Abstract

Equations for spectral peaks and trajectories are found for N superposed one-dimensional gratings. The equations of trajectories are represented using the complex numbers. The number of geometric elements in the spectrum is found under various conditions and in the matrix form. The derivatives of trajectories are obtained. The orthogonal case is investigated in details, in particular, the regular structures (the square and the octagon) are found in the spectrum. The numerical simulation is in a good agreement with the theory. The proposed technique seems to be helpful in estimation of occurrence of moiré patterns in visual displays.

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References

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  1. I. Amidror, The Theory of Moiré Phenomenon (Springer, 2009) vol. 1.
  2. O. Bryngdahl, “Moiré: Formation and interpretation,” J. Opt. Soc. Am.64(10), 1287–1294 (1974).
    [CrossRef]
  3. O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am.66(2), 87–94 (1976).
    [CrossRef]
  4. K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express19(27), 26065–26078 (2011).
    [CrossRef] [PubMed]
  5. S. Yokozeki, Y. Kusaka, and K. Patorski, “Geometric parameters of moiré fringes,” Appl. Opt.15(9), 2223–2227 (1976).
    [CrossRef] [PubMed]
  6. I. Amidror, S. Chosson, and R. D. Hersch, “Moiré methods for the protection of documents and products,” J. Phys.: Conf. Series77, 012001 (2007).
    [CrossRef]
  7. S.-G. Wang and Z. Fan, “Moiré-free color halftoning using 2x2 printer modeling,” Proc. SPIE4300, 397–403 (2000).
    [CrossRef]
  8. 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]
  9. V. Saveljev and S.-K. Kim, “Simulation and measurement of moiré patterns at finite distance,” Opt. Express20(3), 2163–2177 (2012).
    [CrossRef] [PubMed]
  10. M. Dohnal, “Moiré in a scanned image,” Proc. SPIE4016, 166–170 (1999).
    [CrossRef]
  11. V. Saveljev, “Characteristics of moiré spectra in autostereoscopic three-dimensional displays,” J. Displ. Technol.7(5), 259–266 (2011).
    [CrossRef]
  12. C. Huang, J. R. G. Townshend, X. Zhan, M. Hansen, R. DeFries, and R. Solhberg, “Developing the spectral trajectories of major land cover change processes,” Proc. SPIE3502, 155–162 (1998).
    [CrossRef]
  13. P. S. Costa-Pereira and P. Maillard, “Estimating the age of cerrado regeneration using Landsat TM data,” Can. J. Rem. Sens.36(S2), S243–S256 (2010).
    [CrossRef]
  14. Y.-P. Lai and M.-H. Siu, “Hidden Spectral Peak trajectory model for Phone Classification,” Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (2004), I-909 – 912.
  15. S. W. Lee, F. K. Soong, and P. C. Ching, “Iterative trajectory regeneration algorithm for separating mixed speech sources,” Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (2006), I-157 – 160.
  16. J. S. Neto and J. L. A. de Carvalho, F.A. de O. Nascimento, A.F. da Rocha, and L.F. Junqueira Jr., “Trajectories of Spectral Clusters of HRV Related to Myocardial Ischemic Episodes,” Proc. XVIII Congresso Brasileiro de Engenharia Biomédica 5 (2002), 365 −370.
  17. M. Davy, B. Leprettre, C. Doncarli, and N. Martin, Tracking of spectral lines in an ARCAP time-frequency representation,” Proc. 9th Eusipco Conference (1998), 633 −636.
  18. V. Saveljev and S. K. Kim, “Estimation of Moiré Patterns using Spectral Trajectories in the Complex Plane,” Computer Technol. Appl.3, 353–360 (2012).

2012

V. Saveljev and S. K. Kim, “Estimation of Moiré Patterns using Spectral Trajectories in the Complex Plane,” Computer Technol. Appl.3, 353–360 (2012).

V. Saveljev and S.-K. Kim, “Simulation and measurement of moiré patterns at finite distance,” Opt. Express20(3), 2163–2177 (2012).
[CrossRef] [PubMed]

2011

K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express19(27), 26065–26078 (2011).
[CrossRef] [PubMed]

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

2010

P. S. Costa-Pereira and P. Maillard, “Estimating the age of cerrado regeneration using Landsat TM data,” Can. J. Rem. Sens.36(S2), S243–S256 (2010).
[CrossRef]

2009

2007

I. Amidror, S. Chosson, and R. D. Hersch, “Moiré methods for the protection of documents and products,” J. Phys.: Conf. Series77, 012001 (2007).
[CrossRef]

2000

S.-G. Wang and Z. Fan, “Moiré-free color halftoning using 2x2 printer modeling,” Proc. SPIE4300, 397–403 (2000).
[CrossRef]

1999

M. Dohnal, “Moiré in a scanned image,” Proc. SPIE4016, 166–170 (1999).
[CrossRef]

1998

C. Huang, J. R. G. Townshend, X. Zhan, M. Hansen, R. DeFries, and R. Solhberg, “Developing the spectral trajectories of major land cover change processes,” Proc. SPIE3502, 155–162 (1998).
[CrossRef]

1976

1974

Amidror, I.

I. Amidror, S. Chosson, and R. D. Hersch, “Moiré methods for the protection of documents and products,” J. Phys.: Conf. Series77, 012001 (2007).
[CrossRef]

Bryngdahl, O.

Chosson, S.

I. Amidror, S. Chosson, and R. D. Hersch, “Moiré methods for the protection of documents and products,” J. Phys.: Conf. Series77, 012001 (2007).
[CrossRef]

Costa-Pereira, P. S.

P. S. Costa-Pereira and P. Maillard, “Estimating the age of cerrado regeneration using Landsat TM data,” Can. J. Rem. Sens.36(S2), S243–S256 (2010).
[CrossRef]

DeFries, R.

C. Huang, J. R. G. Townshend, X. Zhan, M. Hansen, R. DeFries, and R. Solhberg, “Developing the spectral trajectories of major land cover change processes,” Proc. SPIE3502, 155–162 (1998).
[CrossRef]

Dohnal, M.

M. Dohnal, “Moiré in a scanned image,” Proc. SPIE4016, 166–170 (1999).
[CrossRef]

Fan, Z.

S.-G. Wang and Z. Fan, “Moiré-free color halftoning using 2x2 printer modeling,” Proc. SPIE4300, 397–403 (2000).
[CrossRef]

Hansen, M.

C. Huang, J. R. G. Townshend, X. Zhan, M. Hansen, R. DeFries, and R. Solhberg, “Developing the spectral trajectories of major land cover change processes,” Proc. SPIE3502, 155–162 (1998).
[CrossRef]

Hersch, R. D.

I. Amidror, S. Chosson, and R. D. Hersch, “Moiré methods for the protection of documents and products,” J. Phys.: Conf. Series77, 012001 (2007).
[CrossRef]

Huang, C.

C. Huang, J. R. G. Townshend, X. Zhan, M. Hansen, R. DeFries, and R. Solhberg, “Developing the spectral trajectories of major land cover change processes,” Proc. SPIE3502, 155–162 (1998).
[CrossRef]

Jung, J.-H.

Kim, J.

Kim, S. K.

V. Saveljev and S. K. Kim, “Estimation of Moiré Patterns using Spectral Trajectories in the Complex Plane,” Computer Technol. Appl.3, 353–360 (2012).

Kim, S.-K.

Kim, Y.

Kusaka, Y.

Lee, B.

Maillard, P.

P. S. Costa-Pereira and P. Maillard, “Estimating the age of cerrado regeneration using Landsat TM data,” Can. J. Rem. Sens.36(S2), S243–S256 (2010).
[CrossRef]

Park, G.

Patorski, K.

Pokorski, K.

Saveljev, V.

V. Saveljev and S. K. Kim, “Estimation of Moiré Patterns using Spectral Trajectories in the Complex Plane,” Computer Technol. Appl.3, 353–360 (2012).

V. Saveljev and S.-K. Kim, “Simulation and measurement of moiré patterns at finite distance,” Opt. Express20(3), 2163–2177 (2012).
[CrossRef] [PubMed]

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

Solhberg, R.

C. Huang, J. R. G. Townshend, X. Zhan, M. Hansen, R. DeFries, and R. Solhberg, “Developing the spectral trajectories of major land cover change processes,” Proc. SPIE3502, 155–162 (1998).
[CrossRef]

Townshend, J. R. G.

C. Huang, J. R. G. Townshend, X. Zhan, M. Hansen, R. DeFries, and R. Solhberg, “Developing the spectral trajectories of major land cover change processes,” Proc. SPIE3502, 155–162 (1998).
[CrossRef]

Trusiak, M.

Wang, S.-G.

S.-G. Wang and Z. Fan, “Moiré-free color halftoning using 2x2 printer modeling,” Proc. SPIE4300, 397–403 (2000).
[CrossRef]

Yokozeki, S.

Zhan, X.

C. Huang, J. R. G. Townshend, X. Zhan, M. Hansen, R. DeFries, and R. Solhberg, “Developing the spectral trajectories of major land cover change processes,” Proc. SPIE3502, 155–162 (1998).
[CrossRef]

Appl. Opt.

Can. J. Rem. Sens.

P. S. Costa-Pereira and P. Maillard, “Estimating the age of cerrado regeneration using Landsat TM data,” Can. J. Rem. Sens.36(S2), S243–S256 (2010).
[CrossRef]

Computer Technol. Appl.

V. Saveljev and S. K. Kim, “Estimation of Moiré Patterns using Spectral Trajectories in the Complex Plane,” Computer Technol. Appl.3, 353–360 (2012).

J. Displ. Technol.

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

J. Opt. Soc. Am.

J. Phys.: Conf. Series

I. Amidror, S. Chosson, and R. D. Hersch, “Moiré methods for the protection of documents and products,” J. Phys.: Conf. Series77, 012001 (2007).
[CrossRef]

Opt. Express

Proc. SPIE

M. Dohnal, “Moiré in a scanned image,” Proc. SPIE4016, 166–170 (1999).
[CrossRef]

C. Huang, J. R. G. Townshend, X. Zhan, M. Hansen, R. DeFries, and R. Solhberg, “Developing the spectral trajectories of major land cover change processes,” Proc. SPIE3502, 155–162 (1998).
[CrossRef]

S.-G. Wang and Z. Fan, “Moiré-free color halftoning using 2x2 printer modeling,” Proc. SPIE4300, 397–403 (2000).
[CrossRef]

Other

I. Amidror, The Theory of Moiré Phenomenon (Springer, 2009) vol. 1.

Y.-P. Lai and M.-H. Siu, “Hidden Spectral Peak trajectory model for Phone Classification,” Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (2004), I-909 – 912.

S. W. Lee, F. K. Soong, and P. C. Ching, “Iterative trajectory regeneration algorithm for separating mixed speech sources,” Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (2006), I-157 – 160.

J. S. Neto and J. L. A. de Carvalho, F.A. de O. Nascimento, A.F. da Rocha, and L.F. Junqueira Jr., “Trajectories of Spectral Clusters of HRV Related to Myocardial Ischemic Episodes,” Proc. XVIII Congresso Brasileiro de Engenharia Biomédica 5 (2002), 365 −370.

M. Davy, B. Leprettre, C. Doncarli, and N. Martin, Tracking of spectral lines in an ARCAP time-frequency representation,” Proc. 9th Eusipco Conference (1998), 633 −636.

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Figures (16)

Fig. 1
Fig. 1

Vector sums in a spectrum. The direction of the fundamental wavevectors of two superposed gratings are the real axis and the dotted line.

Fig. 2
Fig. 2

The layout of two orthogonal pairs of gratings.

Fig. 3
Fig. 3

Derivatives of the spectral trajectories in the orthogonal case. Varying α in (a) and (b), ρ in (c) and (d). The combinations of d1 and d2 are as follows, (1, 1) in (a) and (c); (1, 2) in (b) and (d). The squares in (b) and (d) are only shown for the graphical clearness of the illustration.

Fig. 4
Fig. 4

A square moiré patterns: (a) in the spatial domain and (b) in the spectral domain. The spectral peaks arranged into a square are located inside the visibility circle in (b).

Fig. 5
Fig. 5

Eight spectral trajectories arranged along the sides of a square.

Fig. 6
Fig. 6

Squares in the spectral domain and the trajectory of the corner of the square.

Fig. 7
Fig. 7

Spectral trajectories forming the octagon.

Fig. 8
Fig. 8

(a) The repeated but non-periodic structure in the spatial domain. (b) Its spectrum with the octagon inside the visibility circle.

Fig. 9
Fig. 9

Experiment. Two one-dimensional gratings (d1, d2) = (1, 1). Varying angle 0 – 55°; ρ = 0.71, 1.0, 1.4 in (a), (b), (c), resp.

Fig. 10
Fig. 10

Experiment. One-dimensional grating and two-dimensional grid (d1, d2) = (1, 2). Varying angle 0 – 50°; ρ = 0.71, 1.0, 1.4 in (a), (b), (c), resp.

Fig. 11
Fig. 11

Experiment. Two-dimensional grid and one-dimensional grating (d1, d2) = (2, 1). Varying angle 0 – 70°; ρ = 0.71, 1.0, 1.4 in (a), (b), (c), resp.

Fig. 12
Fig. 12

Experiment. Two two-dimensional gratings (d1, d2) = (2, 2). Varying angle 0 – 60°; ρ = 0.71, 1.0, 1.4 in (a), (b), (c), resp.

Fig. 13
Fig. 13

Experimental ((a), (c), (e), (g) at the left side) and corresponding simulated trajectories ((b), (d), (f), (h) at the right side) for the varying ratio ρ from 0.7 to 1.4. The number of gratings by layers (d1, d2) and the angle are as follows, (a), (b): (1, 1) and 15°, (c), (d): (1, 2) and 20°, (e), (f): (2, 1) and 38°, (g), (h): (2, 2) and 42°.

Fig. 14
Fig. 14

Simulated sketches for grid and grating of the square profile with (d1, d2) = (2, 1), σ1 = σ3 = 1 with different ρ: (a) ρ = 1, (b) ρ = 2. The trajectories involved in experimental data are shown by bold lines. The corresponding active centers are shown by bold dots. The non-sinusoidal trajectories are drawn in gray color.

Fig. 15
Fig. 15

Experimental and simulated trajectories together.

Fig. 16
Fig. 16

The square and octagon in the spectrum at α = 45°. The background trajectories are drawn for α from 0 to 50°.

Equations (73)

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T p =pk,
T p (α)=pk e iα ,
T= p 1 k 1 e i α 1 +...+ p N k N e i α N ,
T= n=1 N p n k n e i α n ,
T 12 =k( p 1 σ 1 +i p 2 ).
T 2x2 =k( p 1 σ 1 +i p 2 )+kρ( p 3 σ 3 +i p 4 ) e iα .
T(t)= p 1 k 1 e i α 1 +...+ p m k m (t) e i α m +...+ p N k N e i α N ,
d T k dt = p m e i α m d k m (t) dt .
d T kL dt = A k p m e i α m = A k k m T m .
d T α dt =i p m k m e i α m (t) d α m (t) dt .
d T αL dt =i A α T m .
N tot = j=1 N Q j .
N tot0 = Q 0 N .
N c + N l = N tot =const
T 2x2α (t)=( p 1 σ 1 +i p 2 )k+( p 3 σ 3 +i p 4 )kρ e iα(t) ,
T 2x2ρ (t)=( p 1 σ 1 +i p 2 )k+( p 3 σ 3 +i p 4 )kρ(t) e iα ,
T 2x2σ3 (t)=( p 1 σ 1 +i p 2 )k+( p 3 σ 3 (t)+i p 4 )kρ e iα ,
T 2x2σ1 (t)=( p 1 σ 1 (t)+i p 2 )k+( p 3 σ 3 +i p 4 )kρ e iα(t) .
T α11 (t)= p 1 σ 1 k+ σ 3 kρ e iα(t) ,
T α12 (t)= p 1 σ 1 k+( p 3 σ 3 +i p 4 )kρ e iα(t) ,
T α21 (t)=( p 1 σ 1 +i p 2 )k+ σ 3 kρ e iα(t) ,
T α22 (t)=( p 1 σ 1 +i p 2 )k+( p 3 σ 3 +i p 4 )kρ e iα(t) .
T α '(t)=i( p 3 σ 3 +i p 4 )kρα'(t) e iα(t) ,
T ρ '(t)=( p 3 σ 3 +i p 4 )kρ'(t) e iα ,
T σ3 '(t)= p 3 σ 3 '(t)kρ e iα ,
T σ1 '(t)= p 1 k σ 1 '(t)
T αL '(t)=i A α ( p 3 σ 3 +i p 4 )kρ e iα(t) ,
T ρL '(t)= A ρ ( p 3 σ 3 +i p 4 )k e iα ,
T σ3L '(t)= A σ3 p 3 kρ e iα ,
T σ1L '(t)= A σ1 p 1 k
T σ3 (t)=( p 1 σ 1 +i p 2 +i p 4 ρ e iα )+ p 3 σ 3 (t)ρ e iα ,
T σ1 (t)=( i p 2 +( p 3 σ 3 +i p 4 )ρ e iα(t) )+ p 1 σ 1 (t).
N c = 3 d c ,
M tot = M c + M l ,
( 9 27 27 81 9 27 27 81 )=( 3 3 9 9 3 9 9 27 )+( 6 24 18 72 6 18 18 54 ).
( 3 2 3 3 3 3 3 4 3 2 3 3 3 3 3 4 )=( 3 3 3 2 3 2 3 3 2 3 2 3 3 )+( 23 2 3 3 2 3 2 2 3 3 2 23 2 3 2 2 3 2 2 3 3 ).
N tot / N c = 3 g ,
N l / N c = 3 g 1.
M totC = M tot / M c =( 3 9 3 9 3 3 3 3 ),
M lC = M l / M c =( 2 8 2 8 2 2 2 2 ).
| u 1 v 1 1 u 2 v 2 1 u 3 v 3 1 |=0,
u 1 ( v 2 v 3 )+ u 2 ( v 3 v 1 )+ u 3 ( v 1 v 2 )=0,
z 1 z 2 =0,
v 1 v 2 + u 1 u 2 =0.
S 1 =iiρ e iα ,
S 2 =( σ 1 +i )( σ 3 +i )ρ e iα ,
S 3 = σ 1 σ 3 ρ e iα ,
S 4 =( σ 1 i )( σ 3 i )ρ e iα ,
S 5 =i+iρ e iα ,
S 6 =( σ 1 +i )+( σ 3 +i )ρ e iα ,
S 7 = σ 1 + σ 3 ρ e iα ,
S 8 =( σ 1 i )+( σ 3 i )ρ e iα ,
( σ 1 σ 3 ρcosα+ρsinα )( 1ρcosα+ σ 3 ρsinα1+ρcosα )+ +( σ 1 + σ 3 ρcosα+ρsinα )( 1ρcosα1+ρcosα+ σ 3 ρsinα )+ +( ρsinα )( 1ρcosα σ 3 ρsinα1+ρcosα σ 3 ρsinα )0.
z 1 =iiρ e iα =iiρcosα+ρsinα,
z 2 = σ 1 σ 3 ρ e iα = σ 1 σ 3 ρcosαi σ 3 ρsinα.
z 1 z 2 =( σ 1 σ 3 )ρsinα.
a 1 =2 ρ 2 +12ρcosα .
a 2 =2 σ 3 2 ρ 2 + σ 1 2 2 σ 1 σ 3 ρcosα .
tanθ= 1ρcosα ρsinα .
a=4sin α 2 ,
tanθ= 1cosα sinα =tan α 2 .
O 1 =i+( 1i ) e iα ,
O 2 =1+i e iα ,
O 3 =1+( 1+i ) e iα ,
O 4 =1i+ e iα ,
O 5 =i+( 1+i ) e iα ,
O 6 =1i+i e iα ,
O 7 =1+( 1i ) e iα ,
O 8 =1+ii e iα .
d eb = 106sinα8cosα ,
d bg = 108sinα6cosα .
tan θ 1 = 1sinα 1cosα , tan θ 2 = 1+sinα+cosα cosαsinα ,
θ 2 θ 1 =arctan 2+2sinα+cosα 2+2cosα+sinα .

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