Abstract

The method of generalized projections is used to design quantized phase holograms that generate gray-scale images. As the number of phase quantization levels decreases, the performance typically degrades. However, performance can be significantly improved by constraining the energy in mutually exclusive cliques of image-plane (far-field) pixels instead of constraining the intensity of each individual pixel. Performance is demonstrated with objective measures (error, efficiency, and variance) as well as with a subjective comparison of images.

© 1997 Optical Society of America

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References

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    [CrossRef]
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  15. M. King, “Fourier optics and radar signal processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 209–251.
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    [CrossRef]
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  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1996 (1)

1995 (1)

1994 (1)

1992 (1)

1991 (2)

1990 (1)

1989 (2)

F. B. McCormick, “Generation of large spot arrays from a single laser beam by multipleimaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
[CrossRef]

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

1986 (1)

1984 (2)

1982 (2)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

R. J. Mailloux, “Phased array theory and technology,” Proc. IEEE 70, 246–291 (1982).
[CrossRef]

1979 (1)

1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1974 (1)

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1951 (1)

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
[CrossRef]

Akahori, H.

Allebach, J. P.

Catino, W. C.

H. Stark, W. C. Catino, J. L. LoCicero, “Design of phase gratings by generalized projections,” J. Opt. Soc. Am. A 8, 566–571 (1991).
[CrossRef]

W. C. Catino, “Pure-phase and pure-amplitude hologram design using the method of generalized projections,” Ph.D. dissertation (Illinois Institute of Technology, Chicago, Ill., 1997).

Cohn, R. W.

Collin, R. E.

R. E. Collin, Antenna and Radiowave Propagation (McGraw-Hill, New York, 1985), pp. 107–151.

Compton, R. T.

R. T. Compton, Adaptive Antennas (Prentice-Hall, Englewood Cliffs, N.J., 1988).

Fainman, Y.

Feiner, S. K.

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics: Principles and Practice (Addison-Wesley, Reading, Mass., 1990).

Fisher, A. D.

C. Warde, A. D. Fisher, “Spatial light modulators: applications and functional capabilities,” in Optical Signal Processing, J. Horner, ed. (Academic, San Diego, Calif., 1988), pp. 478–523.

Flannery, D. L.

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Foley, J. D.

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics: Principles and Practice (Addison-Wesley, Reading, Mass., 1990).

Gerchberg, R. W.

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

Horner, J. L.

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Hughes, J. F.

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics: Principles and Practice (Addison-Wesley, Reading, Mass., 1990).

Jennison, B. K.

King, M.

M. King, “Fourier optics and radar signal processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 209–251.

Landweber, L.

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
[CrossRef]

Lee, W. H.

Levi, A.

A. Levi, H. Stark, “Image restoration by the method of generalized projections with applicationto restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

A. Levi, “Image restoration by the method of projections with applications to the phase and magnitude retrieval problems,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1983).

Liang, M.

LoCicero, J. L.

Mailloux, R. J.

R. J. Mailloux, “Phased array theory and technology,” Proc. IEEE 70, 246–291 (1982).
[CrossRef]

Marchand, P.

McCormick, F. B.

F. B. McCormick, “Generation of large spot arrays from a single laser beam by multipleimaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
[CrossRef]

Missig, M. D.

Morris, G. M.

Pang, J.

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Sawchuk, A. A.

A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
[CrossRef]

Stark, H.

Strand, T. C.

A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
[CrossRef]

Sweeney, D. W.

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Urquhart, K. S.

van Dam, A.

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics: Principles and Practice (Addison-Wesley, Reading, Mass., 1990).

Warde, C.

C. Warde, A. D. Fisher, “Spatial light modulators: applications and functional capabilities,” in Optical Signal Processing, J. Horner, ed. (Academic, San Diego, Calif., 1988), pp. 478–523.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Wyrowski, F.

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Am. J. Math. (1)

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
[CrossRef]

Appl. Opt. (5)

IEEE Trans. Circuits Syst. (1)

A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

IEEE Trans. Med. Imaging (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part I. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Eng. (1)

F. B. McCormick, “Generation of large spot arrays from a single laser beam by multipleimaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
[CrossRef]

Proc. IEEE (3)

A. A. Sawchuk, T. C. Strand, “Digital optical computing,” Proc. IEEE 72, 758–779 (1984).
[CrossRef]

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

R. J. Mailloux, “Phased array theory and technology,” Proc. IEEE 70, 246–291 (1982).
[CrossRef]

Other (10)

R. T. Compton, Adaptive Antennas (Prentice-Hall, Englewood Cliffs, N.J., 1988).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

C. Warde, A. D. Fisher, “Spatial light modulators: applications and functional capabilities,” in Optical Signal Processing, J. Horner, ed. (Academic, San Diego, Calif., 1988), pp. 478–523.

R. E. Collin, Antenna and Radiowave Propagation (McGraw-Hill, New York, 1985), pp. 107–151.

M. King, “Fourier optics and radar signal processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 209–251.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

J. D. Foley, A. van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics: Principles and Practice (Addison-Wesley, Reading, Mass., 1990).

H. Stark, Image Recovery: Theory and Application (Academic, Orlando, Fla., 1987).

A. Levi, “Image restoration by the method of projections with applications to the phase and magnitude retrieval problems,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1983).

W. C. Catino, “Pure-phase and pure-amplitude hologram design using the method of generalized projections,” Ph.D. dissertation (Illinois Institute of Technology, Chicago, Ill., 1997).

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Figures (11)

Fig. 1
Fig. 1

Optical imaging. (a) Each far-field pixel is perceived as a single image pixel. (b) Each 2×2 clique of far-field pixels is perceived as a single image pixel.

Fig. 2
Fig. 2

Prescribed Lena pattern for S=1. (a) Entire far-field pattern, (b) Lena portion of the prescribed pattern.

Fig. 3
Fig. 3

Lena results for Q=, S=1.

Fig. 4
Fig. 4

Lena results for Q=2, with a simulated optical system performing the energy summation. (a) S=1, (b) S=2, (c) S=4.

Fig. 5
Fig. 5

Lena results for Q=4, with a simulated optical system performing the energy summation. (a) S=1, (b) S=2, (c) S=4.

Fig. 6
Fig. 6

Prescribed clique geometry for 4×4 uniform-intensity spot array: (a) S=1, (b) S=2.

Fig. 7
Fig. 7

Spot array results for Q=, S=1.

Fig. 8
Fig. 8

Spot array results for Q=2: (a) S=1, (b) S=2.

Fig. 9
Fig. 9

Prescribed clique geometry for block text, S=1.

Fig. 10
Fig. 10

Block-text results for Q=, S=1, with a simulated optical system performing the energy summation.

Fig. 11
Fig. 11

Block-text results for Q=2, with a simulated optical system performing the energy summation. (a) S=1, (b) S=2, (c) S=4.

Tables (3)

Tables Icon

Table 1 Objective Performance Measures for the Lena Experiment

Tables Icon

Table 2 Objective Performance Measures for the Uniform-Intensity Spot Arrays

Tables Icon

Table 3 Objective Performance Measures for the Block-Text Experiment

Equations (50)

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E^i=ΓiI(u)du,
I(u)=Ω exp[jϕ(x)]exp(-j2πux)dx2.
|exp[jϕ(x)]| =1,
ϕ(x)ΦQ,forallxΩ,
fn+1=T1T2TMfn,
Ti  1+λi(Pi-1)(i=1, 2, , M).
fn+1=T1T2fn(f0arbitrary),
J(g)=P1g-g+P2g-g.
J(fn+1)J(T2fn)J(fn)
0λiΓ(fn)(i=1, 2),
CCP  h(x):h(x)=ti,|ti| =1forxΩi,   i=1, , Nc;h(x)=0 forxΩc,
 
CCP=CFSCCP,1CCP,2CCP,Nc,
CCP,i  {h(x):h(x)=ti,
where|ti| =1forxΩi}.
PCP,if(x)=exp[j arg(f¯i)]forxΩif(x)forxΩic,
f¯i  Ωif(x)dxΩi1dx.
PFSf(x)  f(x)forallxΩ0forallxΩc.
ΦQ=n2πQ,n=0, 1, , Q-1.
CQP  h(x):h(x)=tiCQforxΩi, i=1, , Nc;h(x)=0forxΩc.
 
CQP=CFSCQP,1CQP,2CQP,Nc,
CQP,i  {h(x):h(x)=tiCQforxΩi}.
PQP,if(x)=exp(jα)forxΩi,f(x)forxΩic,
|α-arg(f¯i)| =minγΦQ|γ-arg(f¯i)|.
CPM={h(x)H(u):|H(u)| =M(u)},
PPMg(x)M(u)exp[jθG(u)],
Υ=i=1LΓi
ΓiΓj=(ij).
Ei  Γi|F(u)|2du.
CCE,i  {f(x):Ei[E^i-Δi, E^i+Δi]}.
PCE,if(x)  g(x)G(u),
G(u)=E^i+ΔiEi1/2F(u),Ei>E^i+Δi,uΓiF(u)|Ei-E^i|Δi,uΓiF(u) uΓicE^i-ΔiEi1/2F(u),Ei<E^i-Δi,uΓi.
CCE i=1LCCE,i.
PCE  PCE,1PCE,2PCE,L.
f0(x)=1xT=(L/2, L/2)0otherwise,
fn+1=PQPPCEfn=fn.
ci=(c2-c1)ri+c1.
fc(x)=(c2-c1)fr(x)+c1.
Fc(u)=(c2-c1)Fr(u)+c1δ(u).
Fr(-u)=Fr*(u),
|Fr(-u)| = |Fr(u)|.
Fc(-u)=(c2-c1)Fr*(u)+c1δ(u)
|Fc(-u)| = |(c2-c1)||Fr(u)| = |Fc(u)|.
ci=ca+cbri,
fc(x)=ca+cbfr(x),
SAEE  iI|Ei-E^i|jI|Ej|,
η  iIDEijIEj×100%,
σ2  1ND iID(Ei-E¯)2E¯2,
E¯=1ND iIDEi.

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