Abstract

The convergent correlator is widely used but it presents the drawback of the alignment requirements: fine focusing of the input scene Fourier transform on the filter plane, filter centering, scaling the scene Fourier transform to match the filter size, and azimuth matching of the filter with the input scene. We propose a set of tests to obtain a precise alignment of the convergent correlator. These methods are based on frequency filtering properties and they are applicable either for amplitude input or for phase-encoded input. The tests we present allow us to fulfill all the alignment requirements. The theory on which these tests are based is explained. The experimental results obtained during the alignment procedure are presented. We show some additional verifications of the correct alignment of the convergent correlator.

© 2002 Optical Society of America

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References

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  1. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. J. C. Kirsch, D. A. Gregory, “Video rate optical correlation using a magneto-optic spatial light modulator,” Opt. Eng. 29 (9), 1122–1128 (1990).
    [CrossRef]
  11. I. Moreno, J. Campos, M. J. Yzuel, V. Kober, “Implementation of bipolar real-valued input scenes in a real-time optical correlator: application to color pattern recognition,” Opt. Eng. 37, 144–150 (1998).
    [CrossRef]
  12. A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–3564 (2001).
    [CrossRef]
  13. R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
    [CrossRef]
  14. K. Styczynski, J. Campos, M. J. Yzuel, K. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with high diffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
    [CrossRef]
  15. R. W. Cohn, “Pseudorandom encoding of complex-valued functions onto amplitude-coupled phase modulators,” J. Opt. Soc. Am. A 15, 868–883 (1998).
    [CrossRef]
  16. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
    [CrossRef]
  17. A. VanderLugt, “The effects of small displacements of spatial filters,” Appl. Opt. 6, 1221–1225 (1967).
    [CrossRef]
  18. L. Cai, Y. Jin, S. Zhou, P. Yeh, N. Marzwell, H. Liu, “Translational sensitivity adjustable compact optical correlator and its application for fingerprint recognition,” Opt. Eng. 35, 415–422 (1996).
    [CrossRef]
  19. M. Montes-Usategui, S. E. Monroe, R. D. Juday, “Automated self-alignment procedure for optical correlators,” Opt. Eng. 36, 1782–1791 (1997).
    [CrossRef]
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    [CrossRef]
  21. 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]
  22. J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martínez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
    [CrossRef]
  23. A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
    [CrossRef]

2001 (1)

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–3564 (2001).
[CrossRef]

2000 (1)

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

1999 (1)

1998 (2)

R. W. Cohn, “Pseudorandom encoding of complex-valued functions onto amplitude-coupled phase modulators,” J. Opt. Soc. Am. A 15, 868–883 (1998).
[CrossRef]

I. Moreno, J. Campos, M. J. Yzuel, V. Kober, “Implementation of bipolar real-valued input scenes in a real-time optical correlator: application to color pattern recognition,” Opt. Eng. 37, 144–150 (1998).
[CrossRef]

1997 (1)

M. Montes-Usategui, S. E. Monroe, R. D. Juday, “Automated self-alignment procedure for optical correlators,” Opt. Eng. 36, 1782–1791 (1997).
[CrossRef]

1996 (2)

L. Cai, Y. Jin, S. Zhou, P. Yeh, N. Marzwell, H. Liu, “Translational sensitivity adjustable compact optical correlator and its application for fingerprint recognition,” Opt. Eng. 35, 415–422 (1996).
[CrossRef]

J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martínez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
[CrossRef]

1995 (2)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1995).
[CrossRef]

K. Styczynski, J. Campos, M. J. Yzuel, K. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with high diffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
[CrossRef]

1994 (2)

1991 (1)

1990 (3)

B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[CrossRef]

J. C. Kirsch, D. A. Gregory, “Video rate optical correlation using a magneto-optic spatial light modulator,” Opt. Eng. 29 (9), 1122–1128 (1990).
[CrossRef]

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]

1989 (1)

1985 (1)

1984 (1)

1967 (1)

1966 (1)

1964 (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Cai, L.

L. Cai, Y. Jin, S. Zhou, P. Yeh, N. Marzwell, H. Liu, “Translational sensitivity adjustable compact optical correlator and its application for fingerprint recognition,” Opt. Eng. 35, 415–422 (1996).
[CrossRef]

Campos, J.

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–3564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[CrossRef]

I. Moreno, J. Campos, M. J. Yzuel, V. Kober, “Implementation of bipolar real-valued input scenes in a real-time optical correlator: application to color pattern recognition,” Opt. Eng. 37, 144–150 (1998).
[CrossRef]

K. Styczynski, J. Campos, M. J. Yzuel, K. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with high diffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
[CrossRef]

Chalasinska-Macukow, K.

K. Styczynski, J. Campos, M. J. Yzuel, K. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with high diffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
[CrossRef]

Cohn, R. W.

Cottrell, D. M.

Coy, J. A.

J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martínez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
[CrossRef]

Davis, J. A.

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–3564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[CrossRef]

H.-K. Liu, J. A. Davis, R. A. Lilly, “Optical-data-processing properties of a liquid-crystal television spatial light modulator,” Opt. Lett. 10, 635–637 (1985).
[CrossRef] [PubMed]

Gianino, P. D.

Goldstein, D. H.

R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
[CrossRef]

Goodman, J. W.

Gregory, D. A.

J. C. Kirsch, D. A. Gregory, “Video rate optical correlation using a magneto-optic spatial light modulator,” Opt. Eng. 29 (9), 1122–1128 (1990).
[CrossRef]

Grosz, D. F.

J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martínez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
[CrossRef]

Hassebrook, L.

Horner, J. L.

Iemmi, C.

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–3564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Javidi, B.

Jin, Y.

L. Cai, Y. Jin, S. Zhou, P. Yeh, N. Marzwell, H. Liu, “Translational sensitivity adjustable compact optical correlator and its application for fingerprint recognition,” Opt. Eng. 35, 415–422 (1996).
[CrossRef]

Juday, R. D.

M. Montes-Usategui, S. E. Monroe, R. D. Juday, “Automated self-alignment procedure for optical correlators,” Opt. Eng. 36, 1782–1791 (1997).
[CrossRef]

Kallman, R. R.

R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
[CrossRef]

Kirsch, J. C.

J. C. Kirsch, D. A. Gregory, “Video rate optical correlation using a magneto-optic spatial light modulator,” Opt. Eng. 29 (9), 1122–1128 (1990).
[CrossRef]

Kober, V.

I. Moreno, J. Campos, M. J. Yzuel, V. Kober, “Implementation of bipolar real-valued input scenes in a real-time optical correlator: application to color pattern recognition,” Opt. Eng. 37, 144–150 (1998).
[CrossRef]

Laude, V.

Lilly, R. A.

Liu, H.

L. Cai, Y. Jin, S. Zhou, P. Yeh, N. Marzwell, H. Liu, “Translational sensitivity adjustable compact optical correlator and its application for fingerprint recognition,” Opt. Eng. 35, 415–422 (1996).
[CrossRef]

Liu, H.-K.

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]

Márquez, A.

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–3564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Martínez, O. E.

J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martínez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
[CrossRef]

Marzwell, N.

L. Cai, Y. Jin, S. Zhou, P. Yeh, N. Marzwell, H. Liu, “Translational sensitivity adjustable compact optical correlator and its application for fingerprint recognition,” Opt. Eng. 35, 415–422 (1996).
[CrossRef]

Monroe, S. E.

M. Montes-Usategui, S. E. Monroe, R. D. Juday, “Automated self-alignment procedure for optical correlators,” Opt. Eng. 36, 1782–1791 (1997).
[CrossRef]

Montes-Usategui, M.

M. Montes-Usategui, S. E. Monroe, R. D. Juday, “Automated self-alignment procedure for optical correlators,” Opt. Eng. 36, 1782–1791 (1997).
[CrossRef]

Moreno, A.

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Moreno, I.

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–3564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[CrossRef]

I. Moreno, J. Campos, M. J. Yzuel, V. Kober, “Implementation of bipolar real-valued input scenes in a real-time optical correlator: application to color pattern recognition,” Opt. Eng. 37, 144–150 (1998).
[CrossRef]

Réfrégier, P.

Robert, A.

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

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]

Styczynski, K.

K. Styczynski, J. Campos, M. J. Yzuel, K. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with high diffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
[CrossRef]

VanderLugt, A.

A. VanderLugt, “The effects of small displacements of spatial filters,” Appl. Opt. 6, 1221–1225 (1967).
[CrossRef]

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

A. VanderLugt, Optical Signal Processing, (Wiley, New York, 1992).

Vijaya Kumar, B. V. K.

Weaver, C. S.

Yeh, P.

L. Cai, Y. Jin, S. Zhou, P. Yeh, N. Marzwell, H. Liu, “Translational sensitivity adjustable compact optical correlator and its application for fingerprint recognition,” Opt. Eng. 35, 415–422 (1996).
[CrossRef]

Yzuel, M. J.

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–3564 (2001).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[CrossRef]

I. Moreno, J. Campos, M. J. Yzuel, V. Kober, “Implementation of bipolar real-valued input scenes in a real-time optical correlator: application to color pattern recognition,” Opt. Eng. 37, 144–150 (1998).
[CrossRef]

K. Styczynski, J. Campos, M. J. Yzuel, K. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with high diffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
[CrossRef]

Zaldarriaga, M.

J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martínez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
[CrossRef]

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1995).
[CrossRef]

Zhou, S.

L. Cai, Y. Jin, S. Zhou, P. Yeh, N. Marzwell, H. Liu, “Translational sensitivity adjustable compact optical correlator and its application for fingerprint recognition,” Opt. Eng. 35, 415–422 (1996).
[CrossRef]

Appl. Opt. (6)

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

K. Styczynski, J. Campos, M. J. Yzuel, K. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with high diffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
[CrossRef]

Opt. Eng. (9)

L. Cai, Y. Jin, S. Zhou, P. Yeh, N. Marzwell, H. Liu, “Translational sensitivity adjustable compact optical correlator and its application for fingerprint recognition,” Opt. Eng. 35, 415–422 (1996).
[CrossRef]

M. Montes-Usategui, S. E. Monroe, R. D. Juday, “Automated self-alignment procedure for optical correlators,” Opt. Eng. 36, 1782–1791 (1997).
[CrossRef]

J. C. Kirsch, D. A. Gregory, “Video rate optical correlation using a magneto-optic spatial light modulator,” Opt. Eng. 29 (9), 1122–1128 (1990).
[CrossRef]

I. Moreno, J. Campos, M. J. Yzuel, V. Kober, “Implementation of bipolar real-valued input scenes in a real-time optical correlator: application to color pattern recognition,” Opt. Eng. 37, 144–150 (1998).
[CrossRef]

A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, M. J. Yzuel, “Quantitative predictions of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. 40, 2558–3564 (2001).
[CrossRef]

R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
[CrossRef]

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]

J. A. Coy, M. Zaldarriaga, D. F. Grosz, O. E. Martínez, “Characterization of a liquid crystal television as a programmable spatial light modulator,” Opt. Eng. 35, 15–19 (1996).
[CrossRef]

A. Márquez, J. Campos, M. J. Yzuel, I. Moreno, J. A. Davis, C. Iemmi, A. Moreno, A. Robert, “Characterization of edge effects in twisted nematic liquid crystal displays,” Opt. Eng. 39, 3301–3307 (2000).
[CrossRef]

Opt. Lett. (3)

Science (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1995).
[CrossRef]

Other (1)

A. VanderLugt, Optical Signal Processing, (Wiley, New York, 1992).

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Figures (14)

Fig. 1
Fig. 1

Scheme of a convergent optical correlator: L1 and L2 are two convergent lenses, SLM1 and SLM2 are spatial-light modulators; C is the correlation plane. The arrows (x, y, z, w, α) represent the alignment functions.

Fig. 2
Fig. 2

Scheme of a convergent optical correlator, d is the distance between SLM2 and the S′ plane.

Fig. 3
Fig. 3

Filter focusing test and experimental results. (a) Implemented filter. Captures of the correlation plane for different off-focus positions of the filter (b and c), and for the filter on-focus (d).

Fig. 4
Fig. 4

Effect of the inversion of the sign of the diffraction orders: (a) original cosine grating, (b) effect of the π-phase shifting of one fundamental order, (c) contrast inversion as an effect of the π-phase shifting of the zero order.

Fig. 5
Fig. 5

Filter centering: (a) displayed scene, (b) squared magnitude Fourier transform of the scene, (c, d) images of the filtered scene for two off-axis positions of the filter center, (e) image of the filtered scene when the filter is centered in the x coordinate.

Fig. 6
Fig. 6

Azimuth alignment: (a) displayed scene, (b) squared magnitude Fourier transform of the scene. The arrows denote equivalent directions in the scene and its Fourier transform. (c) Proposed filter, (d) filtered image for a 5-deg. misalignment, (e) filtered image for a 1-deg. misalignment, (f) filtered image when the azimuth is well aligned.

Fig. 7
Fig. 7

Scale matching: (a) displayed scene, (b) squared magnitude Fourier transform of the scene, (c) proposed filter, (d) filtered image for a 150% scale mismatch, (e) filtered image for a 101% scale mismatch, (f) filtered image for matched scale.

Fig. 8
Fig. 8

Centering for phase-encoded scene correlator: (a) filtered scene for a 8-pixel misalignment, (b) filtered scene for the aligned system.

Fig. 9
Fig. 9

Azimuth for phase-encoded scene correlator: (a) filtered image for a 5-deg. misalignment, (b) filtered image when the azimuth is aligned.

Fig. 10
Fig. 10

Scale matching for phase-encoded scene correlator: (a) filtered image for a scale mismatch, (b) filtered image when the scale is matched.

Fig. 11
Fig. 11

Microscopic image of the center of the SLM once the filter is focused and centered. A cross has been sent to the filter modulator, and the arrow denotes the central pixel of the SLM.

Fig. 12
Fig. 12

Correlation example: (a) scene, (b) intensity distribution in the correlation plane.

Fig. 13
Fig. 13

Scheme of a SLM working with elliptically polarized light.

Fig. 14
Fig. 14

Complex representation of the SLM’s modulation: (a) amplitude only modulation for the scene SLM, (b) phase only modulation for the scene SLM, (c) phase only modulation for the filter SLM.

Equations (7)

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

ASpheru, v=A0 exp+ik2du2+v2,
ASLM 2u, v=A01-2δu×exp+ik2du2+v2,
ASu, v=A1 exp-i k2du2+v2δuδv-2c0δv,
ACx, y=A2 exp+i k2dCx2+ y2×c1-c2 exp+ikd2dF2 x2,
Ix, y=I0b1-b2 coskd2dF2 x2,
Ax, y=12+12cosax.
Ix, y=141+2 cos ϕ cosax+Δϕ2+cos2ax+Δϕ2,

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