Abstract

The method of generalized projections is used to design pure-amplitude diffraction holograms that generate gray-scale images. Two algorithms are presented: the direct method nonlinearly constrains the hologram transmittance to the range of real values in [0,1]; the indirect method constrains the transmittance values to the real axis and linearly transforms the resulting values to the range [0,1]. Digital amplitude holograms were simulated by quantizing the amplitude holograms resulting from the indirect method. Performance is demonstrated with objective measures (error, efficiency, and variance) as well as with subjective comparison of images. Test images included a photographic quality image of Lena, a uniform intensity spot array, and a binary amplitude block text image.

© 1998 Optical Society of America

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References

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1997 (1)

1996 (2)

1995 (3)

1994 (1)

1991 (2)

1990 (2)

1989 (3)

F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
[CrossRef] [PubMed]

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

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

1988 (1)

J. P. Allebach and D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. SPIE 884, 2–9 (1988).
[CrossRef]

1987 (1)

1984 (2)

1982 (1)

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

Allebach, J. P.

Catino, W. C.

Cohn, R. W.

Dietrich, C. H.

Dou, R.

Fainman, Y.

Flannery, D. L.

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

Gerchberg, R. W.

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

Gianino, P. D.

Giles, M. K.

Horner, J. L.

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

Jennison, B. K.

Khoury, J.

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.

Liang, M.

LoCicero, J. L.

Manner, R.

Marchand, P.

McCormick, F. B.

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

Missig, M. D.

Morris, G. M.

Nochte, S.

Papoulis, A.

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

Sawchuk, A. A.

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

Seidowitz, M. A.

Stark, H.

Strand, T. C.

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

Sweeney, D. W.

Urquhart, K. S.

Webb, H.

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

Woods, C. L.

Wyrowski, F.

Youla, D. C.

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

Zhang, H.

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. (9)

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 and 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 (6)

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 multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
[CrossRef]

Proc. IEEE (2)

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

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

Proc. SPIE (1)

J. P. Allebach and D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. SPIE 884, 2–9 (1988).
[CrossRef]

Other (8)

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).

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).

C. Warde and 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.

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (7)

Fig. 1
Fig. 1

Prescribed Lena pattern. (a) Entire far-field pattern (256×256), (b) Lena portion of the prescribed pattern (127 ×127).

Fig. 2
Fig. 2

Continuous amplitude hologram design results for the Lena image (127×127 with the dc peak blocked). (a) Reconstructed Lena from the direct method, (b) reconstructed Lena from the indirect method.

Fig. 3
Fig. 3

Quantized amplitude hologram design results for the Lena image by the indirect method (127×127 with the dc peak blocked). (a) Q=1024, (b) Q=512, (c) Q=256, (d) Q=128.

Fig. 4
Fig. 4

Prescribed intensity pattern for a 4×4 uniform intensity spot array with Nc=64. The gray lines delineate the elements of the DFT array.

Fig. 5
Fig. 5

Uniform intensity spot arrays generated by the indirect method. (a) Q=, (b) Q=64, (c) Q=16, (d) Q=8.

Fig. 6
Fig. 6

Prescribed intensity pattern for block text with Nc=1024. The gray lines delineate the elements of the DFT array.

Fig. 7
Fig. 7

Block text generated from quantized amplitude hologram design by the indirect method. (a) Q=, (b) Q=64, (c) Q=32, (d) Q=16.

Tables (3)

Tables Icon

Table 1 Normalized Mean Square Error for the Lena Amplitude Holograms Generated by the Indirect Method, Excluding Contributions from the dc Peak

Tables Icon

Table 2 Performance Measures for Uniform Intensity Spot-Array-Amplitude Holograms Generated by the Indirect Methoda

Tables Icon

Table 3 Performance Measures for Block Text Amplitude Holograms Generated by the Indirect Methoda

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

I(u)=Ωa(x)exp(-j2πu·x)dx2,
a(x)[0, 1]forallxΩ.
fn+1=T1T2TMfn,
Ti1+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),
CPM={h(x)H(u) : |H(u)|=M(u)},
PPMg(x)M(u)exp[jθG(u)],
CPA{h(x): h(x)=ai[0, 1]forxΩi,i=1, , Nc,
h(x)=0forxΩc},
CPA=CFSCPA,1CPA,2CPA,Nc,
CPA,i{h(x): h(x)=ai[0, 1]forxΩi}.
PPA,if(x)=f¯RiforxΩiandf¯Ri[0, 1]0forxΩiandf¯Ri<01forxΩiandf¯Ri>1f(x)forxΩic,
f¯RiΩifR(x)dxΩi1·dx
PFSf(x)=f(x)forallxΩ  0forallxΩc.
CPR{h(x): h(x)=riRforxΩi,
i=1, , Nc,
h(x)=0 forxΩc},
CPR=CFSCPR,1CPR,2CPR,Nc,
CPR,i{h(x): h(x)=riRforxΩi}.
PPR,if(x)=f¯RiforxΩif(x)forxΩic,
fs(x)=f(x)-fminfmax-fmin,
ηiID|F(ui)|2allj|F(uj)|2×100%,
ηTiID|F(ui)|2Ein×100%,
NMSEallu[α|F(u)|-M(u)]2alluM2(u),
α=allu[M(u)|F(u)|]allu|F(u)|2.
NMSE=1-M(u),|F(u)|2M(u)2F(u)2,
M(u),|F(u)|alluM(u)|F(u)|,
M(u)2M(u),M(u),
σ21NDiID(|F(ui)|-F¯)2F¯2,
F¯=1NDiID|F(ui)|.

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