Abstract

Moiré phenomena occur when two or more images are nonlinearly combined to create a new superposition image. Moiré patterns are patterns that do not exist in any of the original images but appear in the superposition image, for example as the result of a multiplicative superposition rule. The topic of moiré pattern synthesis deals with creating images that when superimposed will reveal certain desired moiré patterns. Conditions that ensure that a desired moiré pattern will be present in the superposition of two images are known; however, they do not specify these images uniquely. The freedom in choosing the superimposed images can be exploited to produce various degrees of visibility and ensure desired properties. Performance criteria for the images that measure when one superposition is better than another are introduced. These criteria are based on the visibility of the moiré patterns to the human visual system and on the digitization that takes place when the images are presented on discrete displays. We propose to resolve the freedom in moiré synthesis by choosing the images that optimize the chosen criteria.

© 2001 Optical Society of America

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References

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  1. H. Giger, “Moirés,” Comput. Math. Appl. 12, 329–361 (1986).
    [CrossRef]
  2. Lord Rayleigh, “On the manufacture and theory of diffraction-gratings,” Philos. Mag. 81, 81–93 (1874). Published also in G. Indebetouw, R. Czarnek, Selected Papers on Optical Moiré and Applications Vol. 64 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1992), pp. 3–15.
  3. O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am. 66, 87–94 (1976).
    [CrossRef]
  4. I. Amidror, The Theory of the Moiré Phenomenon (Kluwer Academic, Dordecht, The Netherlands, 2000).
  5. M. Wasserman, G. Oster, C. Zwerling, “Theoretical interpretations of moiré patterns,” J. Opt. Soc. Am. 54, 169–175 (1964).
    [CrossRef]
  6. R. Courant, D. Hilbert, Methods in Mathematical Physics (Interscience, New York, 1953), Vol. 1.
  7. W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).
  8. R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).
  9. M. Levine, Vision in Man and Machine (McGraw-Hill, New York, 1985).
  10. M. Taylor, “Visual discrimination and orientation,” J. Appl. Opt. 53, 763–765 (1963).
  11. G. Thomas, R. Finney, Calculus and Analytic Geometry, 9th ed. (Addison-Wesley, Reading, Mass., 1996).
  12. H. Sagan, Introduction to the Calculus of Variations (Dover, New York, 1969).
  13. B. K. P. Horn, M. J. Brooks, “The variational approach to shape from shading,” Comput. Vis. Graph. Image Process. 2, 174–203 (1986).
    [CrossRef]
  14. G. Strang, Introduction to Applied Mathematics (Wellesley–Cambridge Press, Wellesley, Mass., 1986).
  15. R. L. Burden, J. D. Faires, Numerical Analysis, 6th ed. (Brooks Cole, Pacific Grove, Calif., 1997).
  16. M. Naor, A. Shamir, “Visual cryptography,” in EUROCRYPT 1994, Lecture Notes in Computer Science (Springer-Verlag, New York, 1995), Vol. 950, pp. 1–12.

1986

H. Giger, “Moirés,” Comput. Math. Appl. 12, 329–361 (1986).
[CrossRef]

B. K. P. Horn, M. J. Brooks, “The variational approach to shape from shading,” Comput. Vis. Graph. Image Process. 2, 174–203 (1986).
[CrossRef]

1976

1964

1963

M. Taylor, “Visual discrimination and orientation,” J. Appl. Opt. 53, 763–765 (1963).

1874

Lord Rayleigh, “On the manufacture and theory of diffraction-gratings,” Philos. Mag. 81, 81–93 (1874). Published also in G. Indebetouw, R. Czarnek, Selected Papers on Optical Moiré and Applications Vol. 64 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1992), pp. 3–15.

Amidror, I.

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

Brooks, M. J.

B. K. P. Horn, M. J. Brooks, “The variational approach to shape from shading,” Comput. Vis. Graph. Image Process. 2, 174–203 (1986).
[CrossRef]

Bryngdahl, O.

Burden, R. L.

R. L. Burden, J. D. Faires, Numerical Analysis, 6th ed. (Brooks Cole, Pacific Grove, Calif., 1997).

Courant, R.

R. Courant, D. Hilbert, Methods in Mathematical Physics (Interscience, New York, 1953), Vol. 1.

Faires, J. D.

R. L. Burden, J. D. Faires, Numerical Analysis, 6th ed. (Brooks Cole, Pacific Grove, Calif., 1997).

Finney, R.

G. Thomas, R. Finney, Calculus and Analytic Geometry, 9th ed. (Addison-Wesley, Reading, Mass., 1996).

Giger, H.

H. Giger, “Moirés,” Comput. Math. Appl. 12, 329–361 (1986).
[CrossRef]

Hilbert, D.

R. Courant, D. Hilbert, Methods in Mathematical Physics (Interscience, New York, 1953), Vol. 1.

Horn, B. K. P.

B. K. P. Horn, M. J. Brooks, “The variational approach to shape from shading,” Comput. Vis. Graph. Image Process. 2, 174–203 (1986).
[CrossRef]

Levine, M.

M. Levine, Vision in Man and Machine (McGraw-Hill, New York, 1985).

Naor, M.

M. Naor, A. Shamir, “Visual cryptography,” in EUROCRYPT 1994, Lecture Notes in Computer Science (Springer-Verlag, New York, 1995), Vol. 950, pp. 1–12.

Oster, G.

Pratt, W. K.

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).

Rayleigh, Lord

Lord Rayleigh, “On the manufacture and theory of diffraction-gratings,” Philos. Mag. 81, 81–93 (1874). Published also in G. Indebetouw, R. Czarnek, Selected Papers on Optical Moiré and Applications Vol. 64 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1992), pp. 3–15.

Sagan, H.

H. Sagan, Introduction to the Calculus of Variations (Dover, New York, 1969).

Shamir, A.

M. Naor, A. Shamir, “Visual cryptography,” in EUROCRYPT 1994, Lecture Notes in Computer Science (Springer-Verlag, New York, 1995), Vol. 950, pp. 1–12.

Strang, G.

G. Strang, Introduction to Applied Mathematics (Wellesley–Cambridge Press, Wellesley, Mass., 1986).

Taylor, M.

M. Taylor, “Visual discrimination and orientation,” J. Appl. Opt. 53, 763–765 (1963).

Thomas, G.

G. Thomas, R. Finney, Calculus and Analytic Geometry, 9th ed. (Addison-Wesley, Reading, Mass., 1996).

Ulichney, R.

R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).

Wasserman, M.

Zwerling, C.

Comput. Math. Appl.

H. Giger, “Moirés,” Comput. Math. Appl. 12, 329–361 (1986).
[CrossRef]

Comput. Vis. Graph. Image Process.

B. K. P. Horn, M. J. Brooks, “The variational approach to shape from shading,” Comput. Vis. Graph. Image Process. 2, 174–203 (1986).
[CrossRef]

J. Appl. Opt.

M. Taylor, “Visual discrimination and orientation,” J. Appl. Opt. 53, 763–765 (1963).

J. Opt. Soc. Am.

Philos. Mag.

Lord Rayleigh, “On the manufacture and theory of diffraction-gratings,” Philos. Mag. 81, 81–93 (1874). Published also in G. Indebetouw, R. Czarnek, Selected Papers on Optical Moiré and Applications Vol. 64 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 1992), pp. 3–15.

Other

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

R. Courant, D. Hilbert, Methods in Mathematical Physics (Interscience, New York, 1953), Vol. 1.

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).

R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).

M. Levine, Vision in Man and Machine (McGraw-Hill, New York, 1985).

G. Thomas, R. Finney, Calculus and Analytic Geometry, 9th ed. (Addison-Wesley, Reading, Mass., 1996).

H. Sagan, Introduction to the Calculus of Variations (Dover, New York, 1969).

G. Strang, Introduction to Applied Mathematics (Wellesley–Cambridge Press, Wellesley, Mass., 1986).

R. L. Burden, J. D. Faires, Numerical Analysis, 6th ed. (Brooks Cole, Pacific Grove, Calif., 1997).

M. Naor, A. Shamir, “Visual cryptography,” in EUROCRYPT 1994, Lecture Notes in Computer Science (Springer-Verlag, New York, 1995), Vol. 950, pp. 1–12.

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Figures (19)

Fig. 1
Fig. 1

Two types of superposition. The two gratings in (a) and (b) are multiplied in (c) and added in (d). Image brightness is scaled.

Fig. 2
Fig. 2

Nonvisible additive and subtractive moiré patterns.

Fig. 3
Fig. 3

Periodic profiles of a linear function.

Fig. 4
Fig. 4

Superposition in the frequency domain: spectra of the (a) first raised cosine grating, (b) second raised cosine grating, (c) superposition.

Fig. 5
Fig. 5

Linear moiré, initial condition (top), and values after 20 iterations (bottom).

Fig. 6
Fig. 6

Hyperbolic patterns: results for natural boundary conditions.

Fig. 7
Fig. 7

Hyperbolic patterns: results for periodic boundary conditions

Fig. 8
Fig. 8

Decrease in functional value.

Fig. 9
Fig. 9

Result for λ=200.

Fig. 10
Fig. 10

Result for λ=1000.

Fig. 11
Fig. 11

Face image 1.

Fig. 12
Fig. 12

Periodic profile of Fig. 11.

Fig. 13
Fig. 13

ψ and ϕ computed for Fig. 12.

Fig. 14
Fig. 14

Superposition of the images in Fig. 13.

Fig. 15
Fig. 15

Face image 2.

Fig. 16
Fig. 16

Periodic profile of Fig. 15.

Fig. 17
Fig. 17

ψ and ϕ computed for Fig. 16.

Fig. 18
Fig. 18

Superposition of the images in Fig. 17.

Fig. 19
Fig. 19

Low-pass filtering applied to Fig. 18.

Equations (59)

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ψ(x, y)=m,m,
ϕ(x, y)=n,n.
k1m+k2n=l,l,
k1ψ(x, y)+k2ϕ(x, y)=l,l.
g(x, y)=p(ψ(x, y))p(ϕ(x, y))G(u,v)=F [p(ψ(x, y))]*F [p(ϕ(x, y))],
p(x, y)=m=-n=-cm,n exp[2πj(mu0x+nv0y)].
P(u, v)=m=-n=-cm,nδ(u-mu0, v-nv0).
p(x)=m=-cm exp(2πjmfx),
0|cm|=1T Tp(x)exp(2πjmfx)dx
1T T exp(2πjmfx)dx1.
g(x, y)=k1ψ(x, y)+k2ϕ(x, y).
CSF(u)=|5.05[exp(-0.138u)][1-exp(0.1u)]|,
V(f)=H1(f)·H2(angle(f)),
ϕopt=arg minϕH1(f(1, 1))·H2(angle(f(1, 1))),
ψopt=g+ϕopt.
f(1, 1)=fϕ+fψ=ϕ+ψ=2ϕ+g,
g=ψ-ϕψ=g+ϕ.
V(f(1, 1))=H1(f(1, 1))·H2(angle(f(1, 1)))+M(f(1, 1)).
M(f(1, 1))=H1(f(1, 1))·min H2(angle(f(1, 1)))
g=ψ-ϕ,
ψ=g+ϕ,
ψ=g+ϕg+ϕ.
W(f(1, 1))(Ω)=ΩV(f(1, 1)(x, y))dxdy,
W(f(1, 1))(I)=i=1Mj=1NV(f(1, 1)(i, j)).
ϕopt=arg minϕ W(f(1, 1))(I)
subjecttoϕxy=ϕyx.
I1[z]=ΩF(x, y, z, zx, zy)dxdy,
Fz-xFzx-yFzy=0.
I2[z]=ΩF(x, y, z, zx, zy, zxx, zxy, zyy)dxdy,
Fz-xFzx-yFzy+2x2Fzxx+2xyFzxy+2y2Fzyy=0.
I3[p,q]=ΩF(x, y, p, q, px, py, qx, qy)dxdy,
Fp-xFpx-yFpy=0,
Fq-xFqx-yFqy=0.
(Fzx, Fzy)·n=0,
(Fpx, Fpy)·n=0,(Fqx, Fqy)·n=0.
I[p, q]=Ω(V2(p, q)+λ(py-qx)2)dxdy.
-VVp+λ(pyy-qxy)=0,
-VVq+λ(qxx-pyx)=0.
pi, jk+1=p¯i, jk-12q˜i, jk-12λV(pi, jk, qi, jk)Vp(pi, jk, qi, jk),
qi, jk+1=q¯i, jk-12q˜i, jk-12λV(pi, jk, qi, jk)Vq(pi, jk, qi, jk),
p¯i, j=pi, j+1+pi, j-12,
q¯i, j=qi+1, j+qi-1, jx2,
p˜i, j=pi+1, j+1+pi-1, j-1-pi+1, j-1-pi-1, j+14,
q¯i, j=qi+1, j+1+qi-1, j-1-qi+1, j-1-qi-1, j+14.
ϕ(x, y)=ϕ(x0, y0)+Cϕ·dl,
Ω(ϕx-p)2+(ϕy-q)2dxdy.
pi,1k+1=pi,N-1k,pi,Nk+1=pi,2k,
p1,ik+1=pN-1,ik,pN,ik+1=p2,ik,
p(1, j)=p(1, j-1)+j-1jpy(1, t)dt,
p(N, j)=p(N, j-1)+j-1jpy(N, j)dk,
p(i, 1)=p(i, 2)-21py(i, t)dt,
p(i, N)=p(i, N-1)+N-1Npy(i, t)dt.
p(1, j)=p(1, j-1)-12 (py(1, j-1)+py(1, j)),
p(N, j)=p(N, j-1)+12 (py(N, j-1)+py(N, j)),
p(i, 1)=p(i, 2)-12 (py(i, 2)+py(i, 1)),
p(i, N)=p(i, N-1)+12 (py(i, N)+py(i, N-1)).
(x-s)2a2+y2b2=h2,
(x+s)2a2+y2b2=k2.
4x2a2p2-y2(b2p2/4)-b2s2=1,

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