Abstract

The accuracy of depth estimation based on defocus effects has been essentially limited by the depth of field of the imaging system. We show that depth estimation can be improved significantly relative to classical methods by exploiting three-dimensional diffraction effects. We formulate the problem by using information theory analysis and present, to the best of our knowledge, a new paradigm for depth estimation based on spatially rotating point-spread functions (PSFs). Such PSFs are fundamentally more sensitive to defocus thanks to their first-order axial variation. Our system acquires a frame by using a rotating PSF and jointly processes it with an image acquired by using a standard PSF to recover depth information. Analytical, numerical, and experimental evidence suggest that the approach is suitable for applications such as microscopy and machine vision.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. P. Pentland, IEEE Trans. Pattern Anal. Mach. Intell. 9, 523 (1987).
    [CrossRef] [PubMed]
  2. J. Ens and P. Lawrence, IEEE Trans. Pattern Anal. Mach. Intell. 15, 97 (1993).
    [CrossRef]
  3. M. Subbarao and Y. Liu, in Proc. SPIE 3204, 24 (1997).
    [CrossRef]
  4. P. Favaro and S. Soatto, IEEE Trans. Pattern Anal. Mach. Intell. 27, 406 (2005).
    [CrossRef] [PubMed]
  5. A. N. Rajagopalan, S. Chaudhuri, and R. Chellappa, J. Opt. Soc. Am. A 17, 1722 (2000).
    [CrossRef]
  6. D. Rajan and S. Chaudhuri, IEEE Trans. Pattern Anal. Mach. Intell. 25, 1102 (2003).
    [CrossRef]
  7. M. Watanabe and S. K. Nayar, Int. J. Comput. Vis. 27, 203 (1998).
    [CrossRef]
  8. Y. Y. Schechner and N. Kiryati, Int. J. Comput. Vis. 39, 141 (2000).
    [CrossRef]
  9. R. Piestun, Y. Y. Schechner, and J. Shamir, J. Opt. Soc. Am. A 17, 294 (2000).
    [CrossRef]
  10. Y. Y. Schechner, R. Piestun, and S. Shamir, Phys. Rev. E 54, R50 (1996).
    [CrossRef]
  11. R. Piestun and J. Shamir, J. Opt. Soc. Am. A 15, 3039 (1998).
    [CrossRef]
  12. P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, J. Mod. Opt. 45, 2355 (1998).
    [CrossRef]
  13. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
    [CrossRef]
  14. A. Greengard, Y. Y. Schechner, and R. Piestun, in Proc. SPIE 5557, 91 (2004).
    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 5.
  16. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991).
    [CrossRef]
  17. E. R. Dowski and W. T. Cathey, Appl. Opt. 33, 6762 (1994).
    [CrossRef] [PubMed]

2005

P. Favaro and S. Soatto, IEEE Trans. Pattern Anal. Mach. Intell. 27, 406 (2005).
[CrossRef] [PubMed]

2004

A. Greengard, Y. Y. Schechner, and R. Piestun, in Proc. SPIE 5557, 91 (2004).
[CrossRef]

2003

D. Rajan and S. Chaudhuri, IEEE Trans. Pattern Anal. Mach. Intell. 25, 1102 (2003).
[CrossRef]

2000

1998

M. Watanabe and S. K. Nayar, Int. J. Comput. Vis. 27, 203 (1998).
[CrossRef]

R. Piestun and J. Shamir, J. Opt. Soc. Am. A 15, 3039 (1998).
[CrossRef]

P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, J. Mod. Opt. 45, 2355 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

1997

M. Subbarao and Y. Liu, in Proc. SPIE 3204, 24 (1997).
[CrossRef]

1996

Y. Y. Schechner, R. Piestun, and S. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

1994

1993

J. Ens and P. Lawrence, IEEE Trans. Pattern Anal. Mach. Intell. 15, 97 (1993).
[CrossRef]

1987

A. P. Pentland, IEEE Trans. Pattern Anal. Mach. Intell. 9, 523 (1987).
[CrossRef] [PubMed]

Allen, L.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

Cathey, W. T.

Chaudhuri, S.

D. Rajan and S. Chaudhuri, IEEE Trans. Pattern Anal. Mach. Intell. 25, 1102 (2003).
[CrossRef]

A. N. Rajagopalan, S. Chaudhuri, and R. Chellappa, J. Opt. Soc. Am. A 17, 1722 (2000).
[CrossRef]

Chellappa, R.

Courtial, J.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991).
[CrossRef]

Dholakia, K.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

Dowski, E. R.

Ens, J.

J. Ens and P. Lawrence, IEEE Trans. Pattern Anal. Mach. Intell. 15, 97 (1993).
[CrossRef]

Favaro, P.

P. Favaro and S. Soatto, IEEE Trans. Pattern Anal. Mach. Intell. 27, 406 (2005).
[CrossRef] [PubMed]

Friberg, A. T.

P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, J. Mod. Opt. 45, 2355 (1998).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 5.

Greengard, A.

A. Greengard, Y. Y. Schechner, and R. Piestun, in Proc. SPIE 5557, 91 (2004).
[CrossRef]

Honkanen, M.

P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, J. Mod. Opt. 45, 2355 (1998).
[CrossRef]

Khonina, S. N.

P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, J. Mod. Opt. 45, 2355 (1998).
[CrossRef]

Kiryati, N.

Y. Y. Schechner and N. Kiryati, Int. J. Comput. Vis. 39, 141 (2000).
[CrossRef]

Kotlyar, V. V.

P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, J. Mod. Opt. 45, 2355 (1998).
[CrossRef]

Kuittinen, M.

P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, J. Mod. Opt. 45, 2355 (1998).
[CrossRef]

Lautanen, J.

P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, J. Mod. Opt. 45, 2355 (1998).
[CrossRef]

Lawrence, P.

J. Ens and P. Lawrence, IEEE Trans. Pattern Anal. Mach. Intell. 15, 97 (1993).
[CrossRef]

Liu, Y.

M. Subbarao and Y. Liu, in Proc. SPIE 3204, 24 (1997).
[CrossRef]

Nayar, S. K.

M. Watanabe and S. K. Nayar, Int. J. Comput. Vis. 27, 203 (1998).
[CrossRef]

Paakkonen, P.

P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, J. Mod. Opt. 45, 2355 (1998).
[CrossRef]

Padgett, M. J.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

Pentland, A. P.

A. P. Pentland, IEEE Trans. Pattern Anal. Mach. Intell. 9, 523 (1987).
[CrossRef] [PubMed]

Piestun, R.

A. Greengard, Y. Y. Schechner, and R. Piestun, in Proc. SPIE 5557, 91 (2004).
[CrossRef]

R. Piestun, Y. Y. Schechner, and J. Shamir, J. Opt. Soc. Am. A 17, 294 (2000).
[CrossRef]

R. Piestun and J. Shamir, J. Opt. Soc. Am. A 15, 3039 (1998).
[CrossRef]

Y. Y. Schechner, R. Piestun, and S. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

Rajagopalan, A. N.

Rajan, D.

D. Rajan and S. Chaudhuri, IEEE Trans. Pattern Anal. Mach. Intell. 25, 1102 (2003).
[CrossRef]

Robertson, D. A.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

Schechner, Y. Y.

A. Greengard, Y. Y. Schechner, and R. Piestun, in Proc. SPIE 5557, 91 (2004).
[CrossRef]

R. Piestun, Y. Y. Schechner, and J. Shamir, J. Opt. Soc. Am. A 17, 294 (2000).
[CrossRef]

Y. Y. Schechner and N. Kiryati, Int. J. Comput. Vis. 39, 141 (2000).
[CrossRef]

Y. Y. Schechner, R. Piestun, and S. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

Shamir, J.

Shamir, S.

Y. Y. Schechner, R. Piestun, and S. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

Soatto, S.

P. Favaro and S. Soatto, IEEE Trans. Pattern Anal. Mach. Intell. 27, 406 (2005).
[CrossRef] [PubMed]

Soifer, V. A.

P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, J. Mod. Opt. 45, 2355 (1998).
[CrossRef]

Subbarao, M.

M. Subbarao and Y. Liu, in Proc. SPIE 3204, 24 (1997).
[CrossRef]

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991).
[CrossRef]

Turunen, J.

P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, J. Mod. Opt. 45, 2355 (1998).
[CrossRef]

Watanabe, M.

M. Watanabe and S. K. Nayar, Int. J. Comput. Vis. 27, 203 (1998).
[CrossRef]

Appl. Opt.

IEEE Trans. Pattern Anal. Mach. Intell.

A. P. Pentland, IEEE Trans. Pattern Anal. Mach. Intell. 9, 523 (1987).
[CrossRef] [PubMed]

J. Ens and P. Lawrence, IEEE Trans. Pattern Anal. Mach. Intell. 15, 97 (1993).
[CrossRef]

P. Favaro and S. Soatto, IEEE Trans. Pattern Anal. Mach. Intell. 27, 406 (2005).
[CrossRef] [PubMed]

D. Rajan and S. Chaudhuri, IEEE Trans. Pattern Anal. Mach. Intell. 25, 1102 (2003).
[CrossRef]

Int. J. Comput. Vis.

M. Watanabe and S. K. Nayar, Int. J. Comput. Vis. 27, 203 (1998).
[CrossRef]

Y. Y. Schechner and N. Kiryati, Int. J. Comput. Vis. 39, 141 (2000).
[CrossRef]

J. Mod. Opt.

P. Paakkonen, J. Lautanen, M. Honkanen, M. Kuittinen, J. Turunen, S. N. Khonina, V. V. Kotlyar, V. A. Soifer, and A. T. Friberg, J. Mod. Opt. 45, 2355 (1998).
[CrossRef]

J. Opt. Soc. Am. A

Phys. Rev. E

Y. Y. Schechner, R. Piestun, and S. Shamir, Phys. Rev. E 54, R50 (1996).
[CrossRef]

Phys. Rev. Lett.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

Proc. SPIE

A. Greengard, Y. Y. Schechner, and R. Piestun, in Proc. SPIE 5557, 91 (2004).
[CrossRef]

M. Subbarao and Y. Liu, in Proc. SPIE 3204, 24 (1997).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 5.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Intensity of a rotating point-spread function (PSF) (above) and a standard PSF (below) of the same numerical aperture as they go through focus. Note that the standard PSF is essentially constant around focus, while the rotating PSF changes substantially.

Fig. 2
Fig. 2

Normalized FI of the rotating and standard PSFs as they go through focus. The rotating PSF clearly has higher and more constant FI.

Fig. 3
Fig. 3

Frames obtained with (a) rotating PSF and (b) standard PSF. Reconstructed transverse (c) PSF and (d) MTF.

Fig. 4
Fig. 4

Estimated (circles) and actual depths (line) of the resolution chart test object. The average absolute error is 0.66 mm, or 1.6% of the entire depth range. For comparison, the depth of field of a classical optical system of the same dimensions is 5.3 mm.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

ψ = 2 π λ ( 1 z obj focus 1 z obj ) r 2 ,
J j ( ψ ) = h p j [ h ( ψ ) ] { d d ψ log p j [ h ( ψ ) ] } 2 ,
h ̂ rot ( ψ ) = F 1 { H rot ( ψ ) } = F 1 { I rot I ref H ref } = F 1 { I H rot ( ψ ) I H ref H ref } ,
h ̂ rot ( ψ ) = F 1 { I ref * I ref 2 + σ 2 I rot H ref }
z ̂ obj = ( z 0 tan ϕ 2 z img 2 + 1 z obj focus ) 1 ,

Metrics