Abstract

We propose an interpretation of moiré phenomenon in the image domain. The interpretation is basically based on the analysis of the waveform of the line families. The period, angle, and intensity profile of moiré fringes can be obtained directly in the image domain according to this interpretation. Moreover, pseudo-moiré can be interpreted visually with the consideration of the illusional contrast of the human visual system. The interpretation, which is consistent with the Fourier theory when the two superposed gratings are periodic, involves only the image domain and shows remarkable simplicity, just like the indicial equation method.

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References

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  1. M. Abolhassani and M. Mirzaei, “Unification of formulation of moiré fringe spacing in parametric equation and Fourier analysis methods,” Appl. Opt. 46(32), 7924–7926 (2007).
    [CrossRef] [PubMed]
  2. L. Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 4, 81–93 (1874).
  3. G. Oster, M. Wasserman, and C. Zwerling, “Theoretical Interpretation of Moiré patterns,” J. Opt. Soc. Am. A 54(2), 169–175 (1964).
    [CrossRef]
  4. A. J. Durelli and V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, New Jersey, 1970).
  5. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).
  6. O. Bryngdahl, “Moiré: formation and interpretation,” J. Opt. Soc. Am. A 64(10), 1287–1294 (1974).
    [CrossRef]
  7. O. Bryngdahl, “Moiré and higher grating harmonics,” J. Opt. Soc. Am. A 65(6), 685–694 (1975).
    [CrossRef]
  8. I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré′ effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
    [CrossRef]
  9. I. Amidror, The Theory of the Moiré Phenomenon (Springer-Verlag, London, 2009), Chap.2.
  10. G. Lebanon and A. M. Bruckstein, Designing Moiré Patterns (Springer-Verlag, Berlin, 2001).
  11. K. Patorski, S. Yokozeki, and T. Suzuki, “Moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).
    [CrossRef]
  12. I. Amidror and R. D. Hersch, “Mathematical moiré models and their limitations,” J. Mod. Opt. 57(1), 23–36 (2010).
    [CrossRef]

2010 (1)

I. Amidror and R. D. Hersch, “Mathematical moiré models and their limitations,” J. Mod. Opt. 57(1), 23–36 (2010).
[CrossRef]

2009 (1)

I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré′ effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
[CrossRef]

2007 (1)

1976 (1)

K. Patorski, S. Yokozeki, and T. Suzuki, “Moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).
[CrossRef]

1975 (1)

O. Bryngdahl, “Moiré and higher grating harmonics,” J. Opt. Soc. Am. A 65(6), 685–694 (1975).
[CrossRef]

1974 (1)

O. Bryngdahl, “Moiré: formation and interpretation,” J. Opt. Soc. Am. A 64(10), 1287–1294 (1974).
[CrossRef]

1964 (1)

G. Oster, M. Wasserman, and C. Zwerling, “Theoretical Interpretation of Moiré patterns,” J. Opt. Soc. Am. A 54(2), 169–175 (1964).
[CrossRef]

1874 (1)

L. Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 4, 81–93 (1874).

Abolhassani, M.

Amidror, I.

I. Amidror and R. D. Hersch, “Mathematical moiré models and their limitations,” J. Mod. Opt. 57(1), 23–36 (2010).
[CrossRef]

I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré′ effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
[CrossRef]

Bryngdahl, O.

O. Bryngdahl, “Moiré and higher grating harmonics,” J. Opt. Soc. Am. A 65(6), 685–694 (1975).
[CrossRef]

O. Bryngdahl, “Moiré: formation and interpretation,” J. Opt. Soc. Am. A 64(10), 1287–1294 (1974).
[CrossRef]

Hersch, R. D.

I. Amidror and R. D. Hersch, “Mathematical moiré models and their limitations,” J. Mod. Opt. 57(1), 23–36 (2010).
[CrossRef]

I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré′ effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
[CrossRef]

Mirzaei, M.

Oster, G.

G. Oster, M. Wasserman, and C. Zwerling, “Theoretical Interpretation of Moiré patterns,” J. Opt. Soc. Am. A 54(2), 169–175 (1964).
[CrossRef]

Patorski, K.

K. Patorski, S. Yokozeki, and T. Suzuki, “Moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).
[CrossRef]

Rayleigh, L.

L. Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 4, 81–93 (1874).

Suzuki, T.

K. Patorski, S. Yokozeki, and T. Suzuki, “Moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).
[CrossRef]

Wasserman, M.

G. Oster, M. Wasserman, and C. Zwerling, “Theoretical Interpretation of Moiré patterns,” J. Opt. Soc. Am. A 54(2), 169–175 (1964).
[CrossRef]

Yokozeki, S.

K. Patorski, S. Yokozeki, and T. Suzuki, “Moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).
[CrossRef]

Zwerling, C.

G. Oster, M. Wasserman, and C. Zwerling, “Theoretical Interpretation of Moiré patterns,” J. Opt. Soc. Am. A 54(2), 169–175 (1964).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (2)

I. Amidror and R. D. Hersch, “Mathematical moiré models and their limitations,” J. Mod. Opt. 57(1), 23–36 (2010).
[CrossRef]

I. Amidror and R. D. Hersch, “The role of Fourier theory and of modulation in the prediction of visible moiré′ effects,” J. Mod. Opt. 56(9), 1103–1118 (2009).
[CrossRef]

J. Opt. Soc. Am. A (3)

G. Oster, M. Wasserman, and C. Zwerling, “Theoretical Interpretation of Moiré patterns,” J. Opt. Soc. Am. A 54(2), 169–175 (1964).
[CrossRef]

O. Bryngdahl, “Moiré: formation and interpretation,” J. Opt. Soc. Am. A 64(10), 1287–1294 (1974).
[CrossRef]

O. Bryngdahl, “Moiré and higher grating harmonics,” J. Opt. Soc. Am. A 65(6), 685–694 (1975).
[CrossRef]

Jpn. J. Appl. Phys. (1)

K. Patorski, S. Yokozeki, and T. Suzuki, “Moiré profile prediction by using Fourier series formalism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).
[CrossRef]

Philos. Mag. (1)

L. Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 4, 81–93 (1874).

Other (4)

I. Amidror, The Theory of the Moiré Phenomenon (Springer-Verlag, London, 2009), Chap.2.

G. Lebanon and A. M. Bruckstein, Designing Moiré Patterns (Springer-Verlag, Berlin, 2001).

A. J. Durelli and V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, New Jersey, 1970).

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).

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Figures (7)

Fig. 1
Fig. 1

The waveform of the function T with the variable t when φ = 0 and φ = 0.5 .

Fig. 2
Fig. 2

Schematic of the interpretation of the moiré phenomenon in the image domain

Fig. 3
Fig. 3

The multiplicative superposition of two little inclined cosinusoidal gratings with frequency ratio 1:1

Fig. 4
Fig. 4

(a)The multiplicative superposition of two little inclined cosinusoidal gratings with frequency ratio 1:2(b) The multiplicative superposition of two little inclined cosinusoidal gratings with frequency ratio 1:3

Fig. 5
Fig. 5

(a) The multiplicative superposition of two little inclined cosinusoidal gratings with frequency ratio 1:1.1 (b) The multiplicative superposition of two little inclined cosinusoidal gratings with frequency ratio 1:1.9

Fig. 6
Fig. 6

The multiplicative superposition of two little inclined cosinusoidal gratings with frequency ratio 1:1.5

Fig. 7
Fig. 7

(a)the multiplicative superposition of two little inclined binary gratings with frequency ratio 1:1(b) The multiplicative superposition of two binary gratings where the frequency ratio is 1:2 and with little included angle.

Tables (2)

Tables Icon

Table 1 Angles and periods of real and pseudo-moiré in the case of frequency ratio 1.1 and 1.9

Tables Icon

Table 2 Angles and periods of real and pseudo-moiré in the case of frequency ratio 1.5

Equations (19)

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T 1 = 1 2 ( cos 2 π x + 1 ) , T 2 = 1 2 [ cos [ 2 π t ( x + φ ) ] + 1 ] ,
T = T 1 × T 2 = 1 2 ( cos 2 π x + 1 ) × 1 2 [ cos [ 2 π t ( x + φ ) ] + 1 ] = 1 8 cos [ 2 π ( t + 1 t x + φ t ) ] + 1 8 cos [ 2 π ( t 1 t x φ t ) ] + 1 4 cos ( 2 π x ) + 1 4 cos [ 2 π t ( x + φ ) ] + 1 4 .
T = 1 8 cos [ 2 π ( 2 x + φ ) ] + 1 8 cos ( 2 π φ ) + 1 4 cos ( 2 π x ) + 1 4 cos [ 2 π ( x + φ ) ] + 1 4
I a v r = 1 M 0 M T d x = 0 1 [ 1 8 [ cos ( 2 π φ ) + 2 ] d x = 1 8 ( cos 2 π φ + 2 ) .
I a v r = 1 M 0 M T d x = 1 4
P 1 sin γ r = P 2 sin β r
P 2 P 1 = sin β r sin γ r = sin ( π α γ r ) sin γ r = sin ( α + γ r ) sin γ r = sin α cot γ r + cos α
tan γ r = P 1 sin α P 2 P 1 cos α , tan β r = P 2 sin α P 1 P 2 cos α .
tan γ r ' = P 1 sin α ' P 2 P 1 cos α ' = P 1 sin α P 2 + P 1 cos α , tan β r ' = P 2 sin α ' P 1 P 2 cos α ' = P 2 sin α P 1 + P 2 cos α .
R + L = P r cot γ r + P r cot β r = P 1 sin γ r = P 2 sin β r
P r = P 1 P 2 P 1 2 + P 2 2 2 P 1 P 2 cos α , P r ' = P 1 P 2 P 1 2 + P 2 2 + 2 P 1 P 2 cos α .
m P 1 sin γ p = P 2 sin β p
P 2 m P 1 = sin β p sin γ p = sin α cos γ p + cos α sin γ p sin γ p = sin α cot γ p + cos α
tan γ p = m P 1 sin α P 2 m P 1 cos α , tan β p = P 2 sin α m P 1 P 2 cos α .
R + L = P p cot γ p + P p cot β p = P 1 / sin γ p
P p = P 1 P 2 m 2 P 1 2 + P 2 2 2 m P 1 P 2 cos α , P p ' = P 1 P 2 m 2 P 1 2 + P 2 2 + 2 m P 1 P 2 cos α .
tan γ = m P 1 sin α P 2 m P 1 cos α , P m = P 1 P 2 m 2 P 1 2 + P 2 2 2 m P 1 P 2 cos α .
r 1 ( x , y ) = 1 2 cos ( 2 π f 1 x ) + 1 2 , r 1 ( x , y ) = 1 2 cos ( 2 π f 2 [ x cos θ 2 + y sin θ 2 ] ) + 1 2 ,
tan γ p = tan ( π α 2 ) , P p = P 1 2 | sin ( α / 2 ) | .

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