Abstract

The accuracy and flexibility of the technique proposed by Davis et al. [Appl. Opt. 35, 2488 (1996)] for the encoding of the amplitude and the phase of a filter onto a single liquid-crystal spatial light modulator operating in a phase-only regime has been exploited to implement several filter designs in a convergent optical correlator. We have selected some filters, that given their mathematical structure showing some degree of rotational invariance, or having a parameter to regulate their behavior, require a more precise encoding. We present correlation results of outstanding quality for various rotationally invariant filter designs that have never been previously implemented with a real-time optical correlator.

© 2003 Optical Society of America

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  1. B. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  2. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  3. 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]
  4. J. L. Pezzaniti, R. A. Chipman, “Phase-only modulation of a twisted nematic liquid-crystal TV by use of the eigenpolarization states,” Opt. Lett. 18, 1567–1569 (1993).
    [CrossRef] [PubMed]
  5. J. A. Davis, I. Moreno, P. Tsai, “Polarization eigenstates for twisted-nematic liquid-crystal displays,” Appl. Opt. 37, 937–945 (1998).
    [CrossRef]
  6. J. Nicolás, C. Iemmi, J. Campos, M. J. Yzuel, “Real-time correlator with liquid crystal panels: modulation optimization,” in 4th Iberoamerican Meeting on Optics and 7th Latin American Meeting on Optics, Lasers, and Their Applications, V. L. Brudny, S. A. Ledesma, M. C. Marconi, eds., Proc. SPIE4419, 604–607 (2001).
    [CrossRef]
  7. J. Nicolás, J. Campos, M. J. Yzuel, “Phase and amplitude modulation of elliptic polarization states by non-absorbing anisotropic elements: application to liquid crystal devices,” J. Opt. Soc. Am. A 19, 1013–1020 (2002).
    [CrossRef]
  8. 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]
  9. L. G. Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944–1950 (1995).
    [CrossRef] [PubMed]
  10. R. W. Cohn, M. Liang, “Pseudorandom phase-only encoding of real-time spatial light modulators,” Appl. Opt. 35, 2488–2498 (1996).
    [CrossRef] [PubMed]
  11. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Bessel function output from an optical correlator using an inverse filter written onto a phase-only spatial light modulator,” Appl. Opt. 38, 6709–6713 (1999).
    [CrossRef]
  12. J. Campos, A. Marquez, M. J. Yzuel, J. A. Davis, D. M. Cottrell, I. Moreno, “Fully complex synthetic discriminant functions written onto phase-only filters,” Appl. Opt. 39, 5965–5970 (2000).
    [CrossRef]
  13. M. J. Yzuel, J. Campos, A. Marquez, J. C. Escalera, J. A. Davis, C. Iemmi, S. Ledesma, “Inherent apodization of lenses encoded on liquid crystal devices,” Appl. Opt. 39, 6034–6039 (2000).
    [CrossRef]
  14. J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. Yzuel, I. Moreno, “Encoding complex diffractive optical elements onto a phase-only liquid crystal spatial light modulator,” Opt. Eng. 40, 327–329 (2001).
    [CrossRef]
  15. A. Marquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
    [CrossRef]
  16. J. A. Davis, D. A. Smith, D. E. McNamara, D. Cottrell, J. Campos, “Fractional derivatives—Analysis and experimental implementation,” Appl. Opt. 40, 5943–5948 (2001).
    [CrossRef]
  17. Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” App. Opt. 21, 4016–4019 (1982).
    [CrossRef]
  18. O. Gualdrón, H. H. Arsenault, “Improved methods for invariant pattern recognition,” in Real-Time Optical Information Processing, B. Javidi, J. Horner, eds., (Academic, San Diego, Calif., 1994) pp. 89–113.
  19. J. J. Esteve-Taboada, J. García, C. Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” App. Opt. 39, 5998–6005 (2000).
    [CrossRef]
  20. J. E. Guerrero, “Verificación y localización de objetos mediante correlación tridimensional de imágenes de rango,” Ph.D. dissertation (Universidad Industrial de Santander, 2001).
  21. X.-W. Chen, Z.-P. Chen, “Amplitude-modulated circular-harmonic filter for pattern recognition,” Appl. Opt. 34, 879–885 (1995).
    [CrossRef] [PubMed]
  22. L. Leclerc, Y. Sheng, H. H. Arsenault, “Rotation invariant phase-only and binary phase-only correlation,” Appl. Opt. 28, 1251–1256 (1989).
    [CrossRef] [PubMed]
  23. J. A. Davis, L. R. York, D. M. Cottrell, “Rotationally invariant Fresnel lens-encoded circular harmonic binary phase-only filters,” Appl. Opt. 30, 1820–1825 (1991).
    [CrossRef] [PubMed]
  24. L. Leclerc, Y. Sheng, H. H. Arsenault, “Optical binary phase-only filters for circular harmonic correlators,” Appl. Opt. 30, 4643–4649 (1991).
    [CrossRef] [PubMed]
  25. J. P. Drolet, L. Leclerc, Y. Sheng, H. H. Arsenault, “Real-time binary phase-only circular harmonic filters using a liquid crystal television in the Fourier plane of a optical correlator,” Opt. Eng. 31, 939–946 (1992).
    [CrossRef]
  26. Y. Sheng, L. G. Neto, D. Roberge, L. Shen, G. Paulhus, “Phase-only SLMs for optical pattern recognition and security systems,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Proc. SPIECR65, 103–127 (1996).
  27. O. Gualdrón, H. H. Arsenault, “Phase-derived circular harmonic filter,” Opt. Commun. 104, 32–34 (1993).
    [CrossRef]
  28. P. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  29. P. Réfrégier, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
    [CrossRef]

2002 (1)

2001 (3)

2000 (3)

1999 (2)

1998 (1)

1996 (1)

1995 (2)

1993 (2)

1992 (1)

J. P. Drolet, L. Leclerc, Y. Sheng, H. H. Arsenault, “Real-time binary phase-only circular harmonic filters using a liquid crystal television in the Fourier plane of a optical correlator,” Opt. Eng. 31, 939–946 (1992).
[CrossRef]

1991 (4)

1990 (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]

1989 (1)

1982 (1)

Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” App. Opt. 21, 4016–4019 (1982).
[CrossRef]

1966 (1)

1964 (1)

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

Arsenault, H. H.

O. Gualdrón, H. H. Arsenault, “Phase-derived circular harmonic filter,” Opt. Commun. 104, 32–34 (1993).
[CrossRef]

J. P. Drolet, L. Leclerc, Y. Sheng, H. H. Arsenault, “Real-time binary phase-only circular harmonic filters using a liquid crystal television in the Fourier plane of a optical correlator,” Opt. Eng. 31, 939–946 (1992).
[CrossRef]

L. Leclerc, Y. Sheng, H. H. Arsenault, “Optical binary phase-only filters for circular harmonic correlators,” Appl. Opt. 30, 4643–4649 (1991).
[CrossRef] [PubMed]

L. Leclerc, Y. Sheng, H. H. Arsenault, “Rotation invariant phase-only and binary phase-only correlation,” Appl. Opt. 28, 1251–1256 (1989).
[CrossRef] [PubMed]

Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” App. Opt. 21, 4016–4019 (1982).
[CrossRef]

O. Gualdrón, H. H. Arsenault, “Improved methods for invariant pattern recognition,” in Real-Time Optical Information Processing, B. Javidi, J. Horner, eds., (Academic, San Diego, Calif., 1994) pp. 89–113.

Campos, J.

J. Nicolás, J. Campos, M. J. Yzuel, “Phase and amplitude modulation of elliptic polarization states by non-absorbing anisotropic elements: application to liquid crystal devices,” J. Opt. Soc. Am. A 19, 1013–1020 (2002).
[CrossRef]

J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. Yzuel, I. Moreno, “Encoding complex diffractive optical elements onto a phase-only liquid crystal spatial light modulator,” Opt. Eng. 40, 327–329 (2001).
[CrossRef]

A. Marquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
[CrossRef]

J. A. Davis, D. A. Smith, D. E. McNamara, D. Cottrell, J. Campos, “Fractional derivatives—Analysis and experimental implementation,” Appl. Opt. 40, 5943–5948 (2001).
[CrossRef]

J. Campos, A. Marquez, M. J. Yzuel, J. A. Davis, D. M. Cottrell, I. Moreno, “Fully complex synthetic discriminant functions written onto phase-only filters,” Appl. Opt. 39, 5965–5970 (2000).
[CrossRef]

M. J. Yzuel, J. Campos, A. Marquez, J. C. Escalera, J. A. Davis, C. Iemmi, S. Ledesma, “Inherent apodization of lenses encoded on liquid crystal devices,” Appl. Opt. 39, 6034–6039 (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]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Bessel function output from an optical correlator using an inverse filter written onto a phase-only spatial light modulator,” Appl. Opt. 38, 6709–6713 (1999).
[CrossRef]

J. Nicolás, C. Iemmi, J. Campos, M. J. Yzuel, “Real-time correlator with liquid crystal panels: modulation optimization,” in 4th Iberoamerican Meeting on Optics and 7th Latin American Meeting on Optics, Lasers, and Their Applications, V. L. Brudny, S. A. Ledesma, M. C. Marconi, eds., Proc. SPIE4419, 604–607 (2001).
[CrossRef]

Chen, X.-W.

Chen, Z.-P.

Chipman, R. A.

Cohn, R. W.

Cottrell, D.

Cottrell, D. M.

Davis, J. A.

J. A. Davis, D. A. Smith, D. E. McNamara, D. Cottrell, J. Campos, “Fractional derivatives—Analysis and experimental implementation,” Appl. Opt. 40, 5943–5948 (2001).
[CrossRef]

A. Marquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
[CrossRef]

J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. Yzuel, I. Moreno, “Encoding complex diffractive optical elements onto a phase-only liquid crystal spatial light modulator,” Opt. Eng. 40, 327–329 (2001).
[CrossRef]

J. Campos, A. Marquez, M. J. Yzuel, J. A. Davis, D. M. Cottrell, I. Moreno, “Fully complex synthetic discriminant functions written onto phase-only filters,” Appl. Opt. 39, 5965–5970 (2000).
[CrossRef]

M. J. Yzuel, J. Campos, A. Marquez, J. C. Escalera, J. A. Davis, C. Iemmi, S. Ledesma, “Inherent apodization of lenses encoded on liquid crystal devices,” Appl. Opt. 39, 6034–6039 (2000).
[CrossRef]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Bessel function output from an optical correlator using an inverse filter written onto a phase-only spatial light modulator,” Appl. Opt. 38, 6709–6713 (1999).
[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]

J. A. Davis, I. Moreno, P. Tsai, “Polarization eigenstates for twisted-nematic liquid-crystal displays,” Appl. Opt. 37, 937–945 (1998).
[CrossRef]

J. A. Davis, L. R. York, D. M. Cottrell, “Rotationally invariant Fresnel lens-encoded circular harmonic binary phase-only filters,” Appl. Opt. 30, 1820–1825 (1991).
[CrossRef] [PubMed]

Drolet, J. P.

J. P. Drolet, L. Leclerc, Y. Sheng, H. H. Arsenault, “Real-time binary phase-only circular harmonic filters using a liquid crystal television in the Fourier plane of a optical correlator,” Opt. Eng. 31, 939–946 (1992).
[CrossRef]

Escalera, J. C.

Esteve-Taboada, J. J.

J. J. Esteve-Taboada, J. García, C. Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” App. Opt. 39, 5998–6005 (2000).
[CrossRef]

Ferreira, C.

J. J. Esteve-Taboada, J. García, C. Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” App. Opt. 39, 5998–6005 (2000).
[CrossRef]

García, J.

J. J. Esteve-Taboada, J. García, C. Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” App. Opt. 39, 5998–6005 (2000).
[CrossRef]

Goodman, J. W.

Gualdrón, O.

O. Gualdrón, H. H. Arsenault, “Phase-derived circular harmonic filter,” Opt. Commun. 104, 32–34 (1993).
[CrossRef]

O. Gualdrón, H. H. Arsenault, “Improved methods for invariant pattern recognition,” in Real-Time Optical Information Processing, B. Javidi, J. Horner, eds., (Academic, San Diego, Calif., 1994) pp. 89–113.

Guerrero, J. E.

J. E. Guerrero, “Verificación y localización de objetos mediante correlación tridimensional de imágenes de rango,” Ph.D. dissertation (Universidad Industrial de Santander, 2001).

Hsu, Y. N.

Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” App. Opt. 21, 4016–4019 (1982).
[CrossRef]

Iemmi, C.

A. Marquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
[CrossRef]

M. J. Yzuel, J. Campos, A. Marquez, J. C. Escalera, J. A. Davis, C. Iemmi, S. Ledesma, “Inherent apodization of lenses encoded on liquid crystal devices,” Appl. Opt. 39, 6034–6039 (2000).
[CrossRef]

J. Nicolás, C. Iemmi, J. Campos, M. J. Yzuel, “Real-time correlator with liquid crystal panels: modulation optimization,” in 4th Iberoamerican Meeting on Optics and 7th Latin American Meeting on Optics, Lasers, and Their Applications, V. L. Brudny, S. A. Ledesma, M. C. Marconi, eds., Proc. SPIE4419, 604–607 (2001).
[CrossRef]

Leclerc, L.

Ledesma, S.

Liang, M.

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]

Marquez, A.

McNamara, D. E.

J. A. Davis, D. A. Smith, D. E. McNamara, D. Cottrell, J. Campos, “Fractional derivatives—Analysis and experimental implementation,” Appl. Opt. 40, 5943–5948 (2001).
[CrossRef]

J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. Yzuel, I. Moreno, “Encoding complex diffractive optical elements onto a phase-only liquid crystal spatial light modulator,” Opt. Eng. 40, 327–329 (2001).
[CrossRef]

Moreno, I.

Neto, L. G.

L. G. Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944–1950 (1995).
[CrossRef] [PubMed]

Y. Sheng, L. G. Neto, D. Roberge, L. Shen, G. Paulhus, “Phase-only SLMs for optical pattern recognition and security systems,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Proc. SPIECR65, 103–127 (1996).

Nicolás, J.

J. Nicolás, J. Campos, M. J. Yzuel, “Phase and amplitude modulation of elliptic polarization states by non-absorbing anisotropic elements: application to liquid crystal devices,” J. Opt. Soc. Am. A 19, 1013–1020 (2002).
[CrossRef]

J. Nicolás, C. Iemmi, J. Campos, M. J. Yzuel, “Real-time correlator with liquid crystal panels: modulation optimization,” in 4th Iberoamerican Meeting on Optics and 7th Latin American Meeting on Optics, Lasers, and Their Applications, V. L. Brudny, S. A. Ledesma, M. C. Marconi, eds., Proc. SPIE4419, 604–607 (2001).
[CrossRef]

Paulhus, G.

Y. Sheng, L. G. Neto, D. Roberge, L. Shen, G. Paulhus, “Phase-only SLMs for optical pattern recognition and security systems,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Proc. SPIECR65, 103–127 (1996).

Pezzaniti, J. L.

Réfrégier, P.

P. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
[CrossRef] [PubMed]

P. Réfrégier, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
[CrossRef]

Roberge, D.

L. G. Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944–1950 (1995).
[CrossRef] [PubMed]

Y. Sheng, L. G. Neto, D. Roberge, L. Shen, G. Paulhus, “Phase-only SLMs for optical pattern recognition and security systems,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Proc. SPIECR65, 103–127 (1996).

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]

Shen, L.

Y. Sheng, L. G. Neto, D. Roberge, L. Shen, G. Paulhus, “Phase-only SLMs for optical pattern recognition and security systems,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Proc. SPIECR65, 103–127 (1996).

Sheng, Y.

L. G. Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944–1950 (1995).
[CrossRef] [PubMed]

J. P. Drolet, L. Leclerc, Y. Sheng, H. H. Arsenault, “Real-time binary phase-only circular harmonic filters using a liquid crystal television in the Fourier plane of a optical correlator,” Opt. Eng. 31, 939–946 (1992).
[CrossRef]

L. Leclerc, Y. Sheng, H. H. Arsenault, “Optical binary phase-only filters for circular harmonic correlators,” Appl. Opt. 30, 4643–4649 (1991).
[CrossRef] [PubMed]

L. Leclerc, Y. Sheng, H. H. Arsenault, “Rotation invariant phase-only and binary phase-only correlation,” Appl. Opt. 28, 1251–1256 (1989).
[CrossRef] [PubMed]

Y. Sheng, L. G. Neto, D. Roberge, L. Shen, G. Paulhus, “Phase-only SLMs for optical pattern recognition and security systems,” in Optoelectronic Devices and Systems for Processing, B. Javidi, K. M. Johnson, eds., Proc. SPIECR65, 103–127 (1996).

Smith, D. A.

Tsai, P.

VanderLugt, B.

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

Weaver, C. S.

York, L. R.

Yzuel, M.

J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. Yzuel, I. Moreno, “Encoding complex diffractive optical elements onto a phase-only liquid crystal spatial light modulator,” Opt. Eng. 40, 327–329 (2001).
[CrossRef]

Yzuel, M. J.

J. Nicolás, J. Campos, M. J. Yzuel, “Phase and amplitude modulation of elliptic polarization states by non-absorbing anisotropic elements: application to liquid crystal devices,” J. Opt. Soc. Am. A 19, 1013–1020 (2002).
[CrossRef]

A. Marquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
[CrossRef]

M. J. Yzuel, J. Campos, A. Marquez, J. C. Escalera, J. A. Davis, C. Iemmi, S. Ledesma, “Inherent apodization of lenses encoded on liquid crystal devices,” Appl. Opt. 39, 6034–6039 (2000).
[CrossRef]

J. Campos, A. Marquez, M. J. Yzuel, J. A. Davis, D. M. Cottrell, I. Moreno, “Fully complex synthetic discriminant functions written onto phase-only filters,” Appl. Opt. 39, 5965–5970 (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]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Bessel function output from an optical correlator using an inverse filter written onto a phase-only spatial light modulator,” Appl. Opt. 38, 6709–6713 (1999).
[CrossRef]

J. Nicolás, C. Iemmi, J. Campos, M. J. Yzuel, “Real-time correlator with liquid crystal panels: modulation optimization,” in 4th Iberoamerican Meeting on Optics and 7th Latin American Meeting on Optics, Lasers, and Their Applications, V. L. Brudny, S. A. Ledesma, M. C. Marconi, eds., Proc. SPIE4419, 604–607 (2001).
[CrossRef]

App. Opt. (2)

Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” App. Opt. 21, 4016–4019 (1982).
[CrossRef]

J. J. Esteve-Taboada, J. García, C. Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” App. Opt. 39, 5998–6005 (2000).
[CrossRef]

Appl. Opt. (14)

J. A. Davis, I. Moreno, P. Tsai, “Polarization eigenstates for twisted-nematic liquid-crystal displays,” Appl. Opt. 37, 937–945 (1998).
[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]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Bessel function output from an optical correlator using an inverse filter written onto a phase-only spatial light modulator,” Appl. Opt. 38, 6709–6713 (1999).
[CrossRef]

J. Campos, A. Marquez, M. J. Yzuel, J. A. Davis, D. M. Cottrell, I. Moreno, “Fully complex synthetic discriminant functions written onto phase-only filters,” Appl. Opt. 39, 5965–5970 (2000).
[CrossRef]

M. J. Yzuel, J. Campos, A. Marquez, J. C. Escalera, J. A. Davis, C. Iemmi, S. Ledesma, “Inherent apodization of lenses encoded on liquid crystal devices,” Appl. Opt. 39, 6034–6039 (2000).
[CrossRef]

A. Marquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
[CrossRef]

J. A. Davis, D. A. Smith, D. E. McNamara, D. Cottrell, J. Campos, “Fractional derivatives—Analysis and experimental implementation,” Appl. Opt. 40, 5943–5948 (2001).
[CrossRef]

L. Leclerc, Y. Sheng, H. H. Arsenault, “Rotation invariant phase-only and binary phase-only correlation,” Appl. Opt. 28, 1251–1256 (1989).
[CrossRef] [PubMed]

J. A. Davis, L. R. York, D. M. Cottrell, “Rotationally invariant Fresnel lens-encoded circular harmonic binary phase-only filters,” Appl. Opt. 30, 1820–1825 (1991).
[CrossRef] [PubMed]

L. Leclerc, Y. Sheng, H. H. Arsenault, “Optical binary phase-only filters for circular harmonic correlators,” Appl. Opt. 30, 4643–4649 (1991).
[CrossRef] [PubMed]

X.-W. Chen, Z.-P. Chen, “Amplitude-modulated circular-harmonic filter for pattern recognition,” Appl. Opt. 34, 879–885 (1995).
[CrossRef] [PubMed]

L. G. Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944–1950 (1995).
[CrossRef] [PubMed]

R. W. Cohn, M. Liang, “Pseudorandom phase-only encoding of real-time spatial light modulators,” Appl. Opt. 35, 2488–2498 (1996).
[CrossRef] [PubMed]

C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

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

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Figures (9)

Fig. 1
Fig. 1

Convergent correlator setup used for the experiments: WP, wave plate; P, polarizer; L, lens; LCD, liquid-crystal display.

Fig. 2
Fig. 2

Image consisting of an aerial view of a car used for the experiments.

Fig. 3
Fig. 3

Intensity distribution at the correlator output, showing correlation zone (left-hand side) and the DC zone (center).

Fig. 4
Fig. 4

Side view of four correlation patterns from higher to lower: POF, PIF(95), PIF(90), and PIF(85).

Fig. 5
Fig. 5

Simulation result of the correlation intensity between the Porsche and its second-order CHF.

Fig. 6
Fig. 6

Experimental result of the correlation intensity between the Porsche and its second-order CHF.

Fig. 7
Fig. 7

Experimental correlation intensity between the image of the car and three OTCHFs with sigma taking the values of 0.0300 (back), 0.0030 (middle), and 0.0003 (front).

Fig. 8
Fig. 8

Experimental result of the correlation intensity between the image of the car and its second-order PDCHF.

Fig. 9
Fig. 9

Experimental correlation results to verify the rotation-invariant response of the PDCHF. Intensity of the correlation plane (top), the input scene (middle), and side view of the correlation peaks (bottom).

Equations (13)

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Tu=expiMuϕu+2πuA
Tu=n=- Tnuexpinϕu,
Tnu=12π-ππexpiMuϕuexp-inϕudϕu =sinπn-Muπn-Mu.
fkr, θ=fkrexpikθ,
fkr=12π02π fr, θexp-ikθdθ.
fr, θ=k=- fkr, θ.
Fρ, ϕ=k=- Fkρ, ϕ.
Fkρ=2π-jk0 fkrJ02πrρrdr.
Hkρ, ϕ=Fkρexpikϕ.
Hkρ, ϕ=Fkρ|Fkρ|expikϕ.
Hkρ, ϕ=Fkρexpikϕ=expikϕ02πFkρ, ϕ|Fkρ, ϕ|exp-ikϕdϕ.
Hkρ, ϕ=Fkρ02π |Fρ, ϕ|2dϕ+σexpikϕ,
Hu, v=Fu, v|Fu, v|if |Fu, v|<tFu, v|Fu, v|2 tif |Fu, v|t,

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