Abstract

Moiré effects that occur in the superposition of aperiodic layers such as random dot screens are known as Glass patterns. Unlike classical moiré effects between periodic layers, which are periodically repeated throughout the superposition, a Glass pattern is concentrated around a certain point in the superposition, and farther away from this point it fades out and disappears. I show that Glass patterns between aperiodic layers can be analyzed by using an extension of the Fourier-based theory that governs the classical moiré patterns between periodic layers. Surprisingly, even spectral-domain considerations can be extended in a natural way to aperiodic cases, with some straightforward adaptations. These new results allow us to predict quantitatively the intensity profile of Glass patterns; furthermore, they open the way to the synthesis of Glass patterns that have any desired shapes and intensity profiles.

© 2003 Optical Society of America

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References

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  1. L. Glass, “Moiré effect from random dots,” Nature 223, 578–580 (1969).
    [CrossRef] [PubMed]
  2. L. Glass, R. Pérez, “Perception of random dot interference patterns,” Nature 246, 360–362 (1973).
    [CrossRef] [PubMed]
  3. I. Amidror, The Theory of the Moiré Phenomenon (Kluwer Academic, Dordrecht, The Netherlands, 2000).
  4. I. Amidror, “Glass patterns and moiré intensity profiles: new surprising results,” Opt. Lett. 28, 7–9 (2003).
    [CrossRef] [PubMed]
  5. http://lspwww.epfl.ch/books/moire/kit.html .
  6. I. Amidror, “A unified approach for the explanation of stochastic and periodic moirés,” J. Electron. Imaging (to be published).
  7. I. Amidror, “Glass patterns in the superposition of random line gratings,” J. Opt. A, Pure Appl. Opt. 5, 205–215 (2003).
    [CrossRef]
  8. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, Reading, N.Y., 1986).
  9. R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995).
  10. Y. Nishijima, G. Oster, “Moiré patterns: their application to refractive index and refractive index gradient measurements,” J. Opt. Soc. Am. 54, 1–5 (1964).
    [CrossRef]
  11. Note that this impulse is generated in the convolution by the (k1, k2) impulse in the spectrum R1(u, v) of the first image and the (k3, k4) impulse in the spectrum R2(u, v) of the second image.
  12. A. Rosenfeld, A. C. Kak, Digital Picture Processing, Vol. 1, 2nd ed. (Academic, Boca Raton, Fla., 1982).
  13. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  14. It is interesting to note that just like its periodic counterpart (see Sec. 10.9 of Ref. 3), this proposition remains true for nonlinear transformations gi(x, y), too, i.e., when the original aperiodic layers undergo any given geometric transformations. In such cases, part 2 of the proposition simply gives the geometric transformation that is undergone by the resulting Glass pattern.
  15. S. C. Dakin, “The detection of structure in Glass patterns: psychophysics and computational models,” Vision Res. 37, 2227–2246 (1997).
    [CrossRef] [PubMed]
  16. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), pp. 99–139.
  17. I. Amidror, “A new print-based security strategy for the protection of valuable documents and products using moiré intensity profiles,” in Optical Security and Counterfeit Deterrence Techniques IV, R. L. Van Renesse, ed., Proc. SPIE4677, 89–100 (2002).
    [CrossRef]

2003 (2)

I. Amidror, “Glass patterns in the superposition of random line gratings,” J. Opt. A, Pure Appl. Opt. 5, 205–215 (2003).
[CrossRef]

I. Amidror, “Glass patterns and moiré intensity profiles: new surprising results,” Opt. Lett. 28, 7–9 (2003).
[CrossRef] [PubMed]

1997 (1)

S. C. Dakin, “The detection of structure in Glass patterns: psychophysics and computational models,” Vision Res. 37, 2227–2246 (1997).
[CrossRef] [PubMed]

1973 (1)

L. Glass, R. Pérez, “Perception of random dot interference patterns,” Nature 246, 360–362 (1973).
[CrossRef] [PubMed]

1969 (1)

L. Glass, “Moiré effect from random dots,” Nature 223, 578–580 (1969).
[CrossRef] [PubMed]

1964 (1)

Amidror, I.

I. Amidror, “Glass patterns in the superposition of random line gratings,” J. Opt. A, Pure Appl. Opt. 5, 205–215 (2003).
[CrossRef]

I. Amidror, “Glass patterns and moiré intensity profiles: new surprising results,” Opt. Lett. 28, 7–9 (2003).
[CrossRef] [PubMed]

I. Amidror, “A unified approach for the explanation of stochastic and periodic moirés,” J. Electron. Imaging (to be published).

I. Amidror, “A new print-based security strategy for the protection of valuable documents and products using moiré intensity profiles,” in Optical Security and Counterfeit Deterrence Techniques IV, R. L. Van Renesse, ed., Proc. SPIE4677, 89–100 (2002).
[CrossRef]

I. Amidror, The Theory of the Moiré Phenomenon (Kluwer Academic, Dordrecht, The Netherlands, 2000).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, Reading, N.Y., 1986).

R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Dakin, S. C.

S. C. Dakin, “The detection of structure in Glass patterns: psychophysics and computational models,” Vision Res. 37, 2227–2246 (1997).
[CrossRef] [PubMed]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Glass, L.

L. Glass, R. Pérez, “Perception of random dot interference patterns,” Nature 246, 360–362 (1973).
[CrossRef] [PubMed]

L. Glass, “Moiré effect from random dots,” Nature 223, 578–580 (1969).
[CrossRef] [PubMed]

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, Vol. 1, 2nd ed. (Academic, Boca Raton, Fla., 1982).

Nishijima, Y.

Oster, G.

Patorski, K.

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), pp. 99–139.

Pérez, R.

L. Glass, R. Pérez, “Perception of random dot interference patterns,” Nature 246, 360–362 (1973).
[CrossRef] [PubMed]

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, Vol. 1, 2nd ed. (Academic, Boca Raton, Fla., 1982).

J. Opt. A, Pure Appl. Opt. (1)

I. Amidror, “Glass patterns in the superposition of random line gratings,” J. Opt. A, Pure Appl. Opt. 5, 205–215 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (2)

L. Glass, “Moiré effect from random dots,” Nature 223, 578–580 (1969).
[CrossRef] [PubMed]

L. Glass, R. Pérez, “Perception of random dot interference patterns,” Nature 246, 360–362 (1973).
[CrossRef] [PubMed]

Opt. Lett. (1)

Vision Res. (1)

S. C. Dakin, “The detection of structure in Glass patterns: psychophysics and computational models,” Vision Res. 37, 2227–2246 (1997).
[CrossRef] [PubMed]

Other (11)

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), pp. 99–139.

I. Amidror, “A new print-based security strategy for the protection of valuable documents and products using moiré intensity profiles,” in Optical Security and Counterfeit Deterrence Techniques IV, R. L. Van Renesse, ed., Proc. SPIE4677, 89–100 (2002).
[CrossRef]

I. Amidror, The Theory of the Moiré Phenomenon (Kluwer Academic, Dordrecht, The Netherlands, 2000).

http://lspwww.epfl.ch/books/moire/kit.html .

I. Amidror, “A unified approach for the explanation of stochastic and periodic moirés,” J. Electron. Imaging (to be published).

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, Reading, N.Y., 1986).

R. N. Bracewell, Two Dimensional Imaging (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Note that this impulse is generated in the convolution by the (k1, k2) impulse in the spectrum R1(u, v) of the first image and the (k3, k4) impulse in the spectrum R2(u, v) of the second image.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, Vol. 1, 2nd ed. (Academic, Boca Raton, Fla., 1982).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

It is interesting to note that just like its periodic counterpart (see Sec. 10.9 of Ref. 3), this proposition remains true for nonlinear transformations gi(x, y), too, i.e., when the original aperiodic layers undergo any given geometric transformations. In such cases, part 2 of the proposition simply gives the geometric transformation that is undergone by the resulting Glass pattern.

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Figures (15)

Fig. 1
Fig. 1

(a) Superposition of two identical aperiodic dot screens with a small angle difference gives a moiré effect in the form of a Glass pattern around the center of rotation. (b) When the superposed layers are periodic, a Glass pattern is still generated around the center of rotation, but owing to the periodicity of the layers, this pattern is periodically repeated throughout the superposition, thus generating a periodic moiré pattern.

Fig. 2
Fig. 2

Superposition with a small angle difference of a random dot screen consisting of 1-shaped dots and a random dot screen consisting of small white dots (pinholes) on a black background, where the dot locations in both screens are identical, gives a single 1-shaped moiré intensity profile (Glass pattern).

Fig. 3
Fig. 3

Periodic counterpart: The superposition with a small angle difference of a periodic dot screen consisting of 1-shaped dots and a periodic dot screen consisting of small white dots (pinholes) on a black background gives a periodic 1-shaped moiré intensity profile.

Fig. 4
Fig. 4

Periodic line gratings (a) and (b) and their superposition (c) in the image domain; their respective spectra are the infinite impulse combs shown in (d) and (e) and their convolution (f). The circle in the center of the spectrum (f) represents the visibility circle. It contains the impulse pair whose frequency vectors are f1-f2 and f2-f1 and whose indices are (1, -1) and (-1, 1); this is the fundamental impulse pair of the (1, -1) moiré seen in (c). The dotted line in (f) indicates the infinite impulse comb that represents this moiré. This (1, -1) moiré comb is shown isolated in (h), after its extraction from the spectrum convolution (f). The impulse amplitudes of this comb are the term-by-term products of the respective impulse amplitudes from the combs (d) and (e) taken head to tail. In (g) is shown the image domain function that corresponds to the spectrum (h); this is the intensity profile of the (1, -1)-moiré shown in (c). Black dots in the spectra indicate the impulse locations; the straight lines connecting them have been added only to clarify the geometric relations. Impulse amplitudes are not shown.

Fig. 5
Fig. 5

Geometric consideration in the spectral domain for finding the frequency fM and the period TM of the (1, -1) moiré effect between two gratings with identical frequencies f and an angle difference of α. The dotted line indicates the infinite impulse comb that represents the (1, -1) moiré [see Fig. 4(f)]; the fundamental impulse of this moiré has the frequency fM.

Fig. 6
Fig. 6

Superposition of two dot screens with identical frequencies and with an angle difference of α=5° (top) and the corresponding spectrum (bottom). Only impulse locations are shown in the spectrum, but not their amplitudes. Encircled points denote the locations of the fundamental impulses of the two original dot screens. Large points represent convolution impulses of the first order (i.e., (k1, k2, k3, k4) impulses with ki=1, 0, or -1); smaller points represent convolution impulses of higher orders. (Note that impulses of only the first few orders are shown; in reality, each impulse cluster extends in all directions ad infinitum.) The circle around the spectrum origin represents the visibility circle. Note that the spectrum origin is closely surrounded by the impulse cluster of the (1, 0, -1, 0) moiré.

Fig. 7
Fig. 7

(a) Convolution of tiny white dots (from the first screen) with dots of any given shape (from the other screen) gives dots of essentially the same given shape; (b) convolution of tiny black dots (from the first screen) with dots of any given shape (from the other screen) gives dots of essentially the same shape, but in inverse video.

Fig. 8
Fig. 8

Detail from Fig. 6 showing the spectral interpretation (vector diagram) of the (1, 0, -1, 0) moiré between two dot screens with identical frequencies and a small angle difference α. The low-frequency vectorial sums a and b [which are the geometric locations of the two fundamental impulses of the (1, 0, -1, 0) moiré cluster] are closely perpendicular to the directions of the two original screens: a is perpendicular to the bisecting direction between f1 and f3, and b is perpendicular to the bisecting direction between f2 and f4.

Fig. 9
Fig. 9

(a) Superposition of two identical aperiodic line gratings with a small angle difference gives a moiré effect in the form of a linear Glass pattern passing through the center of rotation. (b) When the superposed layers are periodic, a linear Glass pattern is still generated through the center of rotation, but owing to the periodicity of the layers, this pattern is periodically repeated throughout the superposition, thus generating a periodic moiré pattern.

Fig. 10
Fig. 10

Correlated aperiodic line gratings (a) and (b) and their superposition (c) in the image domain; their respective spectra [obtained by 2D fast Fourier transform (FFT)] are the infinite Hermitian blades (d) and (e) and their convolution (f). The white perpendicular line in (f) is the infinite Hermitian blade that represents the Glass pattern seen in (c). This Hermitian blade is shown isolated in (h), after its extraction from the spectrum convolution (f). The amplitude at each point of this blade is the point-by-point product of the respective amplitudes of the blades (d) and (e) taken head to tail. In (g) is shown the image-domain function that corresponds to the spectrum (h), as obtained by inverse FFT of (h); this is the intensity profile of the Glass pattern shown in (c). Note that unlike its periodic counterpart (Fig. 4), the present figure has been obtained by digital simulation; the scaling and angle parameters used here are slightly different in order to reduce aliasing and other FFT artifacts. Only the real parts of the spectra are shown; the complex amplitudes of the blades (d) and (e) are similar to those shown in Fig. 11.

Fig. 11
Fig. 11

Amplitude of the Hermitian blade of Fig. 10(d). (a) Real part of the amplitude, (b) the imaginary part of the amplitude.

Fig. 12
Fig. 12

Geometric consideration in the frequency domain illustrating the scaling ratio between the original line spectra shown in (a) and (b) and the isolated line spectrum shown as a dotted line in (c).

Fig. 13
Fig. 13

Same as in Fig. 10, but with two different aperiodic line gratings (a) and (b). No Glass pattern is generated in their superposition (c).

Fig. 14
Fig. 14

(a) Cross section through the extracted Glass pattern of Fig. 10(g), showing its intensity profile. (b) Cross section through Fig. 13(g). Clearly, when the original gratings are not correlated, no Glass pattern is visible in their superposition nor in the extracted intensity profile.

Fig. 15
Fig. 15

Top row: a series of grating superpositions with a gradually increasing degree of randomness in the element locations. Note the increased fading out of the moiré oscillations as we go from the left (fully periodic) to the right (fully aperiodic, or random). Center row: extracted moiré profiles, as obtained by inverse Fourier transform of the moiré blade extracted from the spectrum of the corresponding grating superpositions. Bottom row: cross sections through the extracted moiré profiles. Note the agreement between these calculated cross sections and the moiré profiles that are visible in the corresponding grating superpositions.

Equations (47)

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r(x, y)=r1(x, y)r2(x, y)  rm(x, y).
R(u, v)=R1(u, v) ** R2(u, v) **  ** Rm(u, v).
R1(u, v)=n=-an(1)δnf1(u, v), R2(u, v)=n=-an(2)δnf2(u, v).
fk1,k2=k1f1+k2f2
ak1,k2=ak1(1)ak2(2),
fuk1,k2=k1f1 cos θ1+k2f2 cos θ2, fvk1,k2=k1f1 sin θ1+k2f2 sin θ2.
f=fu2+fv2,T=1/f,φ=arctan(fv/fu).
sin(α/2)=fM/2f;
fM=2f sin(α/2),
TM=12f sin(α/2)=T2 sin(α/2).
fM=(f12-2f1f2 cos α+f22)1/2,
TM=T1T2(T12+T22-2T1T2 cos α)1/2,
φM=arctanT2 sin θ1-T1 sin θ2T2 cos θ1-T1 cos θ2.
dn=ank1(1)ank2(2),
dn=an(1)a-n(2).
ma+nb=mf1+nf2-mf3-nf4.
dm,n=am,n(1)a-m,-n(2).
a=f1-f3,b=f2-f4.
P(fM)=P1(f)P2(-f),
sin(α/2)=fM/2f;
fM=2f sin(α/2),
c=fM/f=2 sin(α/2).
P(fM)=P1(f1)P2(-f2)
fM=(f12-2f1f2 cos α+f22)1/2.
φM=arctanT2 sin θ1-T1 sin θ2T2 cos θ1-T1 cos θ2.
P(nfM)=P1(nf1)P2(-nf2),n,
P(n)=P1(n)P2(-n),n.
r1(x, y)=p1(g1(x, y)),r2(x, y)=p2(g2(x, y)).
m(x, y)=p(g(x, y)),
p(x)=p1(x)*p2(-x);
g(x, y)=g1(x, y)-g2(x, y).
g1(x, y)=u1x+v1y=f1  x  with f1=(u1, v1),x=(x, y),  g2(x, y)=u2x+v2y=f2  x  with f2=(u2, v2),x=(x, y);
g(x, y)=(u1-u2)x+(v1-v2)y=fM  x with fM=f1-f2,x=(x, y).
p1(x)  p2(x)=p1(x) * p2(-x).
dm,n=am,n(1)a-m,-n(2) withm, n.
P(m, n)=P1(m, n)P2(-m, -n)with m, n,
P(ma+nb)=P1(mf1+nf2)P2(-mf3-nf4).
r1(x)=p1(g1(x)),r2(x)=p2(g2(x)).
m(x)=p(g(x)),
p(x)=p1(x) ** p2(-x);
g(x)=g1(x)-g2(x).
g1(x)=u1x+v1yu2x+v2y=F1  x with F1=u1v1u2v2=f1f2,x=(x, y), g2(x)=u3x+v3yu4x+v4y=F2  x with F2=u3v3u4v4=f3f4,x=(x, y),
g(x)=(u1-u3)x+(v1-v3)y(u2-u4)x+(v2-v4)y=FM  x with FM=u1-u3v1-v3u2-u4v2-v4=ab,
FM=F1-F2,
ab=f1-f3f2-f4.
a=f1-f3, b=f2-f4.
p1(x)  p2(x)=p1(x) ** p2(-x).

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