Abstract

A novel interferometric optical Fourier-transform processor is presented that calculates the complex-valued Fourier transform of an image at preselected points on the spatial-frequency plane. The Fourier spectrum of an arbitrary input image is interfered with that of a reference image in a common-path interferometer. Both the real and the imaginary parts of the complex-valued spectrum are determined. The source and the reference images are easily matched to guarantee good fringe visibility. At least six interferograms are postprocessed to extract the real and the imaginary parts of the Fourier spectrum at preselected points. The proposed hybrid optical–digital technique is computationally appropriate when the number of desired spatial frequencies is small compared with the number of pixels in the image. When the number of desired points is comparable with the number of image pixels, a conventional or pruned two-dimensional fast Fourier transform is more appropriate. The number of digital operations required by the hybrid optical–digital Fourier processor is proportional to the number of desired spatial frequencies rather than the number of pixels in the image. The points may be regularly distributed over the spatial-frequency plane or concentrated in one or several irregularly shaped regions of interest. The interferometric optical Fourier processor is demonstrated in a moving-object trajectory estimation system. The system successfully estimates the trajectory of multiple objects moving over both stationary and white-noise backgrounds. A comparison of performance was made with all-digital computation. With everything else equal, our hybrid optical–digital calculation was more than 3 orders of magnitude faster.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
    [CrossRef]
  2. J. D. Markel, “FFT pruning,” IEEE Trans. Acoust. Speech Signal Process. AU-19, 305–311 (1971).
  3. T. Smit, M. R. Smith, S. T. Nichols, “Efficient sinc function interpolation technique for center padded data,” IEEE Trans. Acoust. Speech Signal Process. 38, 1512–1517 (1990).
    [CrossRef]
  4. K. S. Knudsen, L. T. Bruton, “Moving object detection and trajectory estimation in the transform/spatiotemporal mixed domain,” in Proceedings of the 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, San Francisco, March 1992 (Institute of Electrical and Electronic Engineers, New York, 1992), Vol. 3, pp. 505–508.
  5. K. S. Knudsen, L. T. Bruton, “Moving object nonlinear trajectory estimation in the transform/spatiotemporal mixed domain,” in Proceedings of the 1992 IEEE Custom Integrated Circuits Conference, San Diego, May 1992 (Institute of Electrical and Electronic Engineers, New York, 1992), Vol. 5, pp. 2481–2484.
  6. T. M. Turpin, “Spectrum analysis using optical processing,” Proc. IEEE 69, 79–92 (1981).
    [CrossRef]
  7. M. King, W. R. Bennett, L. B. Lambert, M. Arm, “Real-time electro-optical signal processors with coherent detection,” Appl. Opt. 6, 1367–1375 (1967).
    [CrossRef] [PubMed]
  8. H. R. Carleton, W. T. Maloney, G. Meltz, “Collinear heterodyning in optical processors,” Proc. IEEE 57, 769–775 (1969).
    [CrossRef]
  9. A. VanderLugt, “Interferometric spectrum analyzer,” Appl. Opt. 33, 2770–2779 (1981).
  10. C. C. Aleksoff, N. S. Subotic, “Compact real-time interferometric Fourier transform processors,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. SPIE1347, 427–440 (1990).
    [CrossRef]
  11. A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).
  12. W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS 5, 208–210 (1933) (in Russian).
  13. E. C. Tam, S. W. Tannone, F. T. S. Yu, D. A. Gregory, “Closed-loop binary phase correction of an LCTV using a point diffraction interferometer,” IEEE Photon. Technol. Lett. 2, 143–146 (1990).
    [CrossRef]
  14. Y. Zhang, E. Kanterakis, A. Katz, J.-M. Wang, “Optoelectronic wavelet processors based on Smartt interferometry,” Appl. Opt. 33, 5279–5286 (1994).
    [CrossRef] [PubMed]
  15. P. M. Lane, M. Cada, “An optical Fourier processor and point-diffraction interferometer for moving object trajectory estimation,” Appl. Opt. 38, 4306–4315 (1999).
    [CrossRef]
  16. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1991).
  17. H. Kadono, M. Ogusu, S. Toyooka, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
    [CrossRef]
  18. C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer,” Opt. Lett. 19, 916–918 (1994).
    [CrossRef] [PubMed]
  19. C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt. 35, 1633–1642 (1996).
    [CrossRef] [PubMed]
  20. P. M. Lane, “The complex-valued optical Fourier transform and its application to moving-object trajectory estimation,” Ph.D. dissertation (Dalhousie University, Halifax, N.S., 1999).
  21. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  22. G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge Press, Wellesley, Mass., 1996).
  23. K. S. Knudsen, L. T. Bruton, “Recursive pruning of the 2-D DFT with 3-D signal processing applications,” IEEE Trans. Signal Process. 41, 1340–1356 (1993).
    [CrossRef]
  24. K. S. Knudsen, “Multidimensional mixed domain signal processing,” Ph.D. dissertation (University of Calgary, Calgary, Al., 1992).
  25. K. S. Knudsen, L. T. Bruton, “Mixed domain filtering of multidimensional signals,” IEEE Trans. Circuits Syst. Video Technol. 1, 260–268 (1991).
    [CrossRef]
  26. K. S. Knudsen, L. T. Bruton, “Transform/spatiotemporal mixed domain moving object tracking and enhancement,” in Proceedings of the 1993 European Conference on Circuit Theory and Design, Davos, Switzerland, August 1993 (Elsevier, Amsterdam, 1993), pp. 589–594.
  27. P. M. Lane, K. S. Knudsen, M. Cada, “Moving object trajectory estimation using an optical Fourier processor,” in 1998 International Conference on Applications of Photonic Technology, G. A. Lampropoulos, ed., Proc. SPIE3491, 939–943 (1998).
  28. L. J. Hornbeck, “Digital light processing and MEMS: timely convergence for a bright future (invited plenary paper),” in Micromaching and Microfabrication Process Technology, K. W. Markus, ed., Proc. SPIE2639, p. 2 (abstract only), full paper available from Texas Instruments, Dallas, Tex., 1995.

1999 (1)

1996 (1)

C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt. 35, 1633–1642 (1996).
[CrossRef] [PubMed]

1994 (3)

1993 (1)

K. S. Knudsen, L. T. Bruton, “Recursive pruning of the 2-D DFT with 3-D signal processing applications,” IEEE Trans. Signal Process. 41, 1340–1356 (1993).
[CrossRef]

1991 (1)

K. S. Knudsen, L. T. Bruton, “Mixed domain filtering of multidimensional signals,” IEEE Trans. Circuits Syst. Video Technol. 1, 260–268 (1991).
[CrossRef]

1990 (2)

T. Smit, M. R. Smith, S. T. Nichols, “Efficient sinc function interpolation technique for center padded data,” IEEE Trans. Acoust. Speech Signal Process. 38, 1512–1517 (1990).
[CrossRef]

E. C. Tam, S. W. Tannone, F. T. S. Yu, D. A. Gregory, “Closed-loop binary phase correction of an LCTV using a point diffraction interferometer,” IEEE Photon. Technol. Lett. 2, 143–146 (1990).
[CrossRef]

1981 (2)

A. VanderLugt, “Interferometric spectrum analyzer,” Appl. Opt. 33, 2770–2779 (1981).

T. M. Turpin, “Spectrum analysis using optical processing,” Proc. IEEE 69, 79–92 (1981).
[CrossRef]

1971 (1)

J. D. Markel, “FFT pruning,” IEEE Trans. Acoust. Speech Signal Process. AU-19, 305–311 (1971).

1969 (1)

H. R. Carleton, W. T. Maloney, G. Meltz, “Collinear heterodyning in optical processors,” Proc. IEEE 57, 769–775 (1969).
[CrossRef]

1967 (1)

1965 (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

1933 (1)

W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS 5, 208–210 (1933) (in Russian).

Aleksoff, C. C.

C. C. Aleksoff, N. S. Subotic, “Compact real-time interferometric Fourier transform processors,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. SPIE1347, 427–440 (1990).
[CrossRef]

Arm, M.

Bennett, W. R.

Bruton, L. T.

K. S. Knudsen, L. T. Bruton, “Recursive pruning of the 2-D DFT with 3-D signal processing applications,” IEEE Trans. Signal Process. 41, 1340–1356 (1993).
[CrossRef]

K. S. Knudsen, L. T. Bruton, “Mixed domain filtering of multidimensional signals,” IEEE Trans. Circuits Syst. Video Technol. 1, 260–268 (1991).
[CrossRef]

K. S. Knudsen, L. T. Bruton, “Transform/spatiotemporal mixed domain moving object tracking and enhancement,” in Proceedings of the 1993 European Conference on Circuit Theory and Design, Davos, Switzerland, August 1993 (Elsevier, Amsterdam, 1993), pp. 589–594.

K. S. Knudsen, L. T. Bruton, “Moving object nonlinear trajectory estimation in the transform/spatiotemporal mixed domain,” in Proceedings of the 1992 IEEE Custom Integrated Circuits Conference, San Diego, May 1992 (Institute of Electrical and Electronic Engineers, New York, 1992), Vol. 5, pp. 2481–2484.

K. S. Knudsen, L. T. Bruton, “Moving object detection and trajectory estimation in the transform/spatiotemporal mixed domain,” in Proceedings of the 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, San Francisco, March 1992 (Institute of Electrical and Electronic Engineers, New York, 1992), Vol. 3, pp. 505–508.

Cada, M.

P. M. Lane, M. Cada, “An optical Fourier processor and point-diffraction interferometer for moving object trajectory estimation,” Appl. Opt. 38, 4306–4315 (1999).
[CrossRef]

P. M. Lane, K. S. Knudsen, M. Cada, “Moving object trajectory estimation using an optical Fourier processor,” in 1998 International Conference on Applications of Photonic Technology, G. A. Lampropoulos, ed., Proc. SPIE3491, 939–943 (1998).

Carleton, H. R.

H. R. Carleton, W. T. Maloney, G. Meltz, “Collinear heterodyning in optical processors,” Proc. IEEE 57, 769–775 (1969).
[CrossRef]

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Creath, K.

C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt. 35, 1633–1642 (1996).
[CrossRef] [PubMed]

C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer,” Opt. Lett. 19, 916–918 (1994).
[CrossRef] [PubMed]

Gregory, D. A.

E. C. Tam, S. W. Tannone, F. T. S. Yu, D. A. Gregory, “Closed-loop binary phase correction of an LCTV using a point diffraction interferometer,” IEEE Photon. Technol. Lett. 2, 143–146 (1990).
[CrossRef]

Hornbeck, L. J.

L. J. Hornbeck, “Digital light processing and MEMS: timely convergence for a bright future (invited plenary paper),” in Micromaching and Microfabrication Process Technology, K. W. Markus, ed., Proc. SPIE2639, p. 2 (abstract only), full paper available from Texas Instruments, Dallas, Tex., 1995.

Kadono, H.

H. Kadono, M. Ogusu, S. Toyooka, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[CrossRef]

Kanterakis, E.

Katz, A.

King, M.

Knudsen, K. S.

K. S. Knudsen, L. T. Bruton, “Recursive pruning of the 2-D DFT with 3-D signal processing applications,” IEEE Trans. Signal Process. 41, 1340–1356 (1993).
[CrossRef]

K. S. Knudsen, L. T. Bruton, “Mixed domain filtering of multidimensional signals,” IEEE Trans. Circuits Syst. Video Technol. 1, 260–268 (1991).
[CrossRef]

K. S. Knudsen, “Multidimensional mixed domain signal processing,” Ph.D. dissertation (University of Calgary, Calgary, Al., 1992).

K. S. Knudsen, L. T. Bruton, “Transform/spatiotemporal mixed domain moving object tracking and enhancement,” in Proceedings of the 1993 European Conference on Circuit Theory and Design, Davos, Switzerland, August 1993 (Elsevier, Amsterdam, 1993), pp. 589–594.

P. M. Lane, K. S. Knudsen, M. Cada, “Moving object trajectory estimation using an optical Fourier processor,” in 1998 International Conference on Applications of Photonic Technology, G. A. Lampropoulos, ed., Proc. SPIE3491, 939–943 (1998).

K. S. Knudsen, L. T. Bruton, “Moving object nonlinear trajectory estimation in the transform/spatiotemporal mixed domain,” in Proceedings of the 1992 IEEE Custom Integrated Circuits Conference, San Diego, May 1992 (Institute of Electrical and Electronic Engineers, New York, 1992), Vol. 5, pp. 2481–2484.

K. S. Knudsen, L. T. Bruton, “Moving object detection and trajectory estimation in the transform/spatiotemporal mixed domain,” in Proceedings of the 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, San Francisco, March 1992 (Institute of Electrical and Electronic Engineers, New York, 1992), Vol. 3, pp. 505–508.

Lambert, L. B.

Lane, P. M.

P. M. Lane, M. Cada, “An optical Fourier processor and point-diffraction interferometer for moving object trajectory estimation,” Appl. Opt. 38, 4306–4315 (1999).
[CrossRef]

P. M. Lane, K. S. Knudsen, M. Cada, “Moving object trajectory estimation using an optical Fourier processor,” in 1998 International Conference on Applications of Photonic Technology, G. A. Lampropoulos, ed., Proc. SPIE3491, 939–943 (1998).

P. M. Lane, “The complex-valued optical Fourier transform and its application to moving-object trajectory estimation,” Ph.D. dissertation (Dalhousie University, Halifax, N.S., 1999).

Linnik, W. P.

W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS 5, 208–210 (1933) (in Russian).

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1991).

Maloney, W. T.

H. R. Carleton, W. T. Maloney, G. Meltz, “Collinear heterodyning in optical processors,” Proc. IEEE 57, 769–775 (1969).
[CrossRef]

Markel, J. D.

J. D. Markel, “FFT pruning,” IEEE Trans. Acoust. Speech Signal Process. AU-19, 305–311 (1971).

Meltz, G.

H. R. Carleton, W. T. Maloney, G. Meltz, “Collinear heterodyning in optical processors,” Proc. IEEE 57, 769–775 (1969).
[CrossRef]

Mercer, C. R.

C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt. 35, 1633–1642 (1996).
[CrossRef] [PubMed]

C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer,” Opt. Lett. 19, 916–918 (1994).
[CrossRef] [PubMed]

Nguyen, T.

G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge Press, Wellesley, Mass., 1996).

Nichols, S. T.

T. Smit, M. R. Smith, S. T. Nichols, “Efficient sinc function interpolation technique for center padded data,” IEEE Trans. Acoust. Speech Signal Process. 38, 1512–1517 (1990).
[CrossRef]

Ogusu, M.

H. Kadono, M. Ogusu, S. Toyooka, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Smit, T.

T. Smit, M. R. Smith, S. T. Nichols, “Efficient sinc function interpolation technique for center padded data,” IEEE Trans. Acoust. Speech Signal Process. 38, 1512–1517 (1990).
[CrossRef]

Smith, M. R.

T. Smit, M. R. Smith, S. T. Nichols, “Efficient sinc function interpolation technique for center padded data,” IEEE Trans. Acoust. Speech Signal Process. 38, 1512–1517 (1990).
[CrossRef]

Strang, G.

G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge Press, Wellesley, Mass., 1996).

Subotic, N. S.

C. C. Aleksoff, N. S. Subotic, “Compact real-time interferometric Fourier transform processors,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. SPIE1347, 427–440 (1990).
[CrossRef]

Tam, E. C.

E. C. Tam, S. W. Tannone, F. T. S. Yu, D. A. Gregory, “Closed-loop binary phase correction of an LCTV using a point diffraction interferometer,” IEEE Photon. Technol. Lett. 2, 143–146 (1990).
[CrossRef]

Tannone, S. W.

E. C. Tam, S. W. Tannone, F. T. S. Yu, D. A. Gregory, “Closed-loop binary phase correction of an LCTV using a point diffraction interferometer,” IEEE Photon. Technol. Lett. 2, 143–146 (1990).
[CrossRef]

Toyooka, S.

H. Kadono, M. Ogusu, S. Toyooka, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[CrossRef]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Turpin, T. M.

T. M. Turpin, “Spectrum analysis using optical processing,” Proc. IEEE 69, 79–92 (1981).
[CrossRef]

VanderLugt, A.

A. VanderLugt, “Interferometric spectrum analyzer,” Appl. Opt. 33, 2770–2779 (1981).

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

Wang, J.-M.

Yu, F. T. S.

E. C. Tam, S. W. Tannone, F. T. S. Yu, D. A. Gregory, “Closed-loop binary phase correction of an LCTV using a point diffraction interferometer,” IEEE Photon. Technol. Lett. 2, 143–146 (1990).
[CrossRef]

Zhang, Y.

Appl. Opt. (1)

C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer for wave-front measurements,” Appl. Opt. 35, 1633–1642 (1996).
[CrossRef] [PubMed]

Appl. Opt. (4)

C. R. Acad. Sci. URSS (1)

W. P. Linnik, “A simple interferometer for the investigation of optical systems,” C. R. Acad. Sci. URSS 5, 208–210 (1933) (in Russian).

IEEE Photon. Technol. Lett. (1)

E. C. Tam, S. W. Tannone, F. T. S. Yu, D. A. Gregory, “Closed-loop binary phase correction of an LCTV using a point diffraction interferometer,” IEEE Photon. Technol. Lett. 2, 143–146 (1990).
[CrossRef]

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

T. Smit, M. R. Smith, S. T. Nichols, “Efficient sinc function interpolation technique for center padded data,” IEEE Trans. Acoust. Speech Signal Process. 38, 1512–1517 (1990).
[CrossRef]

IEEE Trans. Signal Process. (1)

K. S. Knudsen, L. T. Bruton, “Recursive pruning of the 2-D DFT with 3-D signal processing applications,” IEEE Trans. Signal Process. 41, 1340–1356 (1993).
[CrossRef]

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

J. D. Markel, “FFT pruning,” IEEE Trans. Acoust. Speech Signal Process. AU-19, 305–311 (1971).

IEEE Trans. Circuits Syst. Video Technol. (1)

K. S. Knudsen, L. T. Bruton, “Mixed domain filtering of multidimensional signals,” IEEE Trans. Circuits Syst. Video Technol. 1, 260–268 (1991).
[CrossRef]

Math. Comput. (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Opt. Commun. (1)

H. Kadono, M. Ogusu, S. Toyooka, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (2)

T. M. Turpin, “Spectrum analysis using optical processing,” Proc. IEEE 69, 79–92 (1981).
[CrossRef]

H. R. Carleton, W. T. Maloney, G. Meltz, “Collinear heterodyning in optical processors,” Proc. IEEE 57, 769–775 (1969).
[CrossRef]

Other (12)

C. C. Aleksoff, N. S. Subotic, “Compact real-time interferometric Fourier transform processors,” in Optical Information Processing Systems and Architectures II, B. Javidi, ed., Proc. SPIE1347, 427–440 (1990).
[CrossRef]

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

K. S. Knudsen, L. T. Bruton, “Moving object detection and trajectory estimation in the transform/spatiotemporal mixed domain,” in Proceedings of the 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing, San Francisco, March 1992 (Institute of Electrical and Electronic Engineers, New York, 1992), Vol. 3, pp. 505–508.

K. S. Knudsen, L. T. Bruton, “Moving object nonlinear trajectory estimation in the transform/spatiotemporal mixed domain,” in Proceedings of the 1992 IEEE Custom Integrated Circuits Conference, San Diego, May 1992 (Institute of Electrical and Electronic Engineers, New York, 1992), Vol. 5, pp. 2481–2484.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, New York, 1991).

P. M. Lane, “The complex-valued optical Fourier transform and its application to moving-object trajectory estimation,” Ph.D. dissertation (Dalhousie University, Halifax, N.S., 1999).

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge Press, Wellesley, Mass., 1996).

K. S. Knudsen, “Multidimensional mixed domain signal processing,” Ph.D. dissertation (University of Calgary, Calgary, Al., 1992).

K. S. Knudsen, L. T. Bruton, “Transform/spatiotemporal mixed domain moving object tracking and enhancement,” in Proceedings of the 1993 European Conference on Circuit Theory and Design, Davos, Switzerland, August 1993 (Elsevier, Amsterdam, 1993), pp. 589–594.

P. M. Lane, K. S. Knudsen, M. Cada, “Moving object trajectory estimation using an optical Fourier processor,” in 1998 International Conference on Applications of Photonic Technology, G. A. Lampropoulos, ed., Proc. SPIE3491, 939–943 (1998).

L. J. Hornbeck, “Digital light processing and MEMS: timely convergence for a bright future (invited plenary paper),” in Micromaching and Microfabrication Process Technology, K. W. Markus, ed., Proc. SPIE2639, p. 2 (abstract only), full paper available from Texas Instruments, Dallas, Tex., 1995.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Construction of a joint image from a source and a reference image: (a) component images, t(m, n), u(m, n), v(m, n), and w(m, n); (b) joint image, f(m, n).

Fig. 2
Fig. 2

(a) Component images assembled to construct (b) the joint image.

Fig. 3
Fig. 3

Modulation-function zeros for horizontal and vertical JTI: (a) real part of spectrum; (b) imaginary part of spectrum.

Fig. 4
Fig. 4

Frequency-domain trajectory estimation: (a) spatiotemporal domain; (b) 3-D frequency domain.

Fig. 5
Fig. 5

Frame 17 of the source image sequence: (a) object-only sequence with simple stationary background; (b) object-only sequence with additive noise.

Fig. 6
Fig. 6

Bessel reference images: (a) L = 5 pixels, (b) L = 17 pixels, (c) L = 41 pixels, (d) L = 117 pixels, (e) L = 229 pixels, (f) L = 449 pixels.

Fig. 7
Fig. 7

Schematic diagram of the experimental setup.

Fig. 8
Fig. 8

Parallel interferometric optical Fourier-transform processor.

Tables (3)

Tables Icon

Table 1 Actual Trajectories for Objects in the Object-Only Image Sequence

Tables Icon

Table 2 Estimated Trajectories for Sequence with Stationary Background

Tables Icon

Table 3 Number of Frames Correctly Estimated for Sequence with White-Noise Background

Equations (23)

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

s p - p s + r p - p r     exp i k · p s S k + exp i k · p r R k ,
I ϕ k = I s k + I r k + 2 exp i ϕ S k R * k = I s k + I r k + 2 cos   ϕ S k R * k - sin   ϕ S k R * k ,
S k R * k = I ϕ k + I - ϕ k - 2 I s k - 2 I r k 4   cos   ϕ .
S k R * k = I ϕ k - I - ϕ k 4   sin   ϕ .
t ˜ m ,   n = t m 2 ,   n 2 n ,   m   even 0 otherwise ,
f wv tu m ,   n = t ˜ m ,   n + ũ m - 1 ,   n + v ˜ m - 1 ,   n - 1 + w ˜ m ,   n - 1 ,
F wv tu k = T 2 k + exp ik x U 2 k + exp i k x + k y V 2 k + exp ik y W 2 k ,
I wv tu k = | F wv tu k | 2 .
I 00 sr k = | S 2 k + exp ik x R 2 k | 2 = I s + I r + 2 cos k x SR * + sin k x SR * ,
I 00 sr = I sr 00 = I s + I r + 2 cos k x SR * + sin k x SR * ,   I 00 rs = I rs 00 = I s + I r + 2 cos k x SR * - sin k x SR * ,   I r 0 s 0 = I 0 r 0 s = I s + I r + 2 cos k y SR * + sin k y SR * ,   I s 0 r 0 = I 0 s 0 r = I s + I r + 2 cos k y SR * - sin k y SR * .
4   sin k x SR * = I 00 sr - I 00 rs ,   4   sin k y SR * = I r 0 s 0 - I s 0 r 0 ,
4   cos k x SR * = I 00 sr + I 00 rs - 2 I s - 2 I r ,   4   cos k y SR * = I r 0 s 0 + I s 0 r 0 - 2 I s - 2 I r .
I s = I s 0 00 = I 0 s 00 = I 00 0 s = I 00 s 0 ,
S 2 k x ,   2 k y = w 1 α vert + w 2 α horz + i w 3 β vert + w 4 β horz ,
w 1 = cos k x 4 cos 2 k x + cos 2 k y ,   w 2 = cos k y 4 cos 2 k x + cos 2 k y ,   w 3 = sin k x 4 sin 2 k x + sin 2 k y ,   w 4 = sin k y 4 sin 2 k x + sin 2 k y ,
α vert = I 00 sr + I 00 rs - 2 I s - 2 I r ,     β vert = I 00 sr - I 00 rs ,   α horz = I r 0 s 0 + I s 0 r 0 - 2 I s - 2 I r ,     β horz = I r 0 s 0 - I s 0 r 0 .
k r circle J 0 k r circle r     2 π δ k r - k r circle ,
I 0 r s 0 = I s + I r + 2 c 2 SR * + s 2 SR * ,   I 0 s r 0 = I s + I r + 2 c 2 SR * - s 2 SR * ,   I s 0 0 r = I s + I r + 2 c 1 SR * + s 1 SR * ,   I r 0 0 s = I s + I r + 2 c 1 SR * - s 1 SR * ,
I s r 0 s = 2 1 + c 1 I s + I r + 2 c 4 SR * + s 4 SR * ,   I s 0 r s = 2 1 + c 1 I s + I r + 2 c 4 SR * - s 4 SR * ,   I s r r 0 = I s + 2 1 + c 2 I r + 2 c 4 SR * + s 3 SR * ,   I 0 r r s = I s + 2 1 + c 2 I r + 2 c 4 SR * - s 3 SR * .
I r s s r = 2 1 + c 2 I s + 2 1 + c 1 I r + 4 c 4 SR * ,   I s r r s = 2 1 + c 1 I s + 2 1 + c 2 I r + 4 c 4 SR * .
I s r s s = 3 + 2 c 5 I s + I r + 2 c 6 SR * + s 6 SR * ,   I s s r s = 3 + 2 c 5 I s + I r + 2 c 6 SR * - s 6 SR * ,   I s r r r = I s + 3 + 2 c 6 I r + 2 c 5 SR * + s 5 SR * ,   I r r r s = I s + 3 + 2 c 6 I r + 2 c 5 SR * - s 5 SR * ,
c 5 = cos k x + cos k y + cos k x - k y ,   c 6 = cos k x + cos k y + cos k x + k y ,
s 5 = sin k x - sin k y + sin k x - k y ,   s 6 = sin k x + sin k y + sin k x + k y .

Metrics