Abstract

The fractional derivative spatial-filtering operator is useful for image-processing applications, particularly for examination of phase objects. Experimental implementation is difficult because the mask function combines both amplitude and phase. We present a simple one-dimensional analysis of the fractional derivative operation and note similarities with the fractional Hilbert transform. We demonstrate how to encode these amplitude and phase masks using a phase-only liquid-crystal spatial light modulator and present experimental results. Finally, we introduce a radially symmetric extension of this operation that is more useful for objects having an arbitrary shape.

© 2001 Optical Society of America

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References

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  1. E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernandez-Alonso, “Phase object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1977).
    [CrossRef]
  2. H. Kasprzak, “Differentiation of a noninteger order and its optical implementation,” Appl. Opt. 21, 3287–3291 (1982).
    [CrossRef] [PubMed]
  3. T. Szoplik, V. Climent, E. Tajahuerce, J. Lancis, M. Fernandez-Alonso, “Phase-change visualization in two-dimensional phase objects with a semiderivative real filter,” Appl. Opt. 37, 5472–5478 (1998).
    [CrossRef]
  4. J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, M. Fernandez-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461–7464 (1997).
    [CrossRef]
  5. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional Hilbert transform,” Opt. Lett. 21, 281–283 (1996).
    [CrossRef] [PubMed]
  6. A. W. Lohmann, E. Tepichin, J. G. Ramirez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” App. Opt. 36, 6620–6626 (1997).
    [CrossRef]
  7. J. A. Davis, D. E. McNamara, D. M. Cottrell, “Analysis of the fractional Hilbert transform,” Appl. Opt. 37, 6911–6913 (1998).
    [CrossRef]
  8. J. A. Davis, D. E. McNamara, D. M. Cottrell, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25, 99–101 (2000).
    [CrossRef]
  9. R. B. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1986), Chap. 12.
  10. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
    [CrossRef]
  11. J. Campos, A. Marquez, M. J. Yzuel, J. A. Davis, D. M. Cottrell, I. Moreno, “Fully complex synthetic discriminant functions written onto phase-only filters,” Appl. Opt. 39, 5965–5970 (2000).
    [CrossRef]
  12. A. Marquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
    [CrossRef]
  13. J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. Yzuel, I. Moreno, “Encoding complex diffractive optical elements onto a phase-only liquid crystal spatial light modulator,” Opt. Eng. 40, 327–329 (2001).
    [CrossRef]
  14. J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
    [CrossRef]

2001

J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. Yzuel, I. Moreno, “Encoding complex diffractive optical elements onto a phase-only liquid crystal spatial light modulator,” Opt. Eng. 40, 327–329 (2001).
[CrossRef]

A. Marquez, C. Iemmi, J. C. Escalera, J. Campos, S. Ledesma, J. A. Davis, M. J. Yzuel, “Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator,” Appl. Opt. 40, 2316–2322 (2001).
[CrossRef]

2000

1999

J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[CrossRef]

1998

1997

A. W. Lohmann, E. Tepichin, J. G. Ramirez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” App. Opt. 36, 6620–6626 (1997).
[CrossRef]

J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, M. Fernandez-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461–7464 (1997).
[CrossRef]

1996

1982

1977

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernandez-Alonso, “Phase object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1977).
[CrossRef]

Amako, J.

J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Bracewell, R. B.

R. B. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1986), Chap. 12.

Campos, J.

Climent, V.

Cottrell, D. M.

Davis, J. A.

Escalera, J. C.

Fernandez-Alonso, M.

Iemmi, C.

Kasprzak, H.

Lancis, J.

Ledesma, S.

Lohmann, A. W.

A. W. Lohmann, E. Tepichin, J. G. Ramirez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” App. Opt. 36, 6620–6626 (1997).
[CrossRef]

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

Marquez, A.

McNamara, D. E.

J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. Yzuel, I. Moreno, “Encoding complex diffractive optical elements onto a phase-only liquid crystal spatial light modulator,” Opt. Eng. 40, 327–329 (2001).
[CrossRef]

J. A. Davis, D. E. McNamara, D. M. Cottrell, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25, 99–101 (2000).
[CrossRef]

J. A. Davis, D. E. McNamara, D. M. Cottrell, “Analysis of the fractional Hilbert transform,” Appl. Opt. 37, 6911–6913 (1998).
[CrossRef]

Mendlovic, D.

Moreno, I.

Ramirez, J. G.

A. W. Lohmann, E. Tepichin, J. G. Ramirez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” App. Opt. 36, 6620–6626 (1997).
[CrossRef]

Sonehara, T.

J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Szoplik, T.

Tajahuerce, E.

Tepichin, E.

A. W. Lohmann, E. Tepichin, J. G. Ramirez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” App. Opt. 36, 6620–6626 (1997).
[CrossRef]

Tsai, P. S.

J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Yzuel, M.

J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. Yzuel, I. Moreno, “Encoding complex diffractive optical elements onto a phase-only liquid crystal spatial light modulator,” Opt. Eng. 40, 327–329 (2001).
[CrossRef]

Yzuel, M. J.

Zalevsky, Z.

App. Opt.

A. W. Lohmann, E. Tepichin, J. G. Ramirez, “Optical implementation of the fractional Hilbert transform for two-dimensional objects,” App. Opt. 36, 6620–6626 (1997).
[CrossRef]

Appl. Opt.

Opt. Eng.

J. A. Davis, D. E. McNamara, D. M. Cottrell, J. Campos, M. Yzuel, I. Moreno, “Encoding complex diffractive optical elements onto a phase-only liquid crystal spatial light modulator,” Opt. Eng. 40, 327–329 (2001).
[CrossRef]

J. A. Davis, P. S. Tsai, D. M. Cottrell, T. Sonehara, J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Opt. Lett.

Pure Appl. Opt.

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, M. Fernandez-Alonso, “Phase object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490 (1977).
[CrossRef]

Other

R. B. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1986), Chap. 12.

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Figures (8)

Fig. 1
Fig. 1

Computer simulation showing convolution of input rectangle function g(x) with the Fourier transform of the amplitude modulation term m β(x) for cases where (a) β = 0, (b) β = 1/3, and (c) β = 1.

Fig. 2
Fig. 2

Computer simulation showing triple convolution of input rectangle function g(x) with [1/πx] and with the Fourier transform of the amplitude modulation term m β(x) for cases where (a) β = 0, (b) β = 1/3, and (c) β = 1.

Fig. 3
Fig. 3

Experimental setup.

Fig. 4
Fig. 4

Experimental results showing the movement of the output image that is due to the linear phase grating written onto the LCSLM for cases in which (a) M = 0, the original slit is imaged; (b) M = 0.5, we see two images of the input slit; and (c) M = 1, the image of the slit is moved.

Fig. 5
Fig. 5

Experimental output showing the results when the Hilbert transform mask is written onto the LCSLM. (a) P = 1 Hilbert transform (both edges are emphasized), (b) P = 1/2 Hilbert transform (the right edge is emphasized), and (c) P = 3/2 Hilbert transform (the left edge is emphasized).

Fig. 6
Fig. 6

Experimental output showing the results when the fractional derivative mask is written onto the LCSLM. (a) α = 1/3 fractional derivative, (b) α = 1/5 fractional derivative, and (c) α = 1/7 fractional derivative.

Fig. 7
Fig. 7

Experimental output comparing the fractional derivative and the fractional Hilbert for the case where α = 1/5. (a) The fractional Hilbert transform and (b) the fractional derivative.

Fig. 8
Fig. 8

Experimental output showing the effects of adding amplitude information to the Hilbert mask. (a) The fractional Hilbert mask, where α = 1 and (b) the effects of adding amplitude information for the case where α = 1, β = 0.5.

Equations (15)

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Dαu=2πiuα.
FGuDαu=FGu2πiuα=αgx/xα.
Dαu=2π|u|αexpiαπ/2Su+exp-iαπ/2S-u.
Hαu=cosαπ/2+i sinαπ/2sgnu=expiϕαu.
Mβu=2π|u|β.
Dα,βu=Mβuexpiϕαu=Mβucosαπ/2+i sinαπ/2sgnu.
g˜x=gx * dα,βx.
g˜x=cosαπ/2gx * mβx+i sinαπ/2gx * mβx * 1/iπx.
gx=rectx/L.
Tu=expiMβuϕαu+2πuA.
Tu=n=- Tnu×expinϕαu+2πuA,
Tnu=sinπn-Mβuπn-Mβu.
g˜x=gx * n=- tnx * δx-nA.
g˜x=(cosαπ/2gx * t1x+i sinαπ/2×gx * t1x * 1/iπx) * δx-A+(cosαπ/2gx * t0x+i sinαπ/2×gx * [t0x] * 1/iπx) * δx.
Dα,βρ, θ=MβρHαρ, θ=2π|ρ|β expiαθ.

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