinary RGB pixels arrangement, T1 = 2.6 mm, displayed by computer; (c) optimum RGB sub pixel arrangement with the same pitch of (b), the yellow dash line indicates that the left part of the figure is superposing with the printed binary grating while the right part does not.
Fig. 7
Fig. 7 Superposition of binary grating and the ordinary RGB sub pixel arrangement.
Fig. 8
Fig. 8 The visual effect of the superposition of binary grating and the optimum RGB arrangement. Left part covers superposition (corresponding to the left part from the yellow dash line in Fig. 6(c)), and the right part is the original image without superposition.

Tables (1)

Tables Icon

Table 1 Optimum randomized sub pixel position (first 20 pixels are shown here)

Equations (7)

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

t 1,1 (x)={ 1n T 1 x<n T 1 + 1 3 T 1 0n T 1 + 1 3 T 1 x<( n+1 ) T 1 n=0,1,2,3
t 1,2 (x)={ 0n T 1 x<n T 1 + 1 3 T 1 1n T 1 + 1 3 T 1 x<n T 1 + 2 3 T 1 0n T 1 + 2 3 T 1 x<( n+1 ) T 1 n=0,1,2,3
t 1,3 (x)={ 0n T 1 x<n T 1 + 2 3 T 1 1n T 1 + 2 3 T 1 x<( n+1 ) T 1 n=0,1,2,3
t 2 (x)={ 1n T 2 x<n T 2 + τ 2 0n T 2 + τ 2 x<( n+1 ) T 2 n=0,1,2,3
t(x)= t 1 (x) t 2 (x)
s i (x)= 1 η T 1 xη T 1 /2 x+η T 1 /2 t(x') dx'
m i = max( s i )min( s i ) max( s i )

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