Abstract

It is shown that both surface tilting and translational motion can be independently estimated by use of the speckle photographic technique by capturing consecutive images in two different fractional Fourier domains. A geometric interpretation, based on use of the Wigner distribution function, is presented to describe this application of the optical fractional Fourier transform when little prior information is known about the motion.

© 2003 Optical Society of America

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References

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  1. P. K. Rastogi, in Speckle Metrology, R. S. Sirohi, ed. (Marcel Dekker, New York, 1993), pp. 50–58.
  2. H. Tiziani, Opt. Commun. 5, 271 (1972).
    [CrossRef]
  3. D. Mendlovic and H. M. Ozatkas, J. Opt. Soc. Am. 10, 1875, 2522 (1993).
  4. A. W. Lohmann, J. Opt. Soc. Am. 10, 2181 (1993).
    [CrossRef]
  5. R. G. Dorsch, Appl. Opt. 34, 6016 (1995).
    [CrossRef] [PubMed]
  6. H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chicester, UK, 2001).
  7. B. Hennelly and J. T. Sheridan, Opt. Lett. 28, 269 (2003).
    [CrossRef] [PubMed]
  8. J. T. Sheridan and R. Patten, Opt. Lett. 25, 448 (2000).
    [CrossRef]
  9. J. T. Sheridan and R. Patten, Optik (Stuttgart) 111, 329 (2000).
  10. R. Patten, J. T. Sheridan, and A. Larkin, Opt. Eng. 40, 1438 (2001).
    [CrossRef]
  11. A. W. Lohmann, Opt. Commun. 115, 427 (1995).
    [CrossRef]
  12. H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, Opt. Commun. 32, 32 (1980).
    [CrossRef]
  13. K.-H. Brenner and A. W. Lohmann, Opt. Commun. 42, 310 (1982).
    [CrossRef]
  14. S. Abe and J. T. Sheridan, Opt. Lett. 19, 1801 (1994).
    [CrossRef]
  15. S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
    [CrossRef]
  16. S. Abe and J. T. Sheridan, J. Opt. Soc. Am. A 16, 1986 (1999).
    [CrossRef]

2003 (1)

2001 (1)

R. Patten, J. T. Sheridan, and A. Larkin, Opt. Eng. 40, 1438 (2001).
[CrossRef]

2000 (2)

J. T. Sheridan and R. Patten, Optik (Stuttgart) 111, 329 (2000).

J. T. Sheridan and R. Patten, Opt. Lett. 25, 448 (2000).
[CrossRef]

1999 (1)

1997 (1)

S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
[CrossRef]

1995 (2)

1994 (1)

1993 (2)

D. Mendlovic and H. M. Ozatkas, J. Opt. Soc. Am. 10, 1875, 2522 (1993).

A. W. Lohmann, J. Opt. Soc. Am. 10, 2181 (1993).
[CrossRef]

1982 (1)

K.-H. Brenner and A. W. Lohmann, Opt. Commun. 42, 310 (1982).
[CrossRef]

1980 (1)

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, Opt. Commun. 32, 32 (1980).
[CrossRef]

1972 (1)

H. Tiziani, Opt. Commun. 5, 271 (1972).
[CrossRef]

Abe, S.

Alper Kutay, M.

H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chicester, UK, 2001).

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, Opt. Commun. 32, 32 (1980).
[CrossRef]

Brenner, K.-H.

K.-H. Brenner and A. W. Lohmann, Opt. Commun. 42, 310 (1982).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, Opt. Commun. 32, 32 (1980).
[CrossRef]

Dorsch, R. G.

Hennelly, B.

Larkin, A.

R. Patten, J. T. Sheridan, and A. Larkin, Opt. Eng. 40, 1438 (2001).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, Opt. Commun. 115, 427 (1995).
[CrossRef]

A. W. Lohmann, J. Opt. Soc. Am. 10, 2181 (1993).
[CrossRef]

K.-H. Brenner and A. W. Lohmann, Opt. Commun. 42, 310 (1982).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, Opt. Commun. 32, 32 (1980).
[CrossRef]

Mendlovic, D.

D. Mendlovic and H. M. Ozatkas, J. Opt. Soc. Am. 10, 1875, 2522 (1993).

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chicester, UK, 2001).

Ozatkas, H. M.

D. Mendlovic and H. M. Ozatkas, J. Opt. Soc. Am. 10, 1875, 2522 (1993).

Patten, R.

R. Patten, J. T. Sheridan, and A. Larkin, Opt. Eng. 40, 1438 (2001).
[CrossRef]

J. T. Sheridan and R. Patten, Opt. Lett. 25, 448 (2000).
[CrossRef]

J. T. Sheridan and R. Patten, Optik (Stuttgart) 111, 329 (2000).

Rastogi, P. K.

P. K. Rastogi, in Speckle Metrology, R. S. Sirohi, ed. (Marcel Dekker, New York, 1993), pp. 50–58.

Sheridan, J. T.

Tiziani, H.

H. Tiziani, Opt. Commun. 5, 271 (1972).
[CrossRef]

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chicester, UK, 2001).

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

D. Mendlovic and H. M. Ozatkas, J. Opt. Soc. Am. 10, 1875, 2522 (1993).

A. W. Lohmann, J. Opt. Soc. Am. 10, 2181 (1993).
[CrossRef]

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

Opt. Commun. (5)

S. Abe and J. T. Sheridan, Opt. Commun. 137, 214 (1997).
[CrossRef]

H. Tiziani, Opt. Commun. 5, 271 (1972).
[CrossRef]

A. W. Lohmann, Opt. Commun. 115, 427 (1995).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, Opt. Commun. 32, 32 (1980).
[CrossRef]

K.-H. Brenner and A. W. Lohmann, Opt. Commun. 42, 310 (1982).
[CrossRef]

Opt. Eng. (1)

R. Patten, J. T. Sheridan, and A. Larkin, Opt. Eng. 40, 1438 (2001).
[CrossRef]

Opt. Lett. (3)

Optik (Stuttgart) (1)

J. T. Sheridan and R. Patten, Optik (Stuttgart) 111, 329 (2000).

Other (2)

P. K. Rastogi, in Speckle Metrology, R. S. Sirohi, ed. (Marcel Dekker, New York, 1993), pp. 50–58.

H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, Chicester, UK, 2001).

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Figures (1)

Fig. 1
Fig. 1

Shift of a WDF in phase space and the Fθ plane.

Equations (13)

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Wx,k=12π-dy exp-jkyu*x-y/2ux+y/2.
uxux-ξexp+jκx.
Uk=FuxUk-κexp+jξk,
Wx,kWx-ξ,k-κ.
Fθuxq=Uθq=12πsin θ1/2×exp-jπ212+Jθπ+j2q2 cot θ-+ux×exp+j2x2 cot θ-jqx csc θdx.
Fθux-ξexpjκxq=12πsin θ1/2×exp-jπ212+Jθπ+j2cot θq2+ξ2-2qξcos θ+2κξ tan θ×-+uyexp+j2y2 cot θ-jq-Qy csc θdy,
UθqUq-Qexp+jΦq,
Φq=q cot θQ-ξcos θ+cot θ2ξ2-Q2+κξ=qQ-QQ2+κξ2.
FTUθq2+Uθq-QexpjΦq2=2 FTIθqqcosQq/2,
ξ=Q sinθ+Δθ-Q+ΔQsinθsinΔθ,
κ=-Q cosθ+Δθ+Q+ΔQcosθsinΔθ.
ξ-ΔQΔθsinθ+Q cosθ-Δθ63Q+ΔQsinθ+OΔθ3-dQdθsinθ+Q cosθ,
κΔQΔθcosθ+Q sinθ+Δθ63Q+ΔQcosθ+OΔθ3dQdθcosθ+Q sinθ.

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