Abstract

When optical signals, like diffraction patterns, are processed by digital means the choice of sampling density and geometry is important during analog-to-digital conversion. Continuous band-limited signals can be sampled and recovered from their samples in accord with the Nyquist sampling criteria. The specific form of the convolution kernel that describes the Fresnel diffraction allows another, alternative, full-reconstruction procedure of an object from the samples of its diffraction pattern when the object is space limited. This alternative procedure is applicable and yields full reconstruction even when the diffraction pattern is undersampled and the Nyquist criteria are severely violated. Application of the new procedure to practical diffraction-related phenomena, like in-line holography, improves the processing efficiency without creating any associated artifacts on the reconstructed-object pattern.

© 2000 Optical Society of America

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References

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  1. D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, New York, 1984), Sec. 1.4.
  2. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.
  3. K. Nishihara, S. Hatano, K. Nagayama, “New method of obtaining particle diameter by the fast fourier transform pattern of the in-line hologram,” Opt. Eng. 36, 2429–2439 (1997).
    [CrossRef]
  4. S. Belaid, D. Lebrun, C. Ozkul, “Application of two-dimensional wavelet transform to hologram analysis—visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947–1951 (1997).
    [CrossRef]
  5. L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
    [CrossRef]
  6. H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdaği, “Digital computation of the fractional Fourier transform,” IEEE Trans. Sig. Process. 44, 2141–2150 (1996).
    [CrossRef]
  7. H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, Chap. 4, pp. 239–291.
    [CrossRef]
  8. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]

1997 (2)

K. Nishihara, S. Hatano, K. Nagayama, “New method of obtaining particle diameter by the fast fourier transform pattern of the in-line hologram,” Opt. Eng. 36, 2429–2439 (1997).
[CrossRef]

S. Belaid, D. Lebrun, C. Ozkul, “Application of two-dimensional wavelet transform to hologram analysis—visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947–1951 (1997).
[CrossRef]

1996 (1)

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdaği, “Digital computation of the fractional Fourier transform,” IEEE Trans. Sig. Process. 44, 2141–2150 (1996).
[CrossRef]

1994 (1)

1987 (1)

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

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdaği, “Digital computation of the fractional Fourier transform,” IEEE Trans. Sig. Process. 44, 2141–2150 (1996).
[CrossRef]

Barshan, B.

Belaid, S.

S. Belaid, D. Lebrun, C. Ozkul, “Application of two-dimensional wavelet transform to hologram analysis—visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947–1951 (1997).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

Bozdagi, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdaği, “Digital computation of the fractional Fourier transform,” IEEE Trans. Sig. Process. 44, 2141–2150 (1996).
[CrossRef]

Dudgeon, D. E.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, New York, 1984), Sec. 1.4.

Hatano, S.

K. Nishihara, S. Hatano, K. Nagayama, “New method of obtaining particle diameter by the fast fourier transform pattern of the in-line hologram,” Opt. Eng. 36, 2429–2439 (1997).
[CrossRef]

Kutay, M. A.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdaği, “Digital computation of the fractional Fourier transform,” IEEE Trans. Sig. Process. 44, 2141–2150 (1996).
[CrossRef]

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, Chap. 4, pp. 239–291.
[CrossRef]

Lebrun, D.

S. Belaid, D. Lebrun, C. Ozkul, “Application of two-dimensional wavelet transform to hologram analysis—visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947–1951 (1997).
[CrossRef]

Mendlovic, D.

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, Chap. 4, pp. 239–291.
[CrossRef]

Mersereau, R. M.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, New York, 1984), Sec. 1.4.

Nagayama, K.

K. Nishihara, S. Hatano, K. Nagayama, “New method of obtaining particle diameter by the fast fourier transform pattern of the in-line hologram,” Opt. Eng. 36, 2429–2439 (1997).
[CrossRef]

Nishihara, K.

K. Nishihara, S. Hatano, K. Nagayama, “New method of obtaining particle diameter by the fast fourier transform pattern of the in-line hologram,” Opt. Eng. 36, 2429–2439 (1997).
[CrossRef]

Onural, L.

Ozaktas, H. M.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdaği, “Digital computation of the fractional Fourier transform,” IEEE Trans. Sig. Process. 44, 2141–2150 (1996).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, Chap. 4, pp. 239–291.
[CrossRef]

Ozkul, C.

S. Belaid, D. Lebrun, C. Ozkul, “Application of two-dimensional wavelet transform to hologram analysis—visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947–1951 (1997).
[CrossRef]

Scott, P. D.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

IEEE Trans. Sig. Process. (1)

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdaği, “Digital computation of the fractional Fourier transform,” IEEE Trans. Sig. Process. 44, 2141–2150 (1996).
[CrossRef]

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

Opt. Eng. (3)

K. Nishihara, S. Hatano, K. Nagayama, “New method of obtaining particle diameter by the fast fourier transform pattern of the in-line hologram,” Opt. Eng. 36, 2429–2439 (1997).
[CrossRef]

S. Belaid, D. Lebrun, C. Ozkul, “Application of two-dimensional wavelet transform to hologram analysis—visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947–1951 (1997).
[CrossRef]

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

Other (3)

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, New York, 1984), Sec. 1.4.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, San Diego, Calif., 1999), Vol. 106, Chap. 4, pp. 239–291.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Reconstruction of a 1-D object from its undersampled diffraction pattern: (a) a 1-D object (slit), (b) its diffraction pattern, (c) the sampled diffraction pattern (the sampling rate is below the Nyquist rate), (d) the reconstruction of the real part from the undersampled diffraction pattern, and (e) the reconstruction of the imaginary part from the undersampled diffraction pattern.

Fig. 2
Fig. 2

Simulated data used in the reconstruction of a 2-D object from its undersampled diffraction pattern: (a) a 2-D object (a transparent circular hole on an opaque background), (b) its diffraction pattern, and (c) reconstruction from the undersampled diffraction field (the black background is shifted to a gray value to permit the observation of negative field values, as well). The darkest values represent the most negative values, whereas the lightest tones correspond to the highest (most positive) values.

Fig. 3
Fig. 3

Simulated data used to show the application of in-line holography: (a) a small 2-D opaque object on a transparent background, (b) its in-line hologram, (c) the conventional reconstruction of an object from its in-line hologram, i.e., the intensity, (d) the reconstructions from the undersampled in-line hologram’s field, and (e) the reconstruction from the undersampled hologram’s intensity.

Fig. 4
Fig. 4

Real data used to show the application to in-line holography: (a) portion of a real optical in-line hologram of a dust particle on a glass substrate, (b) the conventional reconstruction by digital means, and (c) the reconstruction from the undersampled hologram.

Equations (19)

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

hzx, y=1jλzexpj 2πλ zexpj πλzx2+y2,
ψzx, y=fx, y** hzx, y,
Hzu, v=Hzu=expj 2πλ zexp-j λz4πu2+v2=expj 2πλ zexp-j λz4πuTu,
Ψzu, v=Fu, vexpj 2πλ zexp-j λz4πu2+v20 if u, vB else,
ψzsx=n ψzVnδx-Vn,
Ψzsu=ψzsx=1|det V|kΨzu-Uk,
Ψzu, vH-zu, v=Fu, vHzu, vH-zu, v=Fu, v
ΨzsuH-zu=1|det V|kΨzu-UkH-zu=1|det V|k Fu-UkHzu-UkH-zu,
Hzu-UkH-zu=expj λz4π2kTUTu-kTUTUk.
ΨzsuH-zu=1|det V|k Fu-Uk×exp-j λz4πkTUTUkexpj λz2πkTUTu.
-1ΨzsuH-zu=k ckfx-λz2πUkexpjkTUTx,
1/|det V|exp-jλz/4πkTUTUk.
fx=-1ΨzsuH-zuHLPu,
HLPu=|det V|if uB0else
fx=WRxk ckfx-λz2πUkexpjkTUTx,
WRx=1ckif xR0else
limz0k ckfx-λz2πUkexpjkTUTx=fxkexpjkTUTx=fxk δx-Vk.
V=TTT-T.
U=πTπTπT-πT.

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