Abstract

Fractional derivatives of two-dimensional images have been discussed theoretically in terms of Fourier optics and computer simulated. Filters that realize the half-order differentiation can be either complex or real. We prove, in terms of fractional calculus, that the semiderivative filter is useful for the visualization of phase changes in a phase object in such a way that the output-image intensity is directly proportional to the first derivative of the input object. We give computer-simulated results of one-dimensional semidifferentiating.

© 1997 Optical Society of America

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References

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  1. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  2. K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, Orlando, Fla., 1974).
  3. A. C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions (Pitman, New York, 1979).
  4. M. W. Michalski, “Derivatives of noninteger order and their applications,” Diss. Math. 328, 3–47 (1993).
  5. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1986).
  6. G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 424–427.
  7. H. Rabal, L. Zerbino, J. Ojeda-Castaneda, “Continuous order derivative for optical signals and images,” Microwave Opt. Technol. Lett. 1, 64–67 (1988).
    [CrossRef]
  8. H. Kasprzak, “Differentation of a noninteger order and its optical implementation,” Appl. Opt. 21, 3287–3291 (1982).
    [CrossRef] [PubMed]
  9. B. A. Horwitz, “Phase image differentation with linear intensity output,” Appl. Opt. 17, 181–186 (1978).
    [CrossRef] [PubMed]
  10. R. A. Sprague, B. J. Thompson, “Quantitative visualization of large variation phase objects,” Appl. Opt. 11, 1469–1479 (1972).
    [CrossRef] [PubMed]

1993 (2)

M. W. Michalski, “Derivatives of noninteger order and their applications,” Diss. Math. 328, 3–47 (1993).

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

1988 (1)

H. Rabal, L. Zerbino, J. Ojeda-Castaneda, “Continuous order derivative for optical signals and images,” Microwave Opt. Technol. Lett. 1, 64–67 (1988).
[CrossRef]

1982 (1)

1978 (1)

1972 (1)

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1986).

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 424–427.

Horwitz, B. A.

Kasprzak, H.

Lohmann, A. W.

McBride, A. C.

A. C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions (Pitman, New York, 1979).

Michalski, M. W.

M. W. Michalski, “Derivatives of noninteger order and their applications,” Diss. Math. 328, 3–47 (1993).

Ojeda-Castaneda, J.

H. Rabal, L. Zerbino, J. Ojeda-Castaneda, “Continuous order derivative for optical signals and images,” Microwave Opt. Technol. Lett. 1, 64–67 (1988).
[CrossRef]

Oldham, K. B.

K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, Orlando, Fla., 1974).

Parrent, G. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 424–427.

Rabal, H.

H. Rabal, L. Zerbino, J. Ojeda-Castaneda, “Continuous order derivative for optical signals and images,” Microwave Opt. Technol. Lett. 1, 64–67 (1988).
[CrossRef]

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 424–427.

Spanier, J.

K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, Orlando, Fla., 1974).

Sprague, R. A.

Thompson, B. J.

R. A. Sprague, B. J. Thompson, “Quantitative visualization of large variation phase objects,” Appl. Opt. 11, 1469–1479 (1972).
[CrossRef] [PubMed]

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 424–427.

Zerbino, L.

H. Rabal, L. Zerbino, J. Ojeda-Castaneda, “Continuous order derivative for optical signals and images,” Microwave Opt. Technol. Lett. 1, 64–67 (1988).
[CrossRef]

Appl. Opt. (3)

Diss. Math. (1)

M. W. Michalski, “Derivatives of noninteger order and their applications,” Diss. Math. 328, 3–47 (1993).

J. Opt. Soc. Am. A (1)

Microwave Opt. Technol. Lett. (1)

H. Rabal, L. Zerbino, J. Ojeda-Castaneda, “Continuous order derivative for optical signals and images,” Microwave Opt. Technol. Lett. 1, 64–67 (1988).
[CrossRef]

Other (4)

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1986).

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook (SPIE Press, Bellingham, Wash., 1989), pp. 424–427.

K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, Orlando, Fla., 1974).

A. C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions (Pitman, New York, 1979).

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Figures (2)

Fig. 1
Fig. 1

Profile of the amplitude transmittance of the semiderivative real filter.

Fig. 2
Fig. 2

Two phase objects: (a) a super-Gaussian distribution and (b) a cylindrical lenslike distribution characterized by phase shifts with a maximum value equal to π. Intensities of the first derivatives of (c) the super-Gaussian and (d) cylindrical lenslike distributions are also shown.

Equations (8)

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-dfxdxexp-2πiuxdx=2πiuFu.
r1+r2xr1yr2fx, y=--2πiur12πivr2×--fx, yexp-2πiux+vydxdy×exp2πixu+yvdudv,
d1/2fxdx1/2=-2πiuFuexp2πixudu.
d1/2exp2πixu0fxdx1/2=exp2πixu0×-2πiα+u01/2×Fαexp2πixαdα,
d1/2ΨxΘxdx1/2=j=01/2jd1/2-jΨxdx1/2-jdjΘxdxj,
d1/2 exp2πiu0xexpiΦxdx1/2=2πiu01/2exp2πiu0xexpiΦx+i22πiu01/2×exp2iu0xdΦxdxexpiΦx-182πiu031/2×exp2πiu0xexpiΦxid2Φxdx2-dΦxdx2.
Ix=d1/2 exp2πiu0xexpiΦxdx1/22=2πu0+dΦxdx.
Ix=12+14πu0dΦxdx.

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