Abstract

The Hilbert transform is useful for image processing because it can select which edges of an input image are enhanced and to what degree the edge enhancement occurs. However, the transform operation is one dimensional and is not applicable for arbitrarily shaped two-dimensional objects. We introduce a radially symmetric Hilbert transform that permits two-dimensional edge enhancement. We implement one-dimensional, two-dimensional, and radial Hilbert transforms with a programmable phase-only liquid-crystal spatial light modulator. Experimental results are presented.

© 2000 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, 1965), Chap. 12.
  2. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Opt. Lett. 21, 281 (1996).
    [CrossRef] [PubMed]
  3. A. W. Lohmann, E. Tepichin, and J. G. Ramirez, Appl. Opt. 36, 6620 (1997).
    [CrossRef]
  4. J. A. Davis, D. E. McNamara, and D. M. Cottrell, Appl. Opt. 37, 6911 (1999).
    [CrossRef]
  5. A. Jaroszewicz and A. Kolodziejczyk, Opt. Commun. 102, 391 (1993).
    [CrossRef]
  6. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 480.
  7. T. Sonehara and J. Amako, in Spatial Light Modulators, G. Burdge and S. Esener, eds., Vol. 14 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997), pp. 165–168.
  8. J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, Opt. Eng. 38, 1051 (1999).
    [CrossRef]

1999 (2)

J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, Opt. Eng. 38, 1051 (1999).
[CrossRef]

J. A. Davis, D. E. McNamara, and D. M. Cottrell, Appl. Opt. 37, 6911 (1999).
[CrossRef]

1997 (1)

1996 (1)

1993 (1)

A. Jaroszewicz and A. Kolodziejczyk, Opt. Commun. 102, 391 (1993).
[CrossRef]

Amako, J.

J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, Opt. Eng. 38, 1051 (1999).
[CrossRef]

T. Sonehara and J. Amako, in Spatial Light Modulators, G. Burdge and S. Esener, eds., Vol. 14 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997), pp. 165–168.

Bracewell, R. B.

R. B. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1965), Chap. 12.

Cottrell, D. M.

J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, Opt. Eng. 38, 1051 (1999).
[CrossRef]

J. A. Davis, D. E. McNamara, and D. M. Cottrell, Appl. Opt. 37, 6911 (1999).
[CrossRef]

Davis, J. A.

J. A. Davis, D. E. McNamara, and D. M. Cottrell, Appl. Opt. 37, 6911 (1999).
[CrossRef]

J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, Opt. Eng. 38, 1051 (1999).
[CrossRef]

Jaroszewicz, A.

A. Jaroszewicz and A. Kolodziejczyk, Opt. Commun. 102, 391 (1993).
[CrossRef]

Kolodziejczyk, A.

A. Jaroszewicz and A. Kolodziejczyk, Opt. Commun. 102, 391 (1993).
[CrossRef]

Lohmann, A. W.

McNamara, D. E.

Mendlovic, D.

Ramirez, J. G.

Sonehara, T.

J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, Opt. Eng. 38, 1051 (1999).
[CrossRef]

T. Sonehara and J. Amako, in Spatial Light Modulators, G. Burdge and S. Esener, eds., Vol. 14 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997), pp. 165–168.

Tepichin, E.

Tsai, P. S.

J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, Opt. Eng. 38, 1051 (1999).
[CrossRef]

Zalevsky, Z.

Appl. Opt. (2)

Opt. Commun. (1)

A. Jaroszewicz and A. Kolodziejczyk, Opt. Commun. 102, 391 (1993).
[CrossRef]

Opt. Eng. (1)

J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, Opt. Eng. 38, 1051 (1999).
[CrossRef]

Opt. Lett. (1)

Other (3)

R. B. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1965), Chap. 12.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 480.

T. Sonehara and J. Amako, in Spatial Light Modulators, G. Burdge and S. Esener, eds., Vol. 14 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1997), pp. 165–168.

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Figures (4)

Fig. 1
Fig. 1

(a) One-dimensional Hilbert mask. Gray levels represent different phase values. (b) Radial Hilbert mask. Gray levels represent different phase values for P=1.

Fig. 2
Fig. 2

Imaginary component of h1r,θ along the x axis. The real component vanishes along this axis. The horizontal axis is in units of λf/2R.

Fig. 3
Fig. 3

Output when a one-dimensional slit is used as the input object. LCSLM programmed with (a) no pattern, (b) the P=1/2 Hilbert transform (the right-hand edge is emphasized), (c) the P=1 Hilbert transform (both edges are emphasized), (d) the P=3/2 Hilbert transform (the left-hand edge is emphasized).

Fig. 4
Fig. 4

Output when a circular aperture is used as the input object. LCSLM programmed with (a) no pattern, (b) the P=1, Q=1 two-dimensional Hilbert transform, (c) the P=1 radial Hilbert transform, (d) the P=1/2 radial Hilbert transform.

Equations (7)

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

g˜x,y=gx,y*hx,y,
HPu=expiPπ/2Su+exp-iPπ/2S-u,
HPu=cosPπ/2+i sinPπ/2sgnu,
g˜x,y=gx,ycosPπ/2+igx,y*1/iπx×sinPπ/2,
HPρ,θ=expiPθ,
g˜x,y=gx,y*hPr,θ,
h1r,θ=xJ0x+πx/2J1xH0x-J0xH1x,

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