Abstract

We derive an approach for imaging attenuative sample parameters with a confocal scanning system. The technique employs computational processing to form the estimate in a pixel-by-pixel manner from measurements at the Fourier plane, rather than detecting a focused point at a pinhole. While conventional imaging system analysis and design assumes an independent scatterer at each point in the sample, attenuation must be treated with a tomographic approach. We show that a simple estimator may be derived that requires minimal computation and compare it to the conventional pinhole estimate. The method can potentially be used to image attenuation parameters and occlusion with incoherent detection, as well as refractive index variation with coherent detection, and could potentially allow for video rate imaging due to its computational simplicity. We further consider the application to the problem of an unknown gain or phase value, such as in the measurement of phase with a gradient sensor. And we propose a technique to mitigate the effect by computationally imaging off-focus planes. The principles are demonstrated with numerical simulations in two dimensions.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Minsky, “Microscopy apparatus,” U.S. patent 3,013,467, 19 December 1961).
  2. J.-A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2, 920–931 (2005).
    [CrossRef] [PubMed]
  3. M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99, 5788–5792 (2002).
    [CrossRef] [PubMed]
  4. A. M. Zysk, J. J. Reynolds, D. L. Marks, P. S. Carney, and S. A. Boppart, “Projected index computed tomography,” Opt. Lett. 28, 701–703 (2003).
    [CrossRef] [PubMed]
  5. O. Renaud, J. Viña, Y. Yu, C. Machu, A. Trouvé, H. Van der Voort, B. Chalmond, and S. L. Shorte, “High-resolution 3-D imaging of living cells in suspension using confocal axial tomography,” Biotechnol. J. 3, 53–62 (2008).
    [CrossRef]
  6. J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
    [CrossRef] [PubMed]
  7. S. Kikuchi, K. Sonobe, and N. Ohyama, “Three-dimensional microscopic computed tomography based on generalized Radon transform for optical imaging systems,” Opt. Commun. 123, 725–733 (1996).
    [CrossRef]
  8. G. N. Vishnyakov, G. G. Levin, and V. L. Minaev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis 18, 19–21 (2004).
  9. N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Synthetic aperture tomographic phase microscopy for 3D imaging of live cells in translational motion,” Opt. Express 16, 16240–16246 (2008).
    [CrossRef] [PubMed]
  10. J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
  11. T. Wilson, “Coherent methods in confocal microscopy,” IEEE Eng. Med. Biol. Mag. 15, 84–91 (1996).
    [CrossRef]
  12. W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  13. O. K. Ersoy, Diffraction, Fourier Optics, and Imaging (Wiley, 2007).
    [CrossRef]
  14. S. R. Deans, The Radon Transform and Some of its Applications (Wiley, 1983).
  15. K. J. Dillon and Y. Fainman, “Computational confocal scanning tomography,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuC7.
  16. M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
    [CrossRef]
  17. K. J. Dillon and Y. Fainman, “Computational confocal tomography for simultaneous reconstruction of object, occlusions, and aberrations,” Appl. Opt. 49, 2529–2538 (2010).
    [CrossRef]

2010

2009

K. J. Dillon and Y. Fainman, “Computational confocal scanning tomography,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuC7.

2008

O. Renaud, J. Viña, Y. Yu, C. Machu, A. Trouvé, H. Van der Voort, B. Chalmond, and S. L. Shorte, “High-resolution 3-D imaging of living cells in suspension using confocal axial tomography,” Biotechnol. J. 3, 53–62 (2008).
[CrossRef]

N. Lue, W. Choi, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Synthetic aperture tomographic phase microscopy for 3D imaging of live cells in translational motion,” Opt. Express 16, 16240–16246 (2008).
[CrossRef] [PubMed]

2007

O. K. Ersoy, Diffraction, Fourier Optics, and Imaging (Wiley, 2007).
[CrossRef]

2005

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

J.-A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2, 920–931 (2005).
[CrossRef] [PubMed]

2004

G. N. Vishnyakov, G. G. Levin, and V. L. Minaev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis 18, 19–21 (2004).

2003

2002

M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99, 5788–5792 (2002).
[CrossRef] [PubMed]

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

1996

S. Kikuchi, K. Sonobe, and N. Ohyama, “Three-dimensional microscopic computed tomography based on generalized Radon transform for optical imaging systems,” Opt. Commun. 123, 725–733 (1996).
[CrossRef]

T. Wilson, “Coherent methods in confocal microscopy,” IEEE Eng. Med. Biol. Mag. 15, 84–91 (1996).
[CrossRef]

1990

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1983

S. R. Deans, The Radon Transform and Some of its Applications (Wiley, 1983).

M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

Ahlgren, U.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Badizadegan, K.

Baldock, R.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Booth, M. J.

M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Boppart, S. A.

Carney, P. S.

Chalmond, B.

O. Renaud, J. Viña, Y. Yu, C. Machu, A. Trouvé, H. Van der Voort, B. Chalmond, and S. L. Shorte, “High-resolution 3-D imaging of living cells in suspension using confocal axial tomography,” Biotechnol. J. 3, 53–62 (2008).
[CrossRef]

Cheong, W. F.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Choi, W.

Conchello, J. -A.

J.-A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2, 920–931 (2005).
[CrossRef] [PubMed]

Dasari, R. R.

Davidson, D.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Davison, M. E.

M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of its Applications (Wiley, 1983).

Dillon, K. J.

K. J. Dillon and Y. Fainman, “Computational confocal tomography for simultaneous reconstruction of object, occlusions, and aberrations,” Appl. Opt. 49, 2529–2538 (2010).
[CrossRef]

K. J. Dillon and Y. Fainman, “Computational confocal scanning tomography,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuC7.

Ersoy, O. K.

O. K. Ersoy, Diffraction, Fourier Optics, and Imaging (Wiley, 2007).
[CrossRef]

Fainman, Y.

K. J. Dillon and Y. Fainman, “Computational confocal tomography for simultaneous reconstruction of object, occlusions, and aberrations,” Appl. Opt. 49, 2529–2538 (2010).
[CrossRef]

K. J. Dillon and Y. Fainman, “Computational confocal scanning tomography,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuC7.

Feld, M. S.

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Hecksher-Sorensen, J.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Hill, B.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Juškaitis, R.

M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Kikuchi, S.

S. Kikuchi, K. Sonobe, and N. Ohyama, “Three-dimensional microscopic computed tomography based on generalized Radon transform for optical imaging systems,” Opt. Commun. 123, 725–733 (1996).
[CrossRef]

Levin, G. G.

G. N. Vishnyakov, G. G. Levin, and V. L. Minaev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis 18, 19–21 (2004).

Lichtman, J. W.

J.-A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2, 920–931 (2005).
[CrossRef] [PubMed]

Lue, N.

Machu, C.

O. Renaud, J. Viña, Y. Yu, C. Machu, A. Trouvé, H. Van der Voort, B. Chalmond, and S. L. Shorte, “High-resolution 3-D imaging of living cells in suspension using confocal axial tomography,” Biotechnol. J. 3, 53–62 (2008).
[CrossRef]

Marks, D. L.

Minaev, V. L.

G. N. Vishnyakov, G. G. Levin, and V. L. Minaev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis 18, 19–21 (2004).

Minsky, M.

M. Minsky, “Microscopy apparatus,” U.S. patent 3,013,467, 19 December 1961).

Neil, M. A. A.

M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Ohyama, N.

S. Kikuchi, K. Sonobe, and N. Ohyama, “Three-dimensional microscopic computed tomography based on generalized Radon transform for optical imaging systems,” Opt. Commun. 123, 725–733 (1996).
[CrossRef]

Perry, P.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Popescu, G.

Prahl, S. A.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Renaud, O.

O. Renaud, J. Viña, Y. Yu, C. Machu, A. Trouvé, H. Van der Voort, B. Chalmond, and S. L. Shorte, “High-resolution 3-D imaging of living cells in suspension using confocal axial tomography,” Biotechnol. J. 3, 53–62 (2008).
[CrossRef]

Reynolds, J. J.

Ross, A.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Sharpe, J.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Shorte, S. L.

O. Renaud, J. Viña, Y. Yu, C. Machu, A. Trouvé, H. Van der Voort, B. Chalmond, and S. L. Shorte, “High-resolution 3-D imaging of living cells in suspension using confocal axial tomography,” Biotechnol. J. 3, 53–62 (2008).
[CrossRef]

Sonobe, K.

S. Kikuchi, K. Sonobe, and N. Ohyama, “Three-dimensional microscopic computed tomography based on generalized Radon transform for optical imaging systems,” Opt. Commun. 123, 725–733 (1996).
[CrossRef]

Trouvé, A.

O. Renaud, J. Viña, Y. Yu, C. Machu, A. Trouvé, H. Van der Voort, B. Chalmond, and S. L. Shorte, “High-resolution 3-D imaging of living cells in suspension using confocal axial tomography,” Biotechnol. J. 3, 53–62 (2008).
[CrossRef]

Van der Voort, H.

O. Renaud, J. Viña, Y. Yu, C. Machu, A. Trouvé, H. Van der Voort, B. Chalmond, and S. L. Shorte, “High-resolution 3-D imaging of living cells in suspension using confocal axial tomography,” Biotechnol. J. 3, 53–62 (2008).
[CrossRef]

Viña, J.

O. Renaud, J. Viña, Y. Yu, C. Machu, A. Trouvé, H. Van der Voort, B. Chalmond, and S. L. Shorte, “High-resolution 3-D imaging of living cells in suspension using confocal axial tomography,” Biotechnol. J. 3, 53–62 (2008).
[CrossRef]

Vishnyakov, G. N.

G. N. Vishnyakov, G. G. Levin, and V. L. Minaev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis 18, 19–21 (2004).

Welch, A. J.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Wilson, T.

M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99, 5788–5792 (2002).
[CrossRef] [PubMed]

T. Wilson, “Coherent methods in confocal microscopy,” IEEE Eng. Med. Biol. Mag. 15, 84–91 (1996).
[CrossRef]

Yu, Y.

O. Renaud, J. Viña, Y. Yu, C. Machu, A. Trouvé, H. Van der Voort, B. Chalmond, and S. L. Shorte, “High-resolution 3-D imaging of living cells in suspension using confocal axial tomography,” Biotechnol. J. 3, 53–62 (2008).
[CrossRef]

Zysk, A. M.

Appl. Opt.

Biotechnol. J.

O. Renaud, J. Viña, Y. Yu, C. Machu, A. Trouvé, H. Van der Voort, B. Chalmond, and S. L. Shorte, “High-resolution 3-D imaging of living cells in suspension using confocal axial tomography,” Biotechnol. J. 3, 53–62 (2008).
[CrossRef]

IEEE Eng. Med. Biol. Mag.

T. Wilson, “Coherent methods in confocal microscopy,” IEEE Eng. Med. Biol. Mag. 15, 84–91 (1996).
[CrossRef]

IEEE J. Quantum Electron.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Microscopy and Analysis

G. N. Vishnyakov, G. G. Levin, and V. L. Minaev, “Tomographic interference microscopy of living cells,” Microscopy and Analysis 18, 19–21 (2004).

Nat. Methods

J.-A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2, 920–931 (2005).
[CrossRef] [PubMed]

Opt. Commun.

S. Kikuchi, K. Sonobe, and N. Ohyama, “Three-dimensional microscopic computed tomography based on generalized Radon transform for optical imaging systems,” Opt. Commun. 123, 725–733 (1996).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. Natl. Acad. Sci. U.S.A.

M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Science

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

SIAM J. Appl. Math.

M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

Other

O. K. Ersoy, Diffraction, Fourier Optics, and Imaging (Wiley, 2007).
[CrossRef]

S. R. Deans, The Radon Transform and Some of its Applications (Wiley, 1983).

K. J. Dillon and Y. Fainman, “Computational confocal scanning tomography,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuC7.

M. Minsky, “Microscopy apparatus,” U.S. patent 3,013,467, 19 December 1961).

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

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 (5)

Fig. 1
Fig. 1

Confocal system. (a) Conventional and (b) computational systems.

Fig. 2
Fig. 2

System geometry. The sample is assumed stationary, and the optical system is translated relative to it, as determined by the location of the focus at x 0 , y 0 , z 0 . The optical axis is vertical.

Fig. 3
Fig. 3

Focal plane and “off-focus” planes. The focal plane is scanned at high rate, but we may choose to instead computationally form an image at a different plane.

Fig. 4
Fig. 4

(a) Synthetic attenuation image, (b) image formed from depth scan of focal plane, (c) image formed from depth scan of off-focus plane, (d) image formed from depth scan of conventional confocal estimate of focal plane.

Fig. 5
Fig. 5

Simulation of application of random attenuation (at given multiples larger than true signal) to each aperture measurement. (a) Example of random attenuation values (solid curve) versus true attenuation (dashed curve) for 5 × , (b) image formed for 5 × , (c) image formed for 10 × , (d) image formed for 25 × .

Tables (2)

Tables Icon

Table 1 Comparison of Estimates

Tables Icon

Table 2 Coordinates Used

Equations (40)

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

I ( x 0 , y 0 , z 0 ) = | v ( 0 , 0 ; x 0 , y 0 , z 0 ) | 2 = | Aperture u ( x , y ; x 0 , y 0 , z 0 ) d x d y | 2 .
A ( x 0 , y 0 , z 0 ) = Aperture u ( x , y ; x 0 , y 0 , z 0 ) d x d y ,
A ( x 0 , y 0 , z 0 ) = e η ( x 0 , y 0 , z 0 ) Δ x A 0 ,
η ( x 0 , y 0 , z 0 ) = 1 Δ x log   A ( x 0 , y 0 , z 0 ) + 1 Δ x log   A 0 = log   A ( x 0 , y 0 , z 0 ) ,
η ( x 0 , y 0 , z 0 ) = log { Aperture u ( x , y ; x 0 , y 0 , z 0 ) d x d y } .
u ( x , y ; x 0 , y 0 , z 0 ) = { A ( x s , y s , z s ) u 0 ( x , y ) , if   ( x , y ) = f z s z 0 ( x s x 0 , y s y 0 ) u 0 ( x , y ) , otherwise , }
log   u ( x , y ; x 0 , y 0 , z 0 ) = δ ( x f x x 0 z z 0 , y f y y 0 z z 0 ) log   A ( x , y , z ) d x d y d z ,
log   u ( x , y ; x 0 , y 0 , z 0 ) = log   A ( x f ( z z 0 ) + x 0 , y f ( z z 0 ) + y 0 , z ) d z = η ( x f ( z z 0 ) + x 0 , y f ( z z 0 ) + y 0 , z ) d z .
{ log   u ( x , y ; x 0 , y 0 , z 0 ) d x d y } h x ( x 0 , y 0 ) = 2 π η ( x 0 , y 0 , z 0 ) ,
{ log   u ( x , y ; x 0 , y 0 , Δ z f x x 0 , Δ z f y y 0 ) d x } h x ( x 0 , y 0 ) = 2 π η ( x 0 , y 0 , z 0 Δ z ) .
u a ( x , y ) = a ( x 0 , y 0 ) u ( x , y ) .
log   u a ( x , y ) = log   a ( x 0 , y 0 ) + log   u ( x , y ) = log   a ( x 0 , y 0 ) η ( x f ( z z 0 ) + x 0 , y f ( z z 0 ) + y 0 , z ) d z .
{ log   u a ( x , y ; x 0 , y 0 , z 0 ) d x d y } h x ( x 0 , y 0 ) = { log   a ( x 0 , y 0 , z 0 ) d x d y } h x ( x 0 , y 0 ) + { log   u ( x , y ; x 0 , y 0 , z 0 ) d x d y } h x ( x 0 , y 0 ) = N 2   log   a ( x 0 , y 0 , z 0 ) h x ( x 0 , y 0 ) + 2 π η ( x 0 , y 0 , z 0 ) ,
1 ( 2 π ) 2 k x f e j ( Δ z / f ) ( x k x + y k y ) { log   u a ( x , y ) } e j ( k x x 0 + k y y 0 ) d x d k x d k y = error   term + 2 π η ( x 0 , y 0 , z 0 Δ z ) ,
error   term = 1 ( 2 π ) 2 k x f e j ( Δ z / f ) ( x k x + y k y ) { log   a ( x 0 , y 0 ) e j ( k x x 0 + k y y 0 ) d x 0 d y 0 } e j ( k x x 0 + k y y 0 ) d x d k x d k y
= 1 ( 2 π ) 2 k x f e j ( Δ z / f ) y k y { e j ( Δ z / f ) x k x d x } { log   a ( x 0 , y 0 ) e j ( k x x 0 + k y y 0 ) d x 0 d y 0 } e j ( k x x 0 + k y y 0 ) d k x d k y .
error   term = h x ( x 0 , y 0 ) w ( x 0 , y 0 ) log   a ( x 0 , y 0 ) .
log   u ( x , y ; x 0 , y 0 , z 0 ) = η ( x f ( z z 0 ) + x 0 , y f ( z z 0 ) + y 0 , z ) d z ,
η ( x a z , y b z , z ) e j ( k x x + k y y ) d x d y d z = η ̃ ( k x , k y , a k x + b k y ) ,
log   u ( x , y ; x 0 , y 0 , z 0 ) e j ( k x x 0 + k y y 0 ) d x 0 d y 0 = η ̃ ( k x , k y , x f k x y f k y ) e j ( z 0 / f ) ( x k x + y k y ) .
k x f { RHS } d x = k x f η ̃ ( k x , k y , x f k x y f k y ) e j ( z 0 / f ) ( x k x + y k y ) d x
= η ̃ ( k x , k y , ς x ) e j z 0 ς x d ς x ,
ς x = x f k x y f k y ,
d ς x = k x f d x .
1 ( 2 π ) 2 k x f { RHS } e j ( k x x 0 + k y y 0 ) d x d k x d k y = 1 ( 2 π ) 2 η ̃ ( k x , k y , ς x ) e j z 0 ς x d ς x e j ( k x x 0 + k y y 0 ) d k x d k y
= 2 π η ( x 0 , y 0 , z 0 ) .
1 ( 2 π ) 2 k x f { LHS } e j ( k x x 0 + k y y 0 ) d x d k x d k y
= 1 ( 2 π ) 2 k x f { log   u ( x , y ; x 0 , y 0 , z 0 ) e j ( k x x 0 + k y y 0 ) d x 0 d y 0 } e j ( k x x 0 + k y y 0 ) d x d k x d k y
= { log   u ( x , y ; x 0 , y 0 , z 0 ) d x } h x ( x 0 , y 0 ) ,
h x ( x 0 , y 0 ) H x ( k x , k y ) = k x f .
{ log   u ( x , y ; x 0 , y 0 , z 0 ) d x } h x ( x 0 , y 0 ) = 2 π η ( x 0 , y 0 , z 0 ) .
{ log   u ( x , y ; x 0 , y 0 , z 0 ) d x d y } h x ( x 0 , y 0 ) = 2 π η ( x 0 , y 0 , z 0 ) .
log   u ( x , y ; x 0 , y 0 , z 0 ) e j ( k x x 0 + k y y 0 ) d x 0 d y 0 = η ̃ ( k x , k y , x f k x y f k y ) e j ( z 0 / f ) ( x k x + y k y ) ,
1 ( 2 π ) 2 k x f e j ( Δ z / f ) ( x k x + y k y ) { RHS } e j ( k x x 0 + k y y 0 ) d x d k x d k y
= 1 ( 2 π ) 2 k x f e j ( Δ z / f ) ( x k x + y k y ) { η ̃ ( k x , k y , x f k x y f k y ) e j ( z 0 / f ) ( x k x + y k y ) } e j ( k x x 0 + k y y 0 ) d x d k x d k y
= 1 ( 2 π ) 2 k x f { η ̃ ( k x , k y , x f k x y f k y ) e j [ ( z 0 + Δ z ) / f ] ( x k x + y k y ) } e j ( k x x 0 + k y y 0 ) d x d k x d k y
= 1 ( 2 π ) 2 η ̃ ( k x , k y , ς x ) e j ( z 0 + Δ z ) ς x d ς x e j ( k x x 0 + k y y 0 ) d k x d k y
= 2 π η ( x 0 , y 0 , z 0 + Δ z ) .
1 ( 2 π ) 2 k x f e j ( Δ z / f ) ( x k x + y k y ) { LHS } e j ( k x x 0 + k y y 0 ) d x d k x d k y = 1 ( 2 π ) 2 { k x f exp [ j ( Δ z f x x 0 ) k x j ( Δ z f y y 0 ) k y ] { LHS } d k x d k y } d x .
{ log   u ( x , y ; Δ z f x x 0 , Δ z f y y 0 , z 0 ) d x } h x ( x 0 , y 0 ) = 2 π η ( x 0 , y 0 , z 0 Δ z ) .

Metrics