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 (2)

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]

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]

2000 (2)

1999 (2)

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]

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]

1998 (2)

1997 (2)

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]

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]

1996 (1)

1982 (1)

1977 (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]

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. (1)

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. (7)

Opt. Eng. (2)

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. (2)

Pure Appl. Opt. (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]

Other (1)

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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