Abstract

The moiré effect was considered at a finite distance. Formulas for the wavenumber and phase of the visible moiré patterns were found with displacements of the observer and of gratings taken into account. The computer simulation and physical experiment confirm the theory. The method of measurement of the wavevectors of plane waves was proposed basing on the Radon and Fourier transformations.

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References

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  1. G. Bell, R. Craig, and T. Simmiss, “Moiré interference in multilayered displays,” J. Soc. Inf. Disp. 15(11), 883–888 (2007).
    [CrossRef]
  2. R. Martınez-Cuenca, G. Saavedra, M. Martınez-Corral, and B. Javidi, “Progress in 3-D multiperspective display by integral imaging,” Proc. IEEE 97(6), 1067–1077 (2009).
    [CrossRef]
  3. Y. Zhu and T. Zhen, “3D multi-view autostereoscopic display and its key technologie,” in Proceedings of Asia-Pacific Conference on Information Processing (2009), 31 – 35.
  4. I. Amidror, The Theory of the Moiré Phenomenon, Vol. I: Periodic Layers, 2nd ed. (Springer-Verlag, London, UK, 2009) Chaps. 2, 7.
  5. T. Koike, M. Oikawa, and K. Utsugi, “Moiré reduction for integral videography,” in Proceedings of IDW (2006) 1917 - 1918.
  6. K. Oku, Y. Tomizuka, and Y. Tanaka, “Analysis and reduction of moiré in two-layered 3D display,” in SID Symposium Digest (2007) 38, 437–440.
  7. 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]
  8. N. J. Wade, “Vasarely’s vision,” Perception 33, (supplement [27-th European Conference on Visual Perception]), 84 (2004).
  9. K. Creath and J. C. Wyant, “Moiré and fringe projection techniques,” in Optical Shop Testing, Second Edition, (John Wiley & Sons, New York, USA, 1995), 653 - 685.
  10. C. Chiang, “Moiré topography,” Appl. Opt. 14(1), 177–179 (1975).
    [PubMed]
  11. R. E. McCurry, “Multiple source moiré patterns,” J. Appl. Phys. 37(2), 467–472 (1966).
    [CrossRef]
  12. C. A. Sciammarella and F.-P. Chiang, “Gap effect on moiré patterns,” Z. Angew. Math. Phys. 19(2), 326–333 (1968) (ZAMP).
    [CrossRef]
  13. V. Saveljev, “Characteristics of moiré spectra in autostereoscopic three-dimensional displays,” J. Disp. Technol. 7(5), 259–266 (2011).
    [CrossRef]
  14. V. Saveljev and S.-K. Kim, “Simulation of moiré effect in 3D displays,” J. Opt. Soc. Korea 14(4), 310–315 (2010).
    [CrossRef]
  15. L. O. Vargady, “Moiré fringes as visual position indicators,” Appl. Opt. 3(5), 631–636 (1964).
    [CrossRef]
  16. S. R. Deans, The Radon Transform and Some of Its Applications (John Wiley and Sons, Inc., New York, 1983).

2011 (1)

V. Saveljev, “Characteristics of moiré spectra in autostereoscopic three-dimensional displays,” J. Disp. Technol. 7(5), 259–266 (2011).
[CrossRef]

2010 (1)

2009 (2)

2007 (1)

G. Bell, R. Craig, and T. Simmiss, “Moiré interference in multilayered displays,” J. Soc. Inf. Disp. 15(11), 883–888 (2007).
[CrossRef]

1975 (1)

1968 (1)

C. A. Sciammarella and F.-P. Chiang, “Gap effect on moiré patterns,” Z. Angew. Math. Phys. 19(2), 326–333 (1968) (ZAMP).
[CrossRef]

1966 (1)

R. E. McCurry, “Multiple source moiré patterns,” J. Appl. Phys. 37(2), 467–472 (1966).
[CrossRef]

1964 (1)

Bell, G.

G. Bell, R. Craig, and T. Simmiss, “Moiré interference in multilayered displays,” J. Soc. Inf. Disp. 15(11), 883–888 (2007).
[CrossRef]

Chiang, C.

Chiang, F.-P.

C. A. Sciammarella and F.-P. Chiang, “Gap effect on moiré patterns,” Z. Angew. Math. Phys. 19(2), 326–333 (1968) (ZAMP).
[CrossRef]

Craig, R.

G. Bell, R. Craig, and T. Simmiss, “Moiré interference in multilayered displays,” J. Soc. Inf. Disp. 15(11), 883–888 (2007).
[CrossRef]

Javidi, B.

R. Martınez-Cuenca, G. Saavedra, M. Martınez-Corral, and B. Javidi, “Progress in 3-D multiperspective display by integral imaging,” Proc. IEEE 97(6), 1067–1077 (2009).
[CrossRef]

Jung, J.-H.

Kim, J.

Kim, S.-K.

Kim, Y.

Lee, B.

Martinez-Corral, M.

R. Martınez-Cuenca, G. Saavedra, M. Martınez-Corral, and B. Javidi, “Progress in 3-D multiperspective display by integral imaging,” Proc. IEEE 97(6), 1067–1077 (2009).
[CrossRef]

Martinez-Cuenca, R.

R. Martınez-Cuenca, G. Saavedra, M. Martınez-Corral, and B. Javidi, “Progress in 3-D multiperspective display by integral imaging,” Proc. IEEE 97(6), 1067–1077 (2009).
[CrossRef]

McCurry, R. E.

R. E. McCurry, “Multiple source moiré patterns,” J. Appl. Phys. 37(2), 467–472 (1966).
[CrossRef]

Park, G.

Saavedra, G.

R. Martınez-Cuenca, G. Saavedra, M. Martınez-Corral, and B. Javidi, “Progress in 3-D multiperspective display by integral imaging,” Proc. IEEE 97(6), 1067–1077 (2009).
[CrossRef]

Saveljev, V.

V. Saveljev, “Characteristics of moiré spectra in autostereoscopic three-dimensional displays,” J. Disp. Technol. 7(5), 259–266 (2011).
[CrossRef]

V. Saveljev and S.-K. Kim, “Simulation of moiré effect in 3D displays,” J. Opt. Soc. Korea 14(4), 310–315 (2010).
[CrossRef]

Sciammarella, C. A.

C. A. Sciammarella and F.-P. Chiang, “Gap effect on moiré patterns,” Z. Angew. Math. Phys. 19(2), 326–333 (1968) (ZAMP).
[CrossRef]

Simmiss, T.

G. Bell, R. Craig, and T. Simmiss, “Moiré interference in multilayered displays,” J. Soc. Inf. Disp. 15(11), 883–888 (2007).
[CrossRef]

Vargady, L. O.

Appl. Opt. (3)

J. Appl. Phys. (1)

R. E. McCurry, “Multiple source moiré patterns,” J. Appl. Phys. 37(2), 467–472 (1966).
[CrossRef]

J. Disp. Technol. (1)

V. Saveljev, “Characteristics of moiré spectra in autostereoscopic three-dimensional displays,” J. Disp. Technol. 7(5), 259–266 (2011).
[CrossRef]

J. Opt. Soc. Korea (1)

J. Soc. Inf. Disp. (1)

G. Bell, R. Craig, and T. Simmiss, “Moiré interference in multilayered displays,” J. Soc. Inf. Disp. 15(11), 883–888 (2007).
[CrossRef]

Proc. IEEE (1)

R. Martınez-Cuenca, G. Saavedra, M. Martınez-Corral, and B. Javidi, “Progress in 3-D multiperspective display by integral imaging,” Proc. IEEE 97(6), 1067–1077 (2009).
[CrossRef]

Z. Angew. Math. Phys. (1)

C. A. Sciammarella and F.-P. Chiang, “Gap effect on moiré patterns,” Z. Angew. Math. Phys. 19(2), 326–333 (1968) (ZAMP).
[CrossRef]

Other (7)

Y. Zhu and T. Zhen, “3D multi-view autostereoscopic display and its key technologie,” in Proceedings of Asia-Pacific Conference on Information Processing (2009), 31 – 35.

I. Amidror, The Theory of the Moiré Phenomenon, Vol. I: Periodic Layers, 2nd ed. (Springer-Verlag, London, UK, 2009) Chaps. 2, 7.

T. Koike, M. Oikawa, and K. Utsugi, “Moiré reduction for integral videography,” in Proceedings of IDW (2006) 1917 - 1918.

K. Oku, Y. Tomizuka, and Y. Tanaka, “Analysis and reduction of moiré in two-layered 3D display,” in SID Symposium Digest (2007) 38, 437–440.

N. J. Wade, “Vasarely’s vision,” Perception 33, (supplement [27-th European Conference on Visual Perception]), 84 (2004).

K. Creath and J. C. Wyant, “Moiré and fringe projection techniques,” in Optical Shop Testing, Second Edition, (John Wiley & Sons, New York, USA, 1995), 653 - 685.

S. R. Deans, The Radon Transform and Some of Its Applications (John Wiley and Sons, Inc., New York, 1983).

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Figures (14)

Fig. 1
Fig. 1

Layout of gratings (used in simulation and experiment). Camera is installed at z0, the gratings at z1 = 0 and z2 = -d.

Fig. 2
Fig. 2

Symmetrization of image before Radon transformation.

Fig. 3
Fig. 3

Radon transformation of symmetrized image. (The same angles are labeled through Figs. 35(a)).

Fig. 4
Fig. 4

Series of one-dimensional Fourier transformations.

Fig. 5
Fig. 5

Two one-dimensional representations of two-dimensional F(k, j): (a) Radon transformation with a fixed angle (i.e., sum of rows). (b) column of F(k, j0) at one of the angles j0.

Fig. 6
Fig. 6

Experimental photographs of moiré patterns for distance 118 cm: (a) gap 4.25 cm; (b) gap 2.75 cm.

Fig. 7
Fig. 7

Radon transformations of the same image rotated by 37° (a) and 78° (b).

Fig. 8
Fig. 8

Two Radon transformationss of Fourier-Radon transforms for 37° and 78°.

Fig. 9
Fig. 9

Layout of wedge experiment: (a) layout (along lines l1 and l2 the gap is 0.5 cm and 2.75 cm); (b) experimental setup (the planes of the transparent cover glasses are shown by bold lines). N.B. To take this picture, the camera was installed essentially lower than 110 cm.

Fig. 10
Fig. 10

Experimental photographs for wedge experiment. The angle between layers is 9.4°. The camera height is 100 cm, while its lateral displacements are −5, 0, and + 5 cm in (a), (b), and (c), resp.

Fig. 11
Fig. 11

Experimental graphs of profiles of the wedge photographs along the lines l1 (a) and l2 (b). The period of the envelope line correspond to the period of the moiré waves.

Fig. 12
Fig. 12

Geometry of moiré patters depending on ratio ρ, two series of experiments with ρ = 1 and with ρ ≠ 1 (λ1 = 1 mm): (a) period, (b) relative displacement. The dashed line indicates s approximately. For displacement, all theoretical values of the first series with ρ = 1 lye at (1, 1), see Eq. (15).

Fig. 13
Fig. 13

The moiré patterns in an 3D display at the optimal distance: (a) experimental photograph; (b) simulation.

Fig. 14
Fig. 14

Apparent displacement of moiré patterns.

Equations (20)

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G 1 =cos k 1 x=cos 2π λ 1 x,
G 2 =cos k 2 x=cos 2π ρ λ 1 x.
M=( 1 0 x c z c 0 0 1 y c z c 0 0 0 1 0 0 0 1 z c 1 ).
λ 2 '= ρ s λ 1 ,
s=1+ d z c .
x 2 '= d s z c x c .
φ 2 '=2π x 2 ' λ 2 ' =2π x c d z c ρ λ 1 .
G 2 '=cos( k 2 'x+ φ 2 ' )=cos( 2π ρ λ 1 ( sx+ d z c x c ) ).
k m =| 2π λ 2 ' 2π λ 1 |= 2π λ 1 | sρ | ρ ,
λ m = ρ | sρ | λ 1 .
λ m 0 = z c d λ 1 .
x m λ m = x 1 λ 1 ,
G m =cos( k m x+ φ m )=cos( 2π λ 1 | sρ | ρ x+ 2π ρ λ 1 x c d z c ).
x m (0) = s1 sρ x c .
x m = x c .
x m (1) = ρ sρ x 1 .
x m (2) = 1 sρ x 2 .
x m = x m (0) + x m (1) + x m (2) = ( s1 ) x c +ρ x 1 x 2 sρ .
λ m y = 1 | sρ | λ 2 y ,
x m y = s1 sρ y 0 .

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