Abstract

We implement a fully automatic fast face recognition system by using a 1000 frame/s optical parallel correlator designed and assembled by us. The operational speed for the 1:N (i.e., matching one image against N, where N refers to the number of images in the database) identification experiment (4000 face images) amounts to less than 1.5 s, including the preprocessing and postprocessing times. The binary real-only matched filter is devised for the sake of face recognition, and the system is optimized by the false-rejection rate (FRR) and the false-acceptance rate (FAR), according to 300 samples selected by the biometrics guideline. From trial 1:N identification experiments with the optical parallel correlator, we acquired low error rates of 2.6% FRR and 1.3% FAR. Facial images of people wearing thin glasses or heavy makeup that rendered identification difficult were identified with this system.

© 2005 Optical Society of America

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References

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  1. M. Kaneko, O. Hasegawa, “Processing of face images and its applications,” IEICE Trans. Inf. Syst. E82-D, 589–600 (1999).
  2. A. Vanderlugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  3. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two things,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  4. A. Bergeron, “Optical correlator for industrial applications, quality control and target tracking,” Sensor Rev. 20, 316–321 (2000).
    [CrossRef]
  5. Y. Kobayashi, Y. Toyoda, “Development of an optical joint transform correlation system for fingerprint recognition,” Opt. Eng. 38, 1205–1210 (1999).
    [CrossRef]
  6. D. T. Carrott, G. Mallaley, R. Barry Dydyk, S. A. Mills, “Third generation miniature ruggedized optiral correlator (MROC) module,” in Optical Pattern Recognition IX, D. P. Casasent, ed., Proc. SPIE3386, 38–44 (1998).
    [CrossRef]
  7. S. A. Serati, T. K. Ewing, R. A. Serati, “Programmable 128 × 128 ferroelectric-liquid-crystal spatial-light-modulator compact correlator,” in International Conference on Thin Film Physics and Applications, S. Zhou, Y. Wang, eds., Proc. SPIE1519, 55–68 (1993).
  8. R. Thapliya, T. Kamiya, “Optimization of multichannel parallel joint transform correlator accelerated pattern recognition,” Appl. Opt. 39, 5309–5317 (2000).
    [CrossRef]
  9. K. Kodate, A. Hashimoto, R. Thapliya, “Binary zone-plate array for a parallel joint transform correlator applied to face recognition,” Appl. Opt. 38, 3060–3067 (1999).
    [CrossRef]
  10. R. Inaba, E. Watanabe, K. Kodate, “Security applications of optical face recognition system: access control in e-learning,” Opt. Rev. 10, 255–261 (2003).
    [CrossRef]
  11. K. Kodate, R. Inaba, E. Watanabe, T. Kamiya, “Facial recognition by compact parallel optical correlator,” Meas. Sci. Technol. 13, 1756–1766 (2002).
    [CrossRef]
  12. E. Watanabe, N. Arima, K. Kodate, “Facial recognition system with multi-light source optical parallel correlator using vertical cavity surface emitting laser array module,” Jpn. J. Appl. Phys. 43, 5890–5896 (2004).
    [CrossRef]
  13. O. Yamaguchi, K. Fukui, “Smartface—a robust face recognition system under varying facial poses and expressions,” IEICE Trans. Inf. Syst. E86-D, 37–44 (2003).
  14. R. Inaba, A. Hashimoto, K. Kodate, “Discrimination of portraits using a hybrid parallel joint transform correlator system,” in Optical Engineering for Sensing and Nanotechnology (ICOSN’99), I. Yamaguchi, ed., Proc. SPIE3740, 529–532 (1999).
    [CrossRef]
  15. Y. Seto, M. Mimura, “Standardization of accuracy evaluation for biometric authentication in Japan,” IEICE Trans. Inf. Syst. E84-D, 800–805 (2001).
  16. Y. Hsu, H. H. Ausrenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 28, 4725–4727 (1989).
  17. G. Gregory, “Design considerations for low-clutter, distortion invariant correlation filters,” Opt. Eng. 29, 1029–1032 (1990).
    [CrossRef]
  18. B. V. K. Vijaya Kumar, R. D. Juday, “Design of phase-only, binary phase-only, and complex ternary matched filter with increased signal-to-noise ratios for colored noise,” Opt. Lett. 16, 1025–1027 (1991).
    [CrossRef]
  19. S. Yin, M. Lu, C. Chen, F. T. S. Yu, “Design of a bipolar composite filter by a simulated annealing algorithm,” Opt. Lett. 20, 1409–1411 (1995).
    [CrossRef] [PubMed]
  20. J. R. Johansson, D. G. Rabelius, “Experimental results from fusion of binary correlation filters implemented on an optical correlator,” in Optical Pattern Recognition IX, D. P. Casasent, ed., Proc. SPIE3386, 134–143 (1998).
    [CrossRef]
  21. H. V. Sorensen, D. L. Jones, M. T. Heideman, C. S. Burrus, “Real valued fast Fourier transforms,” IEEE Trans. Acoust. Speech Signal Process. 35, 849–863 (1987).
    [CrossRef]
  22. Y. Orihara, W. Klaus, M. Fujino, K. Kodate, “Optimization and application of hybrid-level binary zone plates,” Appl. Opt. 40, 5877–5885 (2001).
    [CrossRef]

2004 (1)

E. Watanabe, N. Arima, K. Kodate, “Facial recognition system with multi-light source optical parallel correlator using vertical cavity surface emitting laser array module,” Jpn. J. Appl. Phys. 43, 5890–5896 (2004).
[CrossRef]

2003 (2)

O. Yamaguchi, K. Fukui, “Smartface—a robust face recognition system under varying facial poses and expressions,” IEICE Trans. Inf. Syst. E86-D, 37–44 (2003).

R. Inaba, E. Watanabe, K. Kodate, “Security applications of optical face recognition system: access control in e-learning,” Opt. Rev. 10, 255–261 (2003).
[CrossRef]

2002 (1)

K. Kodate, R. Inaba, E. Watanabe, T. Kamiya, “Facial recognition by compact parallel optical correlator,” Meas. Sci. Technol. 13, 1756–1766 (2002).
[CrossRef]

2001 (2)

Y. Orihara, W. Klaus, M. Fujino, K. Kodate, “Optimization and application of hybrid-level binary zone plates,” Appl. Opt. 40, 5877–5885 (2001).
[CrossRef]

Y. Seto, M. Mimura, “Standardization of accuracy evaluation for biometric authentication in Japan,” IEICE Trans. Inf. Syst. E84-D, 800–805 (2001).

2000 (2)

A. Bergeron, “Optical correlator for industrial applications, quality control and target tracking,” Sensor Rev. 20, 316–321 (2000).
[CrossRef]

R. Thapliya, T. Kamiya, “Optimization of multichannel parallel joint transform correlator accelerated pattern recognition,” Appl. Opt. 39, 5309–5317 (2000).
[CrossRef]

1999 (3)

Y. Kobayashi, Y. Toyoda, “Development of an optical joint transform correlation system for fingerprint recognition,” Opt. Eng. 38, 1205–1210 (1999).
[CrossRef]

M. Kaneko, O. Hasegawa, “Processing of face images and its applications,” IEICE Trans. Inf. Syst. E82-D, 589–600 (1999).

K. Kodate, A. Hashimoto, R. Thapliya, “Binary zone-plate array for a parallel joint transform correlator applied to face recognition,” Appl. Opt. 38, 3060–3067 (1999).
[CrossRef]

1995 (1)

1991 (1)

1990 (1)

G. Gregory, “Design considerations for low-clutter, distortion invariant correlation filters,” Opt. Eng. 29, 1029–1032 (1990).
[CrossRef]

1989 (1)

1987 (1)

H. V. Sorensen, D. L. Jones, M. T. Heideman, C. S. Burrus, “Real valued fast Fourier transforms,” IEEE Trans. Acoust. Speech Signal Process. 35, 849–863 (1987).
[CrossRef]

1966 (1)

1964 (1)

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

Arima, N.

E. Watanabe, N. Arima, K. Kodate, “Facial recognition system with multi-light source optical parallel correlator using vertical cavity surface emitting laser array module,” Jpn. J. Appl. Phys. 43, 5890–5896 (2004).
[CrossRef]

Ausrenault, H. H.

Barry Dydyk, R.

D. T. Carrott, G. Mallaley, R. Barry Dydyk, S. A. Mills, “Third generation miniature ruggedized optiral correlator (MROC) module,” in Optical Pattern Recognition IX, D. P. Casasent, ed., Proc. SPIE3386, 38–44 (1998).
[CrossRef]

Bergeron, A.

A. Bergeron, “Optical correlator for industrial applications, quality control and target tracking,” Sensor Rev. 20, 316–321 (2000).
[CrossRef]

Burrus, C. S.

H. V. Sorensen, D. L. Jones, M. T. Heideman, C. S. Burrus, “Real valued fast Fourier transforms,” IEEE Trans. Acoust. Speech Signal Process. 35, 849–863 (1987).
[CrossRef]

Carrott, D. T.

D. T. Carrott, G. Mallaley, R. Barry Dydyk, S. A. Mills, “Third generation miniature ruggedized optiral correlator (MROC) module,” in Optical Pattern Recognition IX, D. P. Casasent, ed., Proc. SPIE3386, 38–44 (1998).
[CrossRef]

Chen, C.

Ewing, T. K.

S. A. Serati, T. K. Ewing, R. A. Serati, “Programmable 128 × 128 ferroelectric-liquid-crystal spatial-light-modulator compact correlator,” in International Conference on Thin Film Physics and Applications, S. Zhou, Y. Wang, eds., Proc. SPIE1519, 55–68 (1993).

Fujino, M.

Fukui, K.

O. Yamaguchi, K. Fukui, “Smartface—a robust face recognition system under varying facial poses and expressions,” IEICE Trans. Inf. Syst. E86-D, 37–44 (2003).

Goodman, J. W.

Gregory, G.

G. Gregory, “Design considerations for low-clutter, distortion invariant correlation filters,” Opt. Eng. 29, 1029–1032 (1990).
[CrossRef]

Hasegawa, O.

M. Kaneko, O. Hasegawa, “Processing of face images and its applications,” IEICE Trans. Inf. Syst. E82-D, 589–600 (1999).

Hashimoto, A.

K. Kodate, A. Hashimoto, R. Thapliya, “Binary zone-plate array for a parallel joint transform correlator applied to face recognition,” Appl. Opt. 38, 3060–3067 (1999).
[CrossRef]

R. Inaba, A. Hashimoto, K. Kodate, “Discrimination of portraits using a hybrid parallel joint transform correlator system,” in Optical Engineering for Sensing and Nanotechnology (ICOSN’99), I. Yamaguchi, ed., Proc. SPIE3740, 529–532 (1999).
[CrossRef]

Heideman, M. T.

H. V. Sorensen, D. L. Jones, M. T. Heideman, C. S. Burrus, “Real valued fast Fourier transforms,” IEEE Trans. Acoust. Speech Signal Process. 35, 849–863 (1987).
[CrossRef]

Hsu, Y.

Inaba, R.

R. Inaba, E. Watanabe, K. Kodate, “Security applications of optical face recognition system: access control in e-learning,” Opt. Rev. 10, 255–261 (2003).
[CrossRef]

K. Kodate, R. Inaba, E. Watanabe, T. Kamiya, “Facial recognition by compact parallel optical correlator,” Meas. Sci. Technol. 13, 1756–1766 (2002).
[CrossRef]

R. Inaba, A. Hashimoto, K. Kodate, “Discrimination of portraits using a hybrid parallel joint transform correlator system,” in Optical Engineering for Sensing and Nanotechnology (ICOSN’99), I. Yamaguchi, ed., Proc. SPIE3740, 529–532 (1999).
[CrossRef]

Johansson, J. R.

J. R. Johansson, D. G. Rabelius, “Experimental results from fusion of binary correlation filters implemented on an optical correlator,” in Optical Pattern Recognition IX, D. P. Casasent, ed., Proc. SPIE3386, 134–143 (1998).
[CrossRef]

Jones, D. L.

H. V. Sorensen, D. L. Jones, M. T. Heideman, C. S. Burrus, “Real valued fast Fourier transforms,” IEEE Trans. Acoust. Speech Signal Process. 35, 849–863 (1987).
[CrossRef]

Juday, R. D.

Kamiya, T.

K. Kodate, R. Inaba, E. Watanabe, T. Kamiya, “Facial recognition by compact parallel optical correlator,” Meas. Sci. Technol. 13, 1756–1766 (2002).
[CrossRef]

R. Thapliya, T. Kamiya, “Optimization of multichannel parallel joint transform correlator accelerated pattern recognition,” Appl. Opt. 39, 5309–5317 (2000).
[CrossRef]

Kaneko, M.

M. Kaneko, O. Hasegawa, “Processing of face images and its applications,” IEICE Trans. Inf. Syst. E82-D, 589–600 (1999).

Klaus, W.

Kobayashi, Y.

Y. Kobayashi, Y. Toyoda, “Development of an optical joint transform correlation system for fingerprint recognition,” Opt. Eng. 38, 1205–1210 (1999).
[CrossRef]

Kodate, K.

E. Watanabe, N. Arima, K. Kodate, “Facial recognition system with multi-light source optical parallel correlator using vertical cavity surface emitting laser array module,” Jpn. J. Appl. Phys. 43, 5890–5896 (2004).
[CrossRef]

R. Inaba, E. Watanabe, K. Kodate, “Security applications of optical face recognition system: access control in e-learning,” Opt. Rev. 10, 255–261 (2003).
[CrossRef]

K. Kodate, R. Inaba, E. Watanabe, T. Kamiya, “Facial recognition by compact parallel optical correlator,” Meas. Sci. Technol. 13, 1756–1766 (2002).
[CrossRef]

Y. Orihara, W. Klaus, M. Fujino, K. Kodate, “Optimization and application of hybrid-level binary zone plates,” Appl. Opt. 40, 5877–5885 (2001).
[CrossRef]

K. Kodate, A. Hashimoto, R. Thapliya, “Binary zone-plate array for a parallel joint transform correlator applied to face recognition,” Appl. Opt. 38, 3060–3067 (1999).
[CrossRef]

R. Inaba, A. Hashimoto, K. Kodate, “Discrimination of portraits using a hybrid parallel joint transform correlator system,” in Optical Engineering for Sensing and Nanotechnology (ICOSN’99), I. Yamaguchi, ed., Proc. SPIE3740, 529–532 (1999).
[CrossRef]

Lu, M.

Mallaley, G.

D. T. Carrott, G. Mallaley, R. Barry Dydyk, S. A. Mills, “Third generation miniature ruggedized optiral correlator (MROC) module,” in Optical Pattern Recognition IX, D. P. Casasent, ed., Proc. SPIE3386, 38–44 (1998).
[CrossRef]

Mills, S. A.

D. T. Carrott, G. Mallaley, R. Barry Dydyk, S. A. Mills, “Third generation miniature ruggedized optiral correlator (MROC) module,” in Optical Pattern Recognition IX, D. P. Casasent, ed., Proc. SPIE3386, 38–44 (1998).
[CrossRef]

Mimura, M.

Y. Seto, M. Mimura, “Standardization of accuracy evaluation for biometric authentication in Japan,” IEICE Trans. Inf. Syst. E84-D, 800–805 (2001).

Orihara, Y.

Rabelius, D. G.

J. R. Johansson, D. G. Rabelius, “Experimental results from fusion of binary correlation filters implemented on an optical correlator,” in Optical Pattern Recognition IX, D. P. Casasent, ed., Proc. SPIE3386, 134–143 (1998).
[CrossRef]

Serati, R. A.

S. A. Serati, T. K. Ewing, R. A. Serati, “Programmable 128 × 128 ferroelectric-liquid-crystal spatial-light-modulator compact correlator,” in International Conference on Thin Film Physics and Applications, S. Zhou, Y. Wang, eds., Proc. SPIE1519, 55–68 (1993).

Serati, S. A.

S. A. Serati, T. K. Ewing, R. A. Serati, “Programmable 128 × 128 ferroelectric-liquid-crystal spatial-light-modulator compact correlator,” in International Conference on Thin Film Physics and Applications, S. Zhou, Y. Wang, eds., Proc. SPIE1519, 55–68 (1993).

Seto, Y.

Y. Seto, M. Mimura, “Standardization of accuracy evaluation for biometric authentication in Japan,” IEICE Trans. Inf. Syst. E84-D, 800–805 (2001).

Sorensen, H. V.

H. V. Sorensen, D. L. Jones, M. T. Heideman, C. S. Burrus, “Real valued fast Fourier transforms,” IEEE Trans. Acoust. Speech Signal Process. 35, 849–863 (1987).
[CrossRef]

Thapliya, R.

Toyoda, Y.

Y. Kobayashi, Y. Toyoda, “Development of an optical joint transform correlation system for fingerprint recognition,” Opt. Eng. 38, 1205–1210 (1999).
[CrossRef]

Vanderlugt, A.

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

Vijaya Kumar, B. V. K.

Watanabe, E.

E. Watanabe, N. Arima, K. Kodate, “Facial recognition system with multi-light source optical parallel correlator using vertical cavity surface emitting laser array module,” Jpn. J. Appl. Phys. 43, 5890–5896 (2004).
[CrossRef]

R. Inaba, E. Watanabe, K. Kodate, “Security applications of optical face recognition system: access control in e-learning,” Opt. Rev. 10, 255–261 (2003).
[CrossRef]

K. Kodate, R. Inaba, E. Watanabe, T. Kamiya, “Facial recognition by compact parallel optical correlator,” Meas. Sci. Technol. 13, 1756–1766 (2002).
[CrossRef]

Weaver, C. S.

Yamaguchi, O.

O. Yamaguchi, K. Fukui, “Smartface—a robust face recognition system under varying facial poses and expressions,” IEICE Trans. Inf. Syst. E86-D, 37–44 (2003).

Yin, S.

Yu, F. T. S.

Appl. Opt. (5)

IEEE Trans. Acoust. Speech Signal Process. (1)

H. V. Sorensen, D. L. Jones, M. T. Heideman, C. S. Burrus, “Real valued fast Fourier transforms,” IEEE Trans. Acoust. Speech Signal Process. 35, 849–863 (1987).
[CrossRef]

IEEE Trans. Inf. Theory (1)

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

IEICE Trans. Inf. Syst. (3)

Y. Seto, M. Mimura, “Standardization of accuracy evaluation for biometric authentication in Japan,” IEICE Trans. Inf. Syst. E84-D, 800–805 (2001).

M. Kaneko, O. Hasegawa, “Processing of face images and its applications,” IEICE Trans. Inf. Syst. E82-D, 589–600 (1999).

O. Yamaguchi, K. Fukui, “Smartface—a robust face recognition system under varying facial poses and expressions,” IEICE Trans. Inf. Syst. E86-D, 37–44 (2003).

Jpn. J. Appl. Phys. (1)

E. Watanabe, N. Arima, K. Kodate, “Facial recognition system with multi-light source optical parallel correlator using vertical cavity surface emitting laser array module,” Jpn. J. Appl. Phys. 43, 5890–5896 (2004).
[CrossRef]

Meas. Sci. Technol. (1)

K. Kodate, R. Inaba, E. Watanabe, T. Kamiya, “Facial recognition by compact parallel optical correlator,” Meas. Sci. Technol. 13, 1756–1766 (2002).
[CrossRef]

Opt. Eng. (2)

Y. Kobayashi, Y. Toyoda, “Development of an optical joint transform correlation system for fingerprint recognition,” Opt. Eng. 38, 1205–1210 (1999).
[CrossRef]

G. Gregory, “Design considerations for low-clutter, distortion invariant correlation filters,” Opt. Eng. 29, 1029–1032 (1990).
[CrossRef]

Opt. Lett. (2)

Opt. Rev. (1)

R. Inaba, E. Watanabe, K. Kodate, “Security applications of optical face recognition system: access control in e-learning,” Opt. Rev. 10, 255–261 (2003).
[CrossRef]

Sensor Rev. (1)

A. Bergeron, “Optical correlator for industrial applications, quality control and target tracking,” Sensor Rev. 20, 316–321 (2000).
[CrossRef]

Other (4)

D. T. Carrott, G. Mallaley, R. Barry Dydyk, S. A. Mills, “Third generation miniature ruggedized optiral correlator (MROC) module,” in Optical Pattern Recognition IX, D. P. Casasent, ed., Proc. SPIE3386, 38–44 (1998).
[CrossRef]

S. A. Serati, T. K. Ewing, R. A. Serati, “Programmable 128 × 128 ferroelectric-liquid-crystal spatial-light-modulator compact correlator,” in International Conference on Thin Film Physics and Applications, S. Zhou, Y. Wang, eds., Proc. SPIE1519, 55–68 (1993).

R. Inaba, A. Hashimoto, K. Kodate, “Discrimination of portraits using a hybrid parallel joint transform correlator system,” in Optical Engineering for Sensing and Nanotechnology (ICOSN’99), I. Yamaguchi, ed., Proc. SPIE3740, 529–532 (1999).
[CrossRef]

J. R. Johansson, D. G. Rabelius, “Experimental results from fusion of binary correlation filters implemented on an optical correlator,” in Optical Pattern Recognition IX, D. P. Casasent, ed., Proc. SPIE3386, 134–143 (1998).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Flow chart of our hybrid face recognition system: (a) real-time image capture and preprocessing, (b) optical correlation, (c) postprocessing.

Fig. 2
Fig. 2

Simulation results of the Fourier spectrum of facial images: (a) gray-scale image, (b) edge-enhancement and binarization image. Facial characteristics radiate to the high-frequency region.

Fig. 3
Fig. 3

Eight kinds of matched filter. The images of phase-only or real-part matched filters on (a) axis or (b) off axis.

Fig. 4
Fig. 4

Comparison between a phase-only filter and a real-only filter and an evaluation of the simulation experiment by FAR. An on-axis, phase-only filter (a) shows less than a 1% error rate; however, the real-part filter shows high error rates. In the case of the off-axis filter (b), both phase-only and real-part filters display an accuracy rate of less than 1%.

Fig. 5
Fig. 5

(a) Optical setup of FARCO and (b) photograph of the fabricated FARCO: M, mirror; H. M., half mirror; P, polarizer; A, analyzer; L, lens; PBS, polarized beam splitter; C. L., collimator lens; SLM, spatial light modulator; MLZP, multilevel zone plate; LD, laser diode; PDA, photodiode analyzer; 1/2λ, half-wave plate. See text for other definitions.

Fig. 6
Fig. 6

Time chart of FARCO. (1) It takes ~200 ms for preprocessing. (2) NLC SLM rises in 30 ms for displaying matched filter. (3) LD shines on the SLM as a light pulse. (4) It takes 400 ms to display a database image. (5) PD detects the correlation signals.

Fig. 7
Fig. 7

Design of the control of the FARCO system, which consists of a main control computer for FARCO and a client computer for input images.

Fig. 8
Fig. 8

(a) Optical setup of four-channel correlation process. (b) Four-channel experimental results with the designed matched filter. PD, photodiode.

Fig. 9
Fig. 9

Evaluation accuracy of FARCO according to the biometrics guideline with database images of 300 people. The lowest error rates of the two values were recorded for 300 data, with FAR and FRR error rates of less than 1.3% and 2.6%, respectively.

Fig. 10
Fig. 10

Experimental results of a database with 4000 images (derived from 50 images per person by use of motion pictures). (a) Sample of database images, and (b) registered number area of the input image.

Fig. 11
Fig. 11

Samples of input images that were successfully recognized by the system.

Fig. 12
Fig. 12

Optical parallel correlator with a MLZPA for increasing parallelism. PD, photodiode.

Fig. 13
Fig. 13

Simulation results of error rates (FAR + FRR) with variations in the frequency aperture size of the matched filter. The accuracy level can be maintained at less than 1% within a spatial-frequency range of greater than 0.5.

Tables (2)

Tables Icon

Table 1 Specifications for FARCO

Tables Icon

Table 2 Specifications for the Optical Devices Installed in FARCO

Equations (5)

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

N = 3 / P
C i = j = 1 N [ P i j / ( P i max - 1 ) ] N - 1 .
H m ( v x , v y ) = F ( v x , v y ) exp [ - i ϕ ( v x , v y ) ] ,
H p ( v x , v y ) = exp [ - i ϕ ( v x , v y ) ] ,
H r ( v x , v y ) = real { F ( v x , v y ) exp [ - i ϕ ( v x , v y ) ] } .

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