Abstract

The Hilbert transform is of interest for image-processing applications because it forms an image that is edge enhanced relative to an input object. Recently a fractional Hilbert transform was introduced that can select which edges are enhanced and to what degree the edge enhancement occurs. Although experimental results of this selective edge enhancement were presented, there was no explanation of this phenomenon. We analyze a one-dimensional fractional Hilbert transform acting on a one-dimensional rectangle function and show how it produces an output image that is selectively edge enhanced.

© 1998 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, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
    [CrossRef] [PubMed]
  3. A. W. Lohmann, E. Tepichin, J. G. Ramirez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” Appl. Opt. 36, 6620–6626 (1997).
    [CrossRef]
  4. J. A. Davis, D. M. Cottrell, R. P. Tiangco, “Analysis of the phase-only filter,” in Optical Pattern Recognition VI, D. P. Casasent, T. H. Chao, eds., Proc. SPIE2490, 77–87 (1994).
    [CrossRef]

1997

1996

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, D. M. Cottrell, R. P. Tiangco, “Analysis of the phase-only filter,” in Optical Pattern Recognition VI, D. P. Casasent, T. H. Chao, eds., Proc. SPIE2490, 77–87 (1994).
[CrossRef]

Davis, J. A.

J. A. Davis, D. M. Cottrell, R. P. Tiangco, “Analysis of the phase-only filter,” in Optical Pattern Recognition VI, D. P. Casasent, T. H. Chao, eds., Proc. SPIE2490, 77–87 (1994).
[CrossRef]

Lohmann, A. W.

Mendlovic, D.

Ramirez, J. G.

Tepichin, E.

Tiangco, R. P.

J. A. Davis, D. M. Cottrell, R. P. Tiangco, “Analysis of the phase-only filter,” in Optical Pattern Recognition VI, D. P. Casasent, T. H. Chao, eds., Proc. SPIE2490, 77–87 (1994).
[CrossRef]

Zalevsky, Z.

Appl. Opt.

Opt. Lett.

Other

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

J. A. Davis, D. M. Cottrell, R. P. Tiangco, “Analysis of the phase-only filter,” in Optical Pattern Recognition VI, D. P. Casasent, T. H. Chao, eds., Proc. SPIE2490, 77–87 (1994).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Rectangle function. (b) Convolution of the rectangle function with the Fourier transform of the signum function.

Fig. 2
Fig. 2

(a) Output electric field obtained with the P = 1/2 Hilbert transform (the right-hand edge is emphasized). (b) Output electric field obtained with the P = 1 Hilbert transform (both edges are emphasized). (c) Output electric field obtained with the P = 3/2 Hilbert transform (the left-hand edge is emphasized).

Fig. 3
Fig. 3

(a) Output intensity obtained with the P = 1/2 Hilbert transform (the right-hand edge is emphasized). (b) Output intensity obtained with the P = 1 Hilbert transform (both edges are emphasized). (c) Output intensity obtained with the P = 3/2 Hilbert transform (the left-hand edge is emphasized).

Equations (5)

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g ˜ x = g x   *   h x ,
H P u = exp iP π / 2 S u +   exp - iP π / 2 S - u ,
H P u = cos P π / 2 + i   sin P π / 2 sgn u ,
g ˜ x = g x cos P π / 2 + i g x   *   1 i π x sin P π / 2 ,
g x = rect x L .

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