Abstract

A multicriteria optimization method is introduced in order to find optimal filters for implementation on arbitrary spatial light modulators in the Fourier plane of an optical correlator. This method is applied to the trade-offs between noise robustness, sharpness of the correlation peak, and optical efficiency. A fast and simple algorithm is given in this case, which is independent of the particular form of the spatial light modulator coding constraint. It is used to characterize and to compare typical coding domains through the performances of their associated optimal filters.

© 1994 Optical Society of America

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  7. R. D. Juday, J. L. Lacroix, P. Karivaratha Rajan, “Selection of LCTV operating curves for input and filter,” in Optical Pattern Recognition III, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 78–82 (1992).
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1993 (3)

1992 (6)

1991 (1)

1990 (3)

1989 (3)

1988 (1)

Carlson, D. W.

B. V. K. Vijaya Kumar, C. Hendrix, D. W. Carlson, “Trade-offs in the design of correlation filters,” in Optical Pattern Recognition, J. L. Horner, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.CR40, 191–215 (1992).

Chavel, P.

V. Laude, S. Mazé, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

Cotariu, S. S.

S. S. Cotariu, S. E. Monroe, J. Knopp, “A live input, live filter, liquid crystal correlator,” in Advances in Optical Information Processing V, D. R. Pape, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1704, 248–256 (1992).

Farn, M. W.

Figue, J.

Flannery, D.

Fukushima, S.

T. Kurokawa, S. Fukushima, “Spatial light modulators using ferroelectric liquid crystal,” Opt. Quantum Electron. 24, 1151–1163 (1992).
[CrossRef]

Giles, M.

Goodman, J. W.

Gregory, D. A.

Hassebrook, L.

Hendrix, C.

Ph. Réfrégier, B. V. K. Vijaya Kumar, C. Hendrix, “Multicriteria optimal binary amplitude phase-only filters,” J. Opt. Soc. Am. A 9, 2118–2125 (1992).
[CrossRef]

B. V. K. Vijaya Kumar, C. Hendrix, D. W. Carlson, “Trade-offs in the design of correlation filters,” in Optical Pattern Recognition, J. L. Horner, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.CR40, 191–215 (1992).

Horner, J. L.

Joffre, P.

S. Mazé, P. Joffre, Ph. Réfrégier, “Influence of input information coding for correlation operations,” in Optics for Computers: Architectures and Technologies, G. J. Lebreton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1505, 20–31 (1992).

Juday, R. D.

Karivaratha Rajan, P.

B. V. K. Vijaya Kumar, R. D. Juday, P. Karivaratha Rajan, “Saturated filters,” J. Opt. Soc. Am. A 9, 405–412 (1992).
[CrossRef]

R. D. Juday, J. L. Lacroix, P. Karivaratha Rajan, “Selection of LCTV operating curves for input and filter,” in Optical Pattern Recognition III, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 78–82 (1992).

Kast, B. A.

Kirsch, J. A.

Knopp, J.

S. S. Cotariu, S. E. Monroe, J. Knopp, “A live input, live filter, liquid crystal correlator,” in Advances in Optical Information Processing V, D. R. Pape, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1704, 248–256 (1992).

Konforti, N.

Kurokawa, T.

T. Kurokawa, S. Fukushima, “Spatial light modulators using ferroelectric liquid crystal,” Opt. Quantum Electron. 24, 1151–1163 (1992).
[CrossRef]

Lacroix, J. L.

R. D. Juday, J. L. Lacroix, P. Karivaratha Rajan, “Selection of LCTV operating curves for input and filter,” in Optical Pattern Recognition III, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 78–82 (1992).

Laude, V.

V. Laude, S. Mazé, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

Lindell, S.

Lu, K.

K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

Mait, J. N.

J. N. Mait, J. van der Gracht, S. D. Sarama, “Diffractive filter design for SLM’s in optical processing and pattern recognition systems,” in Optical Pattern Recognition TV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 250–257 (1993).

Marom, E.

Mazé, S.

V. Laude, S. Mazé, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

S. Mazé, P. Joffre, Ph. Réfrégier, “Influence of input information coding for correlation operations,” in Optics for Computers: Architectures and Technologies, G. J. Lebreton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1505, 20–31 (1992).

Monroe, S. E.

S. S. Cotariu, S. E. Monroe, J. Knopp, “A live input, live filter, liquid crystal correlator,” in Advances in Optical Information Processing V, D. R. Pape, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1704, 248–256 (1992).

Réfrégier, Ph.

Saleh, B. E. A.

K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

Sarama, S. D.

J. N. Mait, J. van der Gracht, S. D. Sarama, “Diffractive filter design for SLM’s in optical processing and pattern recognition systems,” in Optical Pattern Recognition TV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 250–257 (1993).

Tarn, E. C.

van der Gracht, J.

J. N. Mait, J. van der Gracht, S. D. Sarama, “Diffractive filter design for SLM’s in optical processing and pattern recognition systems,” in Optical Pattern Recognition TV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 250–257 (1993).

Vijaya Kumar, B. V. K.

Wu, S.-T.

Yaroslavsky, L. P.

Appl. Opt. (9)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

V. Laude, S. Mazé, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

Opt. Eng. (1)

K. Lu, B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

T. Kurokawa, S. Fukushima, “Spatial light modulators using ferroelectric liquid crystal,” Opt. Quantum Electron. 24, 1151–1163 (1992).
[CrossRef]

Other (5)

S. S. Cotariu, S. E. Monroe, J. Knopp, “A live input, live filter, liquid crystal correlator,” in Advances in Optical Information Processing V, D. R. Pape, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1704, 248–256 (1992).

R. D. Juday, J. L. Lacroix, P. Karivaratha Rajan, “Selection of LCTV operating curves for input and filter,” in Optical Pattern Recognition III, D. P. Casasent, T.-H. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 78–82 (1992).

J. N. Mait, J. van der Gracht, S. D. Sarama, “Diffractive filter design for SLM’s in optical processing and pattern recognition systems,” in Optical Pattern Recognition TV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 250–257 (1993).

B. V. K. Vijaya Kumar, C. Hendrix, D. W. Carlson, “Trade-offs in the design of correlation filters,” in Optical Pattern Recognition, J. L. Horner, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.CR40, 191–215 (1992).

S. Mazé, P. Joffre, Ph. Réfrégier, “Influence of input information coding for correlation operations,” in Optics for Computers: Architectures and Technologies, G. J. Lebreton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1505, 20–31 (1992).

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Figures (9)

Fig. 1
Fig. 1

Schematic of the coherent optical correlator. Lens L1 forms the Fourier transform of the input image, displayed on SLM1, in its back focal plane, where the filter is displayed on SLM2. The correlation plane is observed in the back focal plane of lens L2.

Fig. 2
Fig. 2

Examples of coding domains: (a) unit disk; (b) phase only; (c) amplitude only; (d) spiral, parametrized by maximum phase shift K; (e) binary phase only; (f) ternary.

Fig. 3
Fig. 3

Comparison of filters through their criteria values, with an example of three criteria. Set E is composed of all combinations of criteria values generated by filters satisfying the coding-domain constraint. S OT is the set of OT filters that are obtained by a search for planes tangent to E and that are referenced by their normal vector û.

Fig. 4
Fig. 4

(a) 64 × 64 reference image used for simulations; (b) typical complex histogram of an OT filter, generated with the reference in (a) without the coding contraint. Each point of the histogram has to be projected on the coding constraint (here the spiral), which is rotated to find the global optimum.

Fig. 5
Fig. 5

Typical function fu(φ), measuring the quality of projected filters versus rotation angle φ. Optimal angle φS yields the OT filter satisfying the coding-domain contraint.

Fig. 6
Fig. 6

Optimal characteristics surface (OCS) for unit-disk coding. POF, MF, and IF stand for phase-only, matched, and inverse filters, respectively. Boundary curves C1, C2, and C3 joining these points represent, respectively, the optimal trade-offs between SNR and optical efficiency, between PCE and optical efficiency, and between SNR and PCE.

Fig. 7
Fig. 7

Optimal trade-offs between SNR and optical efficiency for the coding domains of Fig. 2.

Fig. 8
Fig. 8

Optimal trade-offs between PCE and optical efficiency for the coding domains of Fig. 2.

Fig. 9
Fig. 9

Optimal trade-offs between SNR and PCE for the coding domains of Fig. 2.

Equations (22)

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h ˆ u ˆ OT = argmin h ˆ D N { α E 1 ( h ˆ ) + β E 2 ( h ˆ ) + γ E 3 ( h ˆ ) } .
η H = | c 0 | 2 ,
SNR = η H MSE ,
PCE = η H CPE .
MSE = k C ˆ k , k | h ˆ k | 2 ,
CPE = k D ˆ k , k | h ˆ k | 2 .
E ( h ˆ ) = α MSE ( h ˆ ) + β CPE ( h ˆ ) 2 γ | c 0 ( h ˆ ) | ,
E ( h ˆ ) = k [ B ˆ u ˆ ] k , k | h ˆ k | 2 2 γ | k h ˆ k x ˆ k | ,
( h ˆ u ˆ 0 ) k = γ x ˆ k * [ B ˆ u ˆ ] k , k .
E φ ( h ˆ ) = k | h ˆ k ( h ˆ u ˆ 0 ) k exp ( i φ ) | 2 ,
h ˆ u ˆ φ = P D [ h ˆ u ˆ 0 exp ( i φ ) ] .
f u ˆ ( φ ) = E ( h ˆ u ˆ φ ) = E { P D [ h ˆ u ˆ 0 exp ( i φ ) ] } ,
E ( h ˆ ) = h ˆ B ˆ u ˆ h ˆ 2 γ | h ˆ T · x ˆ | .
E φ ( h ˆ ) = h ˆ B ˆ u ˆ h ˆ 2 γ [ h ˆ T · x ˆ exp ( i φ ) ] ,
E φ ( h ˆ ) = h ˆ B ˆ u ˆ h ˆ 2 γ | h ˆ T · x ˆ | cos [ φ ( h ˆ ) φ ] ,
E φ ( h ˆ ) E ( h ˆ ) ,
δ E φ ( h ˆ ) = E φ ( h ˆ + δ h ˆ ) E φ ( h ˆ ) ,
δ E φ ( h ˆ ) = 2 [ δ h ˆ B ˆ u ˆ h ˆ γδ h ˆ T · x ˆ exp ( i φ ) ] .
E φ [ h ˆ u ˆ 0 exp ( i φ ) ] = E [ h ˆ u ˆ 0 exp ( i φ ) ] = E ( h ˆ u ˆ 0 )
E φ ( h ˆ ) = [ h ˆ h ˆ u ˆ 0 exp ( i φ ) ] B ˆ u ˆ [ h ˆ h ˆ u ˆ 0 exp ( i φ ) ] + E ( h ˆ u ˆ 0 ) .
E φ ( h ˆ u ˆ φ ) E ( h ˆ u ˆ φ ) E ( h ˆ u ˆ S ) ,
min φ [ 0 , 2 π ] { f u ˆ ( φ ) } E ( h ˆ u ˆ S ) .

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