Abstract

The classical Hilbert transform can be implemented optically as a spatial-filtering process, whereby half the Fourier spectrum is π-phase shifted. Recently the Hilbert transform was generalized. The generalized version, called the fractional Hilbert transform, is quite easy to implement optically if the input is one dimensional. Here we show how to implement the fractional Hilbert transform for two-dimensional inputs. Hence the new transform is now suitable for image processing.

© 1997 Optical Society of America

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References

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  1. R. B. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1995).
  2. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
    [CrossRef] [PubMed]
  3. A. W. Lohmann, J. Ojeda-Castañeda, L. Diaz-Santana, “Fractional Hilbert transform: optical implementation for 1-D objects,” Opt. Mem. Neural Networks 5, 131–135 (1996).
  4. S. Lowenthal, Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. 11, 49–51 (1967).
    [CrossRef]
  5. J. K. T. Eu, A. W. Lohmann, “Isotropic Hilbert spatial filtering,” Opt. Commun. 9, 257–262 (1973).
    [CrossRef]
  6. Y. Belvaux, J. C. Vareille, “Controle de l’état de surface ou d”homogeneïte de materiaux optiques ‘par contraste de phase’ a dephasage quelconque,” Opt. Commun. 2, 101–104 (1971).
    [CrossRef]

1996 (2)

A. W. Lohmann, J. Ojeda-Castañeda, L. Diaz-Santana, “Fractional Hilbert transform: optical implementation for 1-D objects,” Opt. Mem. Neural Networks 5, 131–135 (1996).

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
[CrossRef] [PubMed]

1973 (1)

J. K. T. Eu, A. W. Lohmann, “Isotropic Hilbert spatial filtering,” Opt. Commun. 9, 257–262 (1973).
[CrossRef]

1971 (1)

Y. Belvaux, J. C. Vareille, “Controle de l’état de surface ou d”homogeneïte de materiaux optiques ‘par contraste de phase’ a dephasage quelconque,” Opt. Commun. 2, 101–104 (1971).
[CrossRef]

1967 (1)

S. Lowenthal, Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. 11, 49–51 (1967).
[CrossRef]

Belvaux, Y.

Y. Belvaux, J. C. Vareille, “Controle de l’état de surface ou d”homogeneïte de materiaux optiques ‘par contraste de phase’ a dephasage quelconque,” Opt. Commun. 2, 101–104 (1971).
[CrossRef]

S. Lowenthal, Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. 11, 49–51 (1967).
[CrossRef]

Bracewell, R. B.

R. B. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1995).

Diaz-Santana, L.

A. W. Lohmann, J. Ojeda-Castañeda, L. Diaz-Santana, “Fractional Hilbert transform: optical implementation for 1-D objects,” Opt. Mem. Neural Networks 5, 131–135 (1996).

Eu, J. K. T.

J. K. T. Eu, A. W. Lohmann, “Isotropic Hilbert spatial filtering,” Opt. Commun. 9, 257–262 (1973).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, J. Ojeda-Castañeda, L. Diaz-Santana, “Fractional Hilbert transform: optical implementation for 1-D objects,” Opt. Mem. Neural Networks 5, 131–135 (1996).

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
[CrossRef] [PubMed]

J. K. T. Eu, A. W. Lohmann, “Isotropic Hilbert spatial filtering,” Opt. Commun. 9, 257–262 (1973).
[CrossRef]

Lowenthal, S.

S. Lowenthal, Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. 11, 49–51 (1967).
[CrossRef]

Mendlovic, D.

Ojeda-Castañeda, J.

A. W. Lohmann, J. Ojeda-Castañeda, L. Diaz-Santana, “Fractional Hilbert transform: optical implementation for 1-D objects,” Opt. Mem. Neural Networks 5, 131–135 (1996).

Vareille, J. C.

Y. Belvaux, J. C. Vareille, “Controle de l’état de surface ou d”homogeneïte de materiaux optiques ‘par contraste de phase’ a dephasage quelconque,” Opt. Commun. 2, 101–104 (1971).
[CrossRef]

Zalevsky, Z.

Appl. Phys. Lett. (1)

S. Lowenthal, Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. 11, 49–51 (1967).
[CrossRef]

Opt. Commun. (2)

J. K. T. Eu, A. W. Lohmann, “Isotropic Hilbert spatial filtering,” Opt. Commun. 9, 257–262 (1973).
[CrossRef]

Y. Belvaux, J. C. Vareille, “Controle de l’état de surface ou d”homogeneïte de materiaux optiques ‘par contraste de phase’ a dephasage quelconque,” Opt. Commun. 2, 101–104 (1971).
[CrossRef]

Opt. Lett. (1)

Opt. Mem. Neural Networks (1)

A. W. Lohmann, J. Ojeda-Castañeda, L. Diaz-Santana, “Fractional Hilbert transform: optical implementation for 1-D objects,” Opt. Mem. Neural Networks 5, 131–135 (1996).

Other (1)

R. B. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1995).

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Figures (8)

Fig. 1
Fig. 1

Setup for the two-dimensional FRH. P, polarizer; W, Wollaston prism; Q, quarter-wave plate; ⊙, x polarization; vertical arrow, y polarization.

Fig. 2
Fig. 2

Filter of the setup shown in Fig. 1. The two circles symbolize two shifted versions of the object spectrum. Two razor blades (in gray) eliminate two opposite halves of the object spectra.

Fig. 3
Fig. 3

Fractional Hilbert images of the cross-type input object, shown in the upper left-hand corner. The corresponding fractional degree P appears at the top of each image. For P = 0 the output is an ordinary image; for P = 1, we obtain the classical Hilbert transform.

Fig. 4
Fig. 4

Same as in Fig. 3, but for the circular aperture shown in the upper left-hand corner of this figure.

Fig. 5
Fig. 5

Fractional Hilbert images obtained with the phase version of the input object employed in Fig. 4, which is shown again in the upper left-hand corner of this figure. Despite the noise, the fractional Hilbert images obtained with the pure-amplitude version (Fig. 4) and those obtained with the pure-phase version shown in this figure are practically the same for equal values of P.

Fig. 6
Fig. 6

Same as in Fig. 3, but for the input object shown in the upper left-hand corner, orientated at 0°.

Fig. 7
Fig. 7

Same as in Fig. 6, but for an in-plane rotation of the input object of 90°.

Fig. 8
Fig. 8

Four modifications of the original setup, each of them overcoming a specific handicap: (a) The two Wollaston prisms are not equal. (b), (c) Only one Wollaston prism is available. The missing Wollaston prism is replaced with a grating. (d) Both Wollaston prisms are replaced with gratings.

Equations (15)

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

ϕ=Pπ2.
uxux*-1/πx=vx,
u˜υu˜υH˜υ=υ˜υ,
ũυ= uxexp-2πiυxdx,
H˜υ=-i+i=exp-iπ/2if υ0exp+iπ/2if υ<0,
H˜υ=R˜υexp-iπ/2exp+iπ/2.
F˜υ, P=R˜υexp-iPπ/2exp+Pπ/2=R˜υexp-iϕexp+iϕ.
F˜υ, P=0=R˜υ,
F˜υ, P=1=H˜υ,
F˜υ, PF˜υ, Q=F˜υ, P+Q.
U0x=u0x+υ0x,
Upx=upx+υpx.
F˜υ, ϕ=R˜υcos ϕ+H˜υsin ϕ=F˜υ, 0cos ϕ+F˜υ, π/2sin ϕ.
uPx, y=ux, y; ϕ= ũ0υ, μF˜υ, μ; ϕ×exp2πixυ+yμdυdμ.
u0x, y=Rx/Δx, u0x, y=1+expiα-1Rx/Δx.

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