Abstract

We describe the implementation of the automatic spatial-frequency-selection filter for recognition of patterns obtained with a digital holographic microscope working with a partially coherent source. The microscope provides the complex-optical-amplitude field that allows a refocusing plane-by-plane of the sample under investigation by numerical computation of the optical propagation. By inserting a correlation filter in the propagation equation, the correlation between the filter and the propagated optical field is obtained. In this way, the pattern is located in the direction of the optical axis. Owing to the very weak noise level generated by the partially coherent source, the correlation process is shift invariant. Therefore the samples can be located in the three dimensions. To have a robust recognition process, a generalized version of the automatic spatial-frequency-selection filters has been implemented. The method is experimentally demonstrated in a two-class problem for the recognition of protein crystals.

© 2002 Optical Society of America

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References

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  1. U. Schnars, W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef] [PubMed]
  2. K. Creath, “Temporal Phase Measurement Methods,” in Interferogram Analysis: digital fringe pattern analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Publishing Ltd., London, 1993), pp. 94–140.
  3. I. Yamaguchi, T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
    [CrossRef] [PubMed]
  4. B. Skarman, K. Wozniac, J. Becker, “Simultaneous 3D-PIV and temperature measurement using a New CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1–6 (1996).
    [CrossRef]
  5. T. M. Kreis, W. P. O. Jüptner, “Principle of digital holography,” in Fringe ’97: Automatic Processing of Fringe Patterns, W. Jæptner, W. Osten, eds. (Wiley, New York, 1998), pp. 353–363.
  6. B. Nilsson, T. E. Carlsson, “Direct three-dimensional shape measurement by digital light-in-flight holography,” Appl. Opt. 37, 7954–7959 (1998).
    [CrossRef]
  7. E. Cuche, F. Bevilacqua, C. Depeursinge, “Digital holography for quantitative phase contrast imaging,” Opt. Lett. 24, 291–293 (1999).
    [CrossRef]
  8. D. O. Hogenboom, C. A. Dimarzio, T. J. Gaudette, A. J. Devaney, S. C. Lindberg, “Three-dimensional images generated by quadrature interferometry,” Opt. Lett. 23, 783–785 (1998).
    [CrossRef]
  9. M. Adams, T. M. Kreis, W. P. O. Jüptner, “Particle size and position measurement with digital holography,” in Optical Inspection and Micromeasurements II, C. Gurecki, ed., Proc. SPIE3098, 234–240 (1998).
    [CrossRef]
  10. E. Cuche, F. Belivacqua, C. Despeuringe, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291–293 (1999).
    [CrossRef]
  11. B. Javidi, E. Tajahuerce, “Encrypting three-dimensional information with digital holography,” Opt. Lett. 39, 6595–6601 (2000).
  12. F. Dubois, L. Joannes, O. Dupont, J. L. Dewandel, J. C. Legros, “An integrated optical set-up for fluid-physics experiments under microgravity conditions,” Meas. Sci. Technol. 10, 934–945 (1999).
    [CrossRef]
  13. T. Zhang, I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett. 23, 1221–1223 (1998).
    [CrossRef]
  14. F. Dubois, L. Joannes, J.-C. Legros, “Improved three-dimensional imaging with digital holography microscope using a partial spatial coherent source,” Appl. Opt. 38, 7085–7094 (1999).
    [CrossRef]
  15. Y. Takaki, H. Ohzu, “Hybrid holographic microscopy: Visualization of three-dimensional object information by use of viewing angles,” Appl. Opt. 39, 5302–5308 (2000).
    [CrossRef]
  16. B. Javidi, E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. 25, 610–612 (2000).
    [CrossRef]
  17. E. Tajahuerce, O. Matoba, B. Javidi, “Shift-invariant three-dimensional object recognition by means of digital holography,” Appl. Opt. 40, 3877–3886 (2001).
    [CrossRef]
  18. Y. Frauel, E. Tajahuerce, M.-A. Castro, B. Javidi, “Distortion-tolerant three-dimensional object recognition with digital holography,” Appl. Opt. 40, 3887–3893 (2001).
    [CrossRef]
  19. A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
    [CrossRef]
  20. P. Chavel, “Optical noise and temporal coherence,” J. Opt. Soc. Am. 70, 935–943 (1980).
    [CrossRef]
  21. A. B. Vander Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  22. D. P. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef] [PubMed]
  23. B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. 3, 1579–1584 (1986).
    [CrossRef]
  24. A. Mahalanobis, B. V. K. V. Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  25. A. Mahalanobis, D. Casasent, “Performance evaluation of minimum average correlation filters,” Appl. Opt. 30, 561–572 (1991).
    [CrossRef] [PubMed]
  26. Ph. Réfrégier, “Filter design for optical pattern: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
    [CrossRef]
  27. Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  28. D. L. Flannery, “Optimal trade-off distortion-tolerant constrained-modulation correlation filters,” J. Opt. Soc. Am. A 12, 66–72 (1995).
    [CrossRef]
  29. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  30. A. Vander Lugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 1, 215–222 (1970).
    [CrossRef]
  31. Ph. Réfrégier, V. Laude, B. Javidi, “Nonlinear joint transform correlation: an optimal solution for adaptive image discrimination and input noise robustness,” Opt. Lett. 19, 405–407 (1994).
  32. B. Javidi, D. Painchaud, “Distortion-invariant pattern recognition with Fourier-plane nonlinear filters,” Appl. Opt. 35, 318–331 (1996).
    [CrossRef] [PubMed]
  33. M. Guillaume, Ph. Réfrégier, J. Campos, V. Lashin, “Detection theory approach to multichannel location,” Opt. Lett. 22, 1887–1889 (1997).
    [CrossRef]
  34. F. Guérault, Ph. Réfrégier, “Unified statistically independent region processor for deterministic and fluctuating targets in nonoverlapping background,” Opt. Lett. 23, 412–414 (1998).
    [CrossRef]
  35. B. Javidi, F. Parchekani, G. Zhang, “Minimum-mean-square-error filters for detecting a noisy target in background noise,” Appl. Opt. 35, 6964–6975 (1996).
    [CrossRef] [PubMed]
  36. F. Dubois, “Automatic spatial frequency selection algorithm for pattern recognition by correlation,” Appl. Opt. 32, 4365–4371 (1993).
    [CrossRef] [PubMed]
  37. F. Dubois, “Nonlinear cascaded correlation processes to improve the performances of the automatic spatial-frequency-selective filters in pattern recognition,” Appl. Opt. 35, 4589–4597 (1996).
    [CrossRef] [PubMed]
  38. C. Minetti, F. Dubois, “Reduction in correlation filter sensitivity to background clutter using the automatic spatial frequency selection algorithm,” Appl. Opt. 35, 1900–1903 (1996).
    [CrossRef] [PubMed]
  39. M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
    [CrossRef]
  40. S. P. Almeida, J. Kim-Tzong Eu, “Water pollution monitoring using matched spatial filters,” Appl. Opt. 15, 510–515 (1976).
    [CrossRef] [PubMed]

2001 (2)

2000 (3)

1999 (4)

1998 (4)

1997 (2)

1996 (5)

1995 (1)

1994 (2)

1993 (1)

1991 (2)

1990 (1)

1989 (1)

1987 (1)

1986 (1)

B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. 3, 1579–1584 (1986).
[CrossRef]

1984 (1)

1981 (1)

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

1980 (2)

1976 (1)

1970 (1)

A. Vander Lugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 1, 215–222 (1970).
[CrossRef]

1964 (1)

A. B. Vander Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Adams, M.

M. Adams, T. M. Kreis, W. P. O. Jüptner, “Particle size and position measurement with digital holography,” in Optical Inspection and Micromeasurements II, C. Gurecki, ed., Proc. SPIE3098, 234–240 (1998).
[CrossRef]

Almeida, S. P.

Becker, J.

B. Skarman, K. Wozniac, J. Becker, “Simultaneous 3D-PIV and temperature measurement using a New CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1–6 (1996).
[CrossRef]

Belivacqua, F.

Bevilacqua, F.

Campos, J.

Carlsson, T. E.

Casasent, D.

Casasent, D. P.

Castro, M.-A.

Chavel, P.

Creath, K.

K. Creath, “Temporal Phase Measurement Methods,” in Interferogram Analysis: digital fringe pattern analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Publishing Ltd., London, 1993), pp. 94–140.

Cuche, E.

Depeursinge, C.

Despeuringe, C.

Devaney, A. J.

Dewandel, J. L.

F. Dubois, L. Joannes, O. Dupont, J. L. Dewandel, J. C. Legros, “An integrated optical set-up for fluid-physics experiments under microgravity conditions,” Meas. Sci. Technol. 10, 934–945 (1999).
[CrossRef]

Dimarzio, C. A.

Dubois, F.

Dupont, O.

F. Dubois, L. Joannes, O. Dupont, J. L. Dewandel, J. C. Legros, “An integrated optical set-up for fluid-physics experiments under microgravity conditions,” Meas. Sci. Technol. 10, 934–945 (1999).
[CrossRef]

Flannery, D. L.

Frauel, Y.

Gaudette, T. J.

Guérault, F.

Guillaume, M.

Hogenboom, D. O.

Javidi, B.

Joannes, L.

F. Dubois, L. Joannes, O. Dupont, J. L. Dewandel, J. C. Legros, “An integrated optical set-up for fluid-physics experiments under microgravity conditions,” Meas. Sci. Technol. 10, 934–945 (1999).
[CrossRef]

F. Dubois, L. Joannes, J.-C. Legros, “Improved three-dimensional imaging with digital holography microscope using a partial spatial coherent source,” Appl. Opt. 38, 7085–7094 (1999).
[CrossRef]

Jüptner, W.

Jüptner, W. P. O.

T. M. Kreis, W. P. O. Jüptner, “Principle of digital holography,” in Fringe ’97: Automatic Processing of Fringe Patterns, W. Jæptner, W. Osten, eds. (Wiley, New York, 1998), pp. 353–363.

M. Adams, T. M. Kreis, W. P. O. Jüptner, “Particle size and position measurement with digital holography,” in Optical Inspection and Micromeasurements II, C. Gurecki, ed., Proc. SPIE3098, 234–240 (1998).
[CrossRef]

Kim-Tzong Eu, J.

Kreis, T. M.

M. Adams, T. M. Kreis, W. P. O. Jüptner, “Particle size and position measurement with digital holography,” in Optical Inspection and Micromeasurements II, C. Gurecki, ed., Proc. SPIE3098, 234–240 (1998).
[CrossRef]

T. M. Kreis, W. P. O. Jüptner, “Principle of digital holography,” in Fringe ’97: Automatic Processing of Fringe Patterns, W. Jæptner, W. Osten, eds. (Wiley, New York, 1998), pp. 353–363.

Kumar, B. V. K. V.

Lashin, V.

Laude, V.

Legros, J. C.

F. Dubois, L. Joannes, O. Dupont, J. L. Dewandel, J. C. Legros, “An integrated optical set-up for fluid-physics experiments under microgravity conditions,” Meas. Sci. Technol. 10, 934–945 (1999).
[CrossRef]

Legros, J.-C.

Lim, J. S.

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Lindberg, S. C.

Mahalanobis, A.

Matoba, O.

Minetti, C.

Nazarathy, M.

Nilsson, B.

Ohzu, H.

Oppenheim, A. V.

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Painchaud, D.

Parchekani, F.

Réfrégier, Ph.

Rotz, F. B.

A. Vander Lugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 1, 215–222 (1970).
[CrossRef]

Schnars, U.

Shamir, J.

Skarman, B.

B. Skarman, K. Wozniac, J. Becker, “Simultaneous 3D-PIV and temperature measurement using a New CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1–6 (1996).
[CrossRef]

Tajahuerce, E.

Takaki, Y.

Vander Lugt, A.

A. Vander Lugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 1, 215–222 (1970).
[CrossRef]

Vander Lugt, A. B.

A. B. Vander Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Vijaya Kumar, B. V. K.

B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. 3, 1579–1584 (1986).
[CrossRef]

Wozniac, K.

B. Skarman, K. Wozniac, J. Becker, “Simultaneous 3D-PIV and temperature measurement using a New CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1–6 (1996).
[CrossRef]

Yamaguchi, I.

Zhang, G.

Zhang, T.

Appl. Opt. (17)

S. P. Almeida, J. Kim-Tzong Eu, “Water pollution monitoring using matched spatial filters,” Appl. Opt. 15, 510–515 (1976).
[CrossRef] [PubMed]

D. P. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
[CrossRef] [PubMed]

A. Mahalanobis, B. V. K. V. Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

A. Mahalanobis, D. Casasent, “Performance evaluation of minimum average correlation filters,” Appl. Opt. 30, 561–572 (1991).
[CrossRef] [PubMed]

F. Dubois, “Automatic spatial frequency selection algorithm for pattern recognition by correlation,” Appl. Opt. 32, 4365–4371 (1993).
[CrossRef] [PubMed]

U. Schnars, W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[CrossRef] [PubMed]

B. Nilsson, T. E. Carlsson, “Direct three-dimensional shape measurement by digital light-in-flight holography,” Appl. Opt. 37, 7954–7959 (1998).
[CrossRef]

F. Dubois, L. Joannes, J.-C. Legros, “Improved three-dimensional imaging with digital holography microscope using a partial spatial coherent source,” Appl. Opt. 38, 7085–7094 (1999).
[CrossRef]

B. Javidi, D. Painchaud, “Distortion-invariant pattern recognition with Fourier-plane nonlinear filters,” Appl. Opt. 35, 318–331 (1996).
[CrossRef] [PubMed]

C. Minetti, F. Dubois, “Reduction in correlation filter sensitivity to background clutter using the automatic spatial frequency selection algorithm,” Appl. Opt. 35, 1900–1903 (1996).
[CrossRef] [PubMed]

F. Dubois, “Nonlinear cascaded correlation processes to improve the performances of the automatic spatial-frequency-selective filters in pattern recognition,” Appl. Opt. 35, 4589–4597 (1996).
[CrossRef] [PubMed]

B. Javidi, F. Parchekani, G. Zhang, “Minimum-mean-square-error filters for detecting a noisy target in background noise,” Appl. Opt. 35, 6964–6975 (1996).
[CrossRef] [PubMed]

Y. Takaki, H. Ohzu, “Hybrid holographic microscopy: Visualization of three-dimensional object information by use of viewing angles,” Appl. Opt. 39, 5302–5308 (2000).
[CrossRef]

E. Tajahuerce, O. Matoba, B. Javidi, “Shift-invariant three-dimensional object recognition by means of digital holography,” Appl. Opt. 40, 3877–3886 (2001).
[CrossRef]

Y. Frauel, E. Tajahuerce, M.-A. Castro, B. Javidi, “Distortion-tolerant three-dimensional object recognition with digital holography,” Appl. Opt. 40, 3887–3893 (2001).
[CrossRef]

A. Vander Lugt, F. B. Rotz, “The use of film nonlinearities in optical spatial filtering,” Appl. Opt. 1, 215–222 (1970).
[CrossRef]

Flow Meas. Instrum. (1)

B. Skarman, K. Wozniac, J. Becker, “Simultaneous 3D-PIV and temperature measurement using a New CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1–6 (1996).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. B. Vander Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Meas. Sci. Technol. (1)

F. Dubois, L. Joannes, O. Dupont, J. L. Dewandel, J. C. Legros, “An integrated optical set-up for fluid-physics experiments under microgravity conditions,” Meas. Sci. Technol. 10, 934–945 (1999).
[CrossRef]

Opt. Lett. (12)

B. Javidi, E. Tajahuerce, “Encrypting three-dimensional information with digital holography,” Opt. Lett. 39, 6595–6601 (2000).

I. Yamaguchi, T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
[CrossRef] [PubMed]

M. Guillaume, Ph. Réfrégier, J. Campos, V. Lashin, “Detection theory approach to multichannel location,” Opt. Lett. 22, 1887–1889 (1997).
[CrossRef]

F. Guérault, Ph. Réfrégier, “Unified statistically independent region processor for deterministic and fluctuating targets in nonoverlapping background,” Opt. Lett. 23, 412–414 (1998).
[CrossRef]

D. O. Hogenboom, C. A. Dimarzio, T. J. Gaudette, A. J. Devaney, S. C. Lindberg, “Three-dimensional images generated by quadrature interferometry,” Opt. Lett. 23, 783–785 (1998).
[CrossRef]

T. Zhang, I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett. 23, 1221–1223 (1998).
[CrossRef]

E. Cuche, F. Belivacqua, C. Despeuringe, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291–293 (1999).
[CrossRef]

E. Cuche, F. Bevilacqua, C. Depeursinge, “Digital holography for quantitative phase contrast imaging,” Opt. Lett. 24, 291–293 (1999).
[CrossRef]

Ph. Réfrégier, “Filter design for optical pattern: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
[CrossRef]

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

B. Javidi, E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. 25, 610–612 (2000).
[CrossRef]

Ph. Réfrégier, V. Laude, B. Javidi, “Nonlinear joint transform correlation: an optimal solution for adaptive image discrimination and input noise robustness,” Opt. Lett. 19, 405–407 (1994).

Proc. IEEE (1)

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Other (3)

K. Creath, “Temporal Phase Measurement Methods,” in Interferogram Analysis: digital fringe pattern analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Publishing Ltd., London, 1993), pp. 94–140.

T. M. Kreis, W. P. O. Jüptner, “Principle of digital holography,” in Fringe ’97: Automatic Processing of Fringe Patterns, W. Jæptner, W. Osten, eds. (Wiley, New York, 1998), pp. 353–363.

M. Adams, T. M. Kreis, W. P. O. Jüptner, “Particle size and position measurement with digital holography,” in Optical Inspection and Micromeasurements II, C. Gurecki, ed., Proc. SPIE3098, 234–240 (1998).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Scheme of the microscope implemented in a Mach–Zehnder interferometer with a spatially partial coherent source.

Fig. 2
Fig. 2

Refocusing by digital holography on a metric scale (100 divisions/mm). (a) Image of the defocused intensity. (b) Image of the computer refocused intensity. The refocus distance is 400 µm.

Fig. 3
Fig. 3

(a) Intensity image and (b) phase image of the crystal to be rejected.

Fig. 4
Fig. 4

(a) Intensity image and (b) phase image of the crystal to be recognized.

Fig. 5
Fig. 5

Phase-corrected image of the crystal of Fig. 4(b).

Fig. 6
Fig. 6

Central region of the primary reference (a) intensity image and (b) phase image of the crystal to be recognized.

Fig. 7
Fig. 7

Defocused image by a distance of -150 µm.

Fig. 8
Fig. 8

Correlation intensities for the crystal to be recognized (a) series 3 and (b) series 1. Defocus distance 100 µm, reconstruction distance 100 µm.

Fig. 9
Fig. 9

Horizontal correlation-intensity profile crossing the correlation peak as a function of the digital holographic reconstruction distance d (a) series 3 and (b) series 4. Defocus distance D = 0 µm.

Fig. 10
Fig. 10

Horizontal correlation intensity profile crossing the correlation center as a function of the digital holographic reconstruction distance d (a) series 1 and (b) series 2. Defocus distance D = 0 µm.

Fig. 11
Fig. 11

Maximum correlation intensities as a function of the digital reconstruction distance d for the samples of series 1 and 3.

Fig. 12
Fig. 12

Maximum correlation intensities as a function of the digital reconstruction distance d for the samples of series 2 and 4.

Fig. 13
Fig. 13

Defocus distance D as a function of the reconstruction distance d when the maximum correlation intensity is reached for the samples of series 3.

Fig. 14
Fig. 14

Maximum correlation intensities between the ASFS filter and the in-plane rotated amplitude fields of the focus phase-corrected amplitudes corresponding to the crystals shown in Figs. 3 and 4.

Equations (20)

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

d2f2f4ρf3νmax,
ϕs, t=tan-1I4s, t-I2s, tI1s, t-I3s, t.
u0x, y=expikdFCνx,νy-1 exp-jkdλ22νx2+νy2 FCx,y+1uix, y,
FCαβ±1gα, β=--exp2jπαη+βξgα, βdαdβ.
u0sΔ, tΔ=expjkdFU,V-1 exp-jkλ2d2N2Δ2U2+V2Fs,t+1uisΔ, tΔ,
Fk,l±1gm, n=1Nk,l=0N-1exp2πjNmk+nlgk, l,
x0s, t=expjkdFU,V-1 exp-jkλ2d2N2Δ2U2+V2Fs,t+1xis, t,
x0s, t=u0sΔ, tΔ, xis, t=uisΔ, tΔ.
x0hs, t=FU,V-1H*U, VX0U, V,
x0hs, t=expjkdFU,V-1H*U, V×exp-jkλ2d2N2Δ2U2+V2 Fs,t+1xis, t.
rkq=U,V=0n-1 H*U, VXkqU, V.
F=U,V=0N-1k,q,i |H*U, VXkqU, V-YkqiU, V|2
X+H=r,
F=H+EH.
EZ, Z=k,q,i |XkqZ-YkqiZ|2,
H=E-1XX+E-1X-1r.
ϕcs, t=mod2πϕs, t-ϕBs, t.
xks, t=Iks, t expjϕks, t,
xkqs, t=R-10°+q.4°xks, t,
ykqis, t=R2°xkqs, t

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