Abstract

The Cohen bilinear class of shift-invariant space-frequency representations provides an automated means for extracting three-dimensional particle locations from in-line holograms without any focusing. For two-dimensional holograms a fixed-frequency slice technique, based on examining, concurrently, the zero-frequency slice and a nonzero-frequency slice of the two-dimensional representation used, is developed for particle-location analysis. The trade-off between auto-term sharpness and cross-term suppression for a multiple-particle hologram is achieved by relating kernel parameters of the representation to the smallest planar interparticle distance determined from the hologram by visual inspection with simple rules that result from an ambiguity-function-domain analysis. In addition, one-dimensional Cohen class representations are used to obtain complete space-frequency patterns that display object-location information and illustrate the cross-term suppression, for multiple-object one-dimensional holograms. The proposed techniques are implemented digitally, and the results are presented.

© 1998 Optical Society of America

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References

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  1. L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. (Bellingham) 26, 1124–1132 (1987).
    [CrossRef]
  2. G. Liu, P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A 4, 159–165 (1987).
    [CrossRef]
  3. H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. (Bellingham) 24, 462–463 (1985).
    [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 62–74, 207–208.
  5. L. Onural, M. T. Özgen, “Extraction of three-dimensional object location information directly from in-line holograms using Wigner analysis,” J. Opt. Soc. Am. A 9, 252–260 (1992).
    [CrossRef]
  6. L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction, and holography,” IEEE Trans. Signal Process. 43, 1568–1578 (1995).
    [CrossRef]
  7. L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography, D. Parsons, transl. (Consultants Bureau, New York, 1980).
  8. L. Onural, “Digital decoding of in-line holograms,” Ph.D. dissertation (State University of New York at Buffalo, Buffalo, New York, 1985).
  9. T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals; Part 2: discrete-time signals,” Philips J. Res. 35, 217–250, 276–300 (1980).
  10. L. Jacobson, H. Wechsler, “A theory for invariant object recognition in the frontoparallel plane,” IEEE Trans. Pattern. Anal. Mach. Intell. 6, 325–331 (1984).
    [CrossRef] [PubMed]
  11. S. Stanković, L. Stanković, Z. Uskoković, “On the local frequency, group shift, and cross-terms in some multidimensional time-frequency distributions: a method for multidimensional time-frequency analysis,” IEEE Trans. Signal Process. 43, 1719–1724 (1995).
    [CrossRef]
  12. S. Kadambe, G. F. Boudreaux-Bartels, “A comparison of the existence of cross terms in the Wigner distribution and the squared magnitude of the wavelet transform and the short time Fourier transform,” IEEE Trans. Signal Process. 40, 2498–2517 (1992).
    [CrossRef]
  13. B. Boashash, B. Escudie, “Wigner–Ville analysis of asymptotic signals and applications,” Signal Process. 8, 315–327 (1985).
    [CrossRef]
  14. P. Flandrin, “Some features of time-frequency representations of multicomponent signals,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing ’84 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1984), Papers 41B.4.1–41B.4.4.
  15. T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 3: relations with other time-frequency signal transformations,” Philips J. Res. 35, 372–389 (1980).
  16. L. Jacobson, H. Wechsler, “Joint space/spatial-frequency representation,” Signal Process. 14, 37–68 (1988).
    [CrossRef]
  17. L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
    [CrossRef]
  18. F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (April1992).
    [CrossRef]
  19. W. Martin, P. Flandrin, “Wigner–Ville spectral analysis of nonstationary processes,” IEEE Trans. Acoust. Speech, Signal Process. 33, 1461–1470 (1985).
    [CrossRef]
  20. H. I. Choi, W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Trans. Acoust. Speech, Signal Process. 37, 862–871 (1989).
    [CrossRef]
  21. J. Jeong, W. J. Williams, “Alias-free generalized discrete-time time-frequency distributions,” IEEE Trans. Signal Process. 40, 2757–2764 (1992).
    [CrossRef]
  22. A. Papandreou, G. F. Boudreaux-Bartels, “Generalization of the Choi–Williams distribution and the Butterworth distribution for time-frequency analysis,” IEEE Trans. Signal Process. 41, 463–472 (1993).
    [CrossRef]

1995 (2)

L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction, and holography,” IEEE Trans. Signal Process. 43, 1568–1578 (1995).
[CrossRef]

S. Stanković, L. Stanković, Z. Uskoković, “On the local frequency, group shift, and cross-terms in some multidimensional time-frequency distributions: a method for multidimensional time-frequency analysis,” IEEE Trans. Signal Process. 43, 1719–1724 (1995).
[CrossRef]

1993 (1)

A. Papandreou, G. F. Boudreaux-Bartels, “Generalization of the Choi–Williams distribution and the Butterworth distribution for time-frequency analysis,” IEEE Trans. Signal Process. 41, 463–472 (1993).
[CrossRef]

1992 (4)

S. Kadambe, G. F. Boudreaux-Bartels, “A comparison of the existence of cross terms in the Wigner distribution and the squared magnitude of the wavelet transform and the short time Fourier transform,” IEEE Trans. Signal Process. 40, 2498–2517 (1992).
[CrossRef]

L. Onural, M. T. Özgen, “Extraction of three-dimensional object location information directly from in-line holograms using Wigner analysis,” J. Opt. Soc. Am. A 9, 252–260 (1992).
[CrossRef]

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (April1992).
[CrossRef]

J. Jeong, W. J. Williams, “Alias-free generalized discrete-time time-frequency distributions,” IEEE Trans. Signal Process. 40, 2757–2764 (1992).
[CrossRef]

1989 (2)

H. I. Choi, W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Trans. Acoust. Speech, Signal Process. 37, 862–871 (1989).
[CrossRef]

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

1988 (1)

L. Jacobson, H. Wechsler, “Joint space/spatial-frequency representation,” Signal Process. 14, 37–68 (1988).
[CrossRef]

1987 (2)

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. (Bellingham) 26, 1124–1132 (1987).
[CrossRef]

G. Liu, P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A 4, 159–165 (1987).
[CrossRef]

1985 (3)

H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. (Bellingham) 24, 462–463 (1985).
[CrossRef]

B. Boashash, B. Escudie, “Wigner–Ville analysis of asymptotic signals and applications,” Signal Process. 8, 315–327 (1985).
[CrossRef]

W. Martin, P. Flandrin, “Wigner–Ville spectral analysis of nonstationary processes,” IEEE Trans. Acoust. Speech, Signal Process. 33, 1461–1470 (1985).
[CrossRef]

1984 (1)

L. Jacobson, H. Wechsler, “A theory for invariant object recognition in the frontoparallel plane,” IEEE Trans. Pattern. Anal. Mach. Intell. 6, 325–331 (1984).
[CrossRef] [PubMed]

1980 (2)

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 3: relations with other time-frequency signal transformations,” Philips J. Res. 35, 372–389 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals; Part 2: discrete-time signals,” Philips J. Res. 35, 217–250, 276–300 (1980).

Boashash, B.

B. Boashash, B. Escudie, “Wigner–Ville analysis of asymptotic signals and applications,” Signal Process. 8, 315–327 (1985).
[CrossRef]

Boudreaux-Bartels, G. F.

A. Papandreou, G. F. Boudreaux-Bartels, “Generalization of the Choi–Williams distribution and the Butterworth distribution for time-frequency analysis,” IEEE Trans. Signal Process. 41, 463–472 (1993).
[CrossRef]

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (April1992).
[CrossRef]

S. Kadambe, G. F. Boudreaux-Bartels, “A comparison of the existence of cross terms in the Wigner distribution and the squared magnitude of the wavelet transform and the short time Fourier transform,” IEEE Trans. Signal Process. 40, 2498–2517 (1992).
[CrossRef]

Caulfield, H. J.

H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. (Bellingham) 24, 462–463 (1985).
[CrossRef]

Choi, H. I.

H. I. Choi, W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Trans. Acoust. Speech, Signal Process. 37, 862–871 (1989).
[CrossRef]

Claasen, T. A. C. M.

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 3: relations with other time-frequency signal transformations,” Philips J. Res. 35, 372–389 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals; Part 2: discrete-time signals,” Philips J. Res. 35, 217–250, 276–300 (1980).

Cohen, L.

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Escudie, B.

B. Boashash, B. Escudie, “Wigner–Ville analysis of asymptotic signals and applications,” Signal Process. 8, 315–327 (1985).
[CrossRef]

Flandrin, P.

W. Martin, P. Flandrin, “Wigner–Ville spectral analysis of nonstationary processes,” IEEE Trans. Acoust. Speech, Signal Process. 33, 1461–1470 (1985).
[CrossRef]

P. Flandrin, “Some features of time-frequency representations of multicomponent signals,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing ’84 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1984), Papers 41B.4.1–41B.4.4.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 62–74, 207–208.

Hlawatsch, F.

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (April1992).
[CrossRef]

Jacobson, L.

L. Jacobson, H. Wechsler, “Joint space/spatial-frequency representation,” Signal Process. 14, 37–68 (1988).
[CrossRef]

L. Jacobson, H. Wechsler, “A theory for invariant object recognition in the frontoparallel plane,” IEEE Trans. Pattern. Anal. Mach. Intell. 6, 325–331 (1984).
[CrossRef] [PubMed]

Jeong, J.

J. Jeong, W. J. Williams, “Alias-free generalized discrete-time time-frequency distributions,” IEEE Trans. Signal Process. 40, 2757–2764 (1992).
[CrossRef]

Kadambe, S.

S. Kadambe, G. F. Boudreaux-Bartels, “A comparison of the existence of cross terms in the Wigner distribution and the squared magnitude of the wavelet transform and the short time Fourier transform,” IEEE Trans. Signal Process. 40, 2498–2517 (1992).
[CrossRef]

Kocatepe, M.

L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction, and holography,” IEEE Trans. Signal Process. 43, 1568–1578 (1995).
[CrossRef]

Liu, G.

Martin, W.

W. Martin, P. Flandrin, “Wigner–Ville spectral analysis of nonstationary processes,” IEEE Trans. Acoust. Speech, Signal Process. 33, 1461–1470 (1985).
[CrossRef]

Mecklenbrauker, W. F. G.

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 3: relations with other time-frequency signal transformations,” Philips J. Res. 35, 372–389 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals; Part 2: discrete-time signals,” Philips J. Res. 35, 217–250, 276–300 (1980).

Merzlyakov, N. S.

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography, D. Parsons, transl. (Consultants Bureau, New York, 1980).

Onural, L.

L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction, and holography,” IEEE Trans. Signal Process. 43, 1568–1578 (1995).
[CrossRef]

L. Onural, M. T. Özgen, “Extraction of three-dimensional object location information directly from in-line holograms using Wigner analysis,” J. Opt. Soc. Am. A 9, 252–260 (1992).
[CrossRef]

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. (Bellingham) 26, 1124–1132 (1987).
[CrossRef]

L. Onural, “Digital decoding of in-line holograms,” Ph.D. dissertation (State University of New York at Buffalo, Buffalo, New York, 1985).

Özgen, M. T.

Papandreou, A.

A. Papandreou, G. F. Boudreaux-Bartels, “Generalization of the Choi–Williams distribution and the Butterworth distribution for time-frequency analysis,” IEEE Trans. Signal Process. 41, 463–472 (1993).
[CrossRef]

Scott, P. D.

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. (Bellingham) 26, 1124–1132 (1987).
[CrossRef]

G. Liu, P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A 4, 159–165 (1987).
[CrossRef]

Stankovic, L.

S. Stanković, L. Stanković, Z. Uskoković, “On the local frequency, group shift, and cross-terms in some multidimensional time-frequency distributions: a method for multidimensional time-frequency analysis,” IEEE Trans. Signal Process. 43, 1719–1724 (1995).
[CrossRef]

Stankovic, S.

S. Stanković, L. Stanković, Z. Uskoković, “On the local frequency, group shift, and cross-terms in some multidimensional time-frequency distributions: a method for multidimensional time-frequency analysis,” IEEE Trans. Signal Process. 43, 1719–1724 (1995).
[CrossRef]

Uskokovic, Z.

S. Stanković, L. Stanković, Z. Uskoković, “On the local frequency, group shift, and cross-terms in some multidimensional time-frequency distributions: a method for multidimensional time-frequency analysis,” IEEE Trans. Signal Process. 43, 1719–1724 (1995).
[CrossRef]

Wechsler, H.

L. Jacobson, H. Wechsler, “Joint space/spatial-frequency representation,” Signal Process. 14, 37–68 (1988).
[CrossRef]

L. Jacobson, H. Wechsler, “A theory for invariant object recognition in the frontoparallel plane,” IEEE Trans. Pattern. Anal. Mach. Intell. 6, 325–331 (1984).
[CrossRef] [PubMed]

Williams, W. J.

J. Jeong, W. J. Williams, “Alias-free generalized discrete-time time-frequency distributions,” IEEE Trans. Signal Process. 40, 2757–2764 (1992).
[CrossRef]

H. I. Choi, W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Trans. Acoust. Speech, Signal Process. 37, 862–871 (1989).
[CrossRef]

Yaroslavskii, L. P.

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography, D. Parsons, transl. (Consultants Bureau, New York, 1980).

IEEE Signal Process. Mag. (1)

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (April1992).
[CrossRef]

IEEE Trans. Acoust. Speech, Signal Process. (2)

W. Martin, P. Flandrin, “Wigner–Ville spectral analysis of nonstationary processes,” IEEE Trans. Acoust. Speech, Signal Process. 33, 1461–1470 (1985).
[CrossRef]

H. I. Choi, W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Trans. Acoust. Speech, Signal Process. 37, 862–871 (1989).
[CrossRef]

IEEE Trans. Pattern. Anal. Mach. Intell. (1)

L. Jacobson, H. Wechsler, “A theory for invariant object recognition in the frontoparallel plane,” IEEE Trans. Pattern. Anal. Mach. Intell. 6, 325–331 (1984).
[CrossRef] [PubMed]

IEEE Trans. Signal Process. (5)

S. Stanković, L. Stanković, Z. Uskoković, “On the local frequency, group shift, and cross-terms in some multidimensional time-frequency distributions: a method for multidimensional time-frequency analysis,” IEEE Trans. Signal Process. 43, 1719–1724 (1995).
[CrossRef]

S. Kadambe, G. F. Boudreaux-Bartels, “A comparison of the existence of cross terms in the Wigner distribution and the squared magnitude of the wavelet transform and the short time Fourier transform,” IEEE Trans. Signal Process. 40, 2498–2517 (1992).
[CrossRef]

J. Jeong, W. J. Williams, “Alias-free generalized discrete-time time-frequency distributions,” IEEE Trans. Signal Process. 40, 2757–2764 (1992).
[CrossRef]

A. Papandreou, G. F. Boudreaux-Bartels, “Generalization of the Choi–Williams distribution and the Butterworth distribution for time-frequency analysis,” IEEE Trans. Signal Process. 41, 463–472 (1993).
[CrossRef]

L. Onural, M. Kocatepe, “Family of scaling chirp functions, diffraction, and holography,” IEEE Trans. Signal Process. 43, 1568–1578 (1995).
[CrossRef]

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

Opt. Eng. (Bellingham) (2)

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. (Bellingham) 26, 1124–1132 (1987).
[CrossRef]

H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. (Bellingham) 24, 462–463 (1985).
[CrossRef]

Philips J. Res. (2)

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 3: relations with other time-frequency signal transformations,” Philips J. Res. 35, 372–389 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals; Part 2: discrete-time signals,” Philips J. Res. 35, 217–250, 276–300 (1980).

Proc. IEEE (1)

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Signal Process. (2)

L. Jacobson, H. Wechsler, “Joint space/spatial-frequency representation,” Signal Process. 14, 37–68 (1988).
[CrossRef]

B. Boashash, B. Escudie, “Wigner–Ville analysis of asymptotic signals and applications,” Signal Process. 8, 315–327 (1985).
[CrossRef]

Other (4)

P. Flandrin, “Some features of time-frequency representations of multicomponent signals,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing ’84 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1984), Papers 41B.4.1–41B.4.4.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 62–74, 207–208.

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography, D. Parsons, transl. (Consultants Bureau, New York, 1980).

L. Onural, “Digital decoding of in-line holograms,” Ph.D. dissertation (State University of New York at Buffalo, Buffalo, New York, 1985).

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Figures (7)

Fig. 1
Fig. 1

(a) Simulated object distribution, with right-triangle-shaped particle. (b) The simulated in-line hologram of the object distribution given in (a); normalized depth parameter, α=1.0. (c) The WD slice, (m1, m2)=(0, 0), for the hologram. The particle replica is at the same location as the original particle shown in (a). (d) The WD slice, (m1, m2)=(16, 0), for the hologram. There are two replicas; the distance between them is 16 samples.

Fig. 2
Fig. 2

(a) Simulated hologram with three planar particles; normalized depth parameter, α=1.0. (b) The WD slice, (m1, m2)=(0, 0), for the hologram. There are three true replicas each at a corner and three false replicas located between them. (c) The WD slice (m1, m2)=(16, 16). There are three true pairs and three false pairs. Within each pair the discrete-distance vector is (Δxd, Δyd)=(16, 16) samples.

Fig. 3
Fig. 3

(a) Simulated hologram with two particles located at different depths. Upper particle, α=3.0; lower particle, α=1.0. (b) The WD slice, (m1, m2)=(0, 0), for this hologram. The cross term is apparent in the middle. True replicas indicate the particle locations in the individual object planes. (c) The WD slice (m1, m2)=(24, 0). For the lower pair of replicas the discrete-distance vector is (Δxd, Δyd)=(24, 0) samples; for the upper pair of replicas, (Δxd, Δyd)=(8, 0) samples. Cross terms appear in the middle.

Fig. 4
Fig. 4

(a) Simulated 1-D hologram of three objects located at the same depth. There is no variation along the vertical direction in the object plane. Normalized depth parameter, α=1.0. (b) The 1-D Wigner distribution for the hologram. The horizontal axis indicated by the horizontal bright line is the discrete-space axis. The vertical axis indicates discrete-frequency indices. Two solid crosses that represent side objects and three oscillating crosses between them as cross terms are apparent. The middle auto term is obscured. (c) Untilted Gaussian representation for the hologram. Kernel parameters are αc=50 samples and ξc=π. Cross terms are significantly suppressed; hence the auto term for the middle object is now revealed.

Fig. 5
Fig. 5

(a) Simulated 2-D hologram of two square-shaped 4×4 particles; normalized depth parameter, α=1.0. (b) The WD slice, (m1, m2)=(0, 0), for the hologram. The cross term appears as the pulse in the middle. (c) The WD slice, (m1, m2)=(16, 0). Cross terms appear as the pair in the middle. Within each pair, the discrete distance vector is (Δxd, Δyd)=(16, 0) samples. (d) The (m1, m2)=(0, 0) slice of the RBMR for the hologram. Kernel parameters are αcx=αcy=32 samples, ξcx=ξcy=π. The cross term in the middle is eliminated. (e) The (m1, m2)=(16, 0) slice of the RBMR. Cross terms in the middle are eliminated. (Δxd, Δyd)=(16, 0) samples again, for both pairs.

Fig. 6
Fig. 6

(a) Two-dimensional hologram of four square-shaped 4×4 particles; normalized depth parameter, α=1.0. (b) The (m1, m2)=(0, 0) slice of the pseudo-Wigner distribution with rectangular window for the hologram. Kernel parameters are αcx=αcy=16 samples. (c) The (m1, m2)=(16, 0) slice. The discrete-distance vector is (Δxd, Δyd)=(16, 0) samples, for each pair.

Fig. 7
Fig. 7

(a) Simulated 2-D hologram of five particles with different depths, shapes, sizes, and orientations. There are two back-to-back particles represented by the diffraction pattern on the lower right. (b) The (m1, m2)=(0, 0) slice of the smoothed pseudo-Wigner distribution with rectangular windows for the hologram. Kernel parameters are αcx=αcy=20 samples, xcx=xcy=1.0. (c) The (m1, m2)=(9, 9) slice. Back-to-back particles on the lower right are resolved in this slice.

Tables (1)

Tables Icon

Table 1 Representations of the Cohen Class Used in This Paper

Equations (51)

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

Iz(x, y)=1-a(x, y) ** 2 Re{hz(x, y)}
+|a(x, y) ** hz(x, y)|2,
gz(x, y)=2 Re{hz(x, y)}=2λzsinπλz(x2+y2).
I(x, y)=1-kak(x, y) ** gzk(x, y),
Iz(x)=1-a(x) * gz(x),
α2/N=X2/(λz),
Wf(x, w)=R2f(x+α/2)f*(x-α/2)×exp(-jw·α)dα,
Wf(n1, n2, θ1, θ2)=4k1,k2f(n1+k1, n2+k2)×f*(n1-k1, n2-k2)×exp[-j2(θ1k1+θ2k2)]
Wf ** g=Wf ** Wg.
Wf+g=Wf+Wg+2 Re{Wf,g},
Af(ξ, α)=f(x+α/2)f*(x-α/2)exp(jξx)dx=Wf(x, w)exp[j(xξ+wα)]dxdw,
Af(ξ, α)=R2f(x+α/2)f*(x-α/2)exp(jξ·x)dx,
Cf(x, w; ϕ1a)
=12π exp[jξ(u-x)-jwα]×ϕ1a(ξ, α)f(u+α/2)f*(u-α/2)dξdudα=12π[Af(ξ, α)ϕ1a(ξ, α)]×exp[-j(ξx+αw)]dξdα,
Cf(n, θ)=2k=- exp(-j2θk)p=-G1a(p-n, 2k)×f(p+k)f*(p-k),
G1a(x, α)=12π-ϕ1a(ξ, α)exp(jξx)dξ.
Cf(x, w; ϕa)=14π2R2R2R2exp[jξ·(u-x)-jw·α]ϕa(ξ, α)×f(u+α/2)f*(u-α/2)dξdudα,
ϕa(ξx, ξy, αx, αy)=ϕ1a(ξx, αx)ϕ1a(ξy, αy).
Ga(x, y, αx, αy)=F(ξx, ξy)(x, y)-1[ϕa(ξx, ξy, αx, αy)]=G1a(x, αx)G1a(y, αy),
Wgz(x, y, wx, wy)
=4π2λ2z2δwx-2πλzx, wy-2πλzy+δwx+2πλzx, wy+2πλzy+4λzsin2πλz(x2+y2)-λz2π(wx2+wy2),
WJz(x, y, wx, wy)
=-Wa(α, β, wx, wy)×Wgz(x-α, y-β, wx, wy)dαdβ,
WJz(x, y, wxf, wyf)
=δx-x0+λz2πwxf, y-y0+λz2πwyf+δx-x0-λz2πwxf, y-y0-λz2πwyf
WJz(x, y, 0, 0)=2δ(x-x0, y-y0)
Wa(x, w)=i=1nδ(x-xi)+i=1n-1j=i+1n2 cos[w·(xi-xj)]×δ[x-(xi+xj)/2].
WJz(x, wf)=i=1nδx-xi+λz2πwf+δx-xi-λz2πwf+i=1n-1j=i+1n2 cos[wf·(xi-xj)]×δx-(xi+xj)/2+λz2πwf+δx-(xi+xj)/2-λz2πwf,
WJz(x, 0)=i=1n2δ(x-xi)+i=1n-1j=i+1n4δ[x-(xi+xj)/2].
WJ(x, w)=kWJzk(x, w)+km>k2 Re{WJzkJzm(x,w)},
WJ(x, w)kWJzk(x, w).
WJn1, n2, πNm1, πNm2
=1N2k1=02N-1k2=02N-1J˜2N(m1+k1, m2+k2)×J˜2N*(m1-k1, m2-k2)×expj 2πN(k1n1+k2n2)
(Δxd, Δyd)=(Δx, Δy)/X=λzNX2(m1, m2)
(Δxd, Δyd)=1α2(m1, m2).
Δz=NX2λmi,
AJz(ξ, α)=Aa(ξ, α) ** Agz(ξ, α),
Agz(ξ, α)=16π4λ2z2δξ+2πλzα+δξ-2πλzα,
AJz(ξ, α)=[exp(jξ·x1)+exp(jξ·x2)]Agz(ξ, α)+exp[jξ·(x1+x2)/2]×[Agz(ξ, α+x1-x2)+Agz(ξ, α+x2-x1)].
AJz(0, α)=16π2δ(α)+8π2δ(α+x1-x2)+8π2δ(α+x2-x1),
αcxdx,min,αcydy,min,
αcxLx,max,αcyLy,max.
CJn, πNm=2k=-N/2N/2-1 exp-j 2πNmk×p=max(0,n-M)min(N-1,n+M)G1a(p-n, 2k)×J(p+k)J(p-k)
CJn, πNm
=4π2N4kRN expj 2πNk·n×pR2NJ˜2N(p+k)J˜2N*(p-k)×ψ˜d,N(k, p-m)
ψ˜d,N(k, l)=ψ˜1d,N(k1, l1)ψ˜1d,N(k2, l2).
ψ1a(ξ, w)=12π-ϕ1a(ξ, α)exp(jαw)dα.
ψ˜1d,N(k, l)=12ψ1a-2πNk, πNl+ψ1a-2πNk, πNl-π,
0kN2
=12ψ1a2π-2πNk, πNl+ψ1a2π-2πNk, πNl-π,
N2+1k<N

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