Abstract

This paper proposes an all-numerical robust method to compensate for the chromatic aberrations induced by the optical elements in digital color holographic imaging. It combines a zero-padding algorithm and a convolution approach with adjustable magnification, using a single recording of a reference rectangular grid. Experimental results confirm and validate the proposed approach.

© 2013 Optical Society of America

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References

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  1. S. Yeom, B. Javidi, P. Ferraro, D. Alfieri, S. Denicola, and A. Finizio, “Three-dimensional color object visualization and recognition using multi-wavelength computational holography,” Opt. Express 15(15), 9394–9402 (2007).
    [Crossref] [PubMed]
  2. C. J. Mann, P. R. Bingham, V. C. Paquit, and K. W. Tobin, “Quantitative phase imaging by three-wavelength digital holography,” Opt. Express 16(13), 9753–9764 (2008).
    [Crossref] [PubMed]
  3. P. Picart, P. Tankam, D. Mounier, Z. J. Peng, and J. C. Li, “Spatial bandwidth extended reconstruction for digital color Fresnel holograms,” Opt. Express 17(11), 9145–9156 (2009).
    [Crossref] [PubMed]
  4. P. Xia, Y. Shimozato, Y. Ito, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Improvement of color reproduction in color digital holography by using spectral estimation technique,” Appl. Opt. 50(34), H177–H182 (2011).
    [Crossref] [PubMed]
  5. J. Garcia-Sucerquia, “Color lensless digital holographic microscopy with micrometer resolution,” Opt. Lett. 37(10), 1724–1726 (2012).
    [Crossref] [PubMed]
  6. Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
    [Crossref]
  7. A. Kowalczyk, M. Bieda, M. Makowski, M. Sypek, and A. Kolodziejczyk, “Fiber-based real-time color digital in-line holography,” Appl. Opt. 52(19), 4743–4748 (2013).
    [Crossref] [PubMed]
  8. M. K. Kim, “Full color natural light holographic camera,” Opt. Express 21(8), 9636–9642 (2013).
    [Crossref] [PubMed]
  9. R. Kingslake, Lens Design Fundamentals, (Academic Pr, 1978).
  10. P. Tankam, Q. Song, M. Karray, J. C. Li, J.-M. Desse, and P. Picart, “Real-time three-sensitivity measurements based on three-color digital Fresnel holographic interferometry,” Opt. Lett. 35(12), 2055–2057 (2010).
    [Crossref] [PubMed]
  11. U. Schnars, T. M. Kreis, and W. O. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35(4), 977–982 (1996).
    [Crossref]
  12. A. Stadelmaier and J. H. Massig, “Compensation of lens aberrations in digital holography,” Opt. Lett. 25(22), 1630–1632 (2000).
    [Crossref] [PubMed]
  13. T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23(12), 3177–3190 (2006).
    [Crossref] [PubMed]
  14. S. De Nicola, A. Finizio, G. Pierattini, D. Alfieri, S. Grilli, L. Sansone, and P. Ferraro, “Recovering correct phase information in multiwavelength digital holographic microscopy by compensation for chromatic aberrations,” Opt. Lett. 30(20), 2706–2708 (2005).
    [Crossref] [PubMed]
  15. P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
    [Crossref]
  16. T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41(4), 771–778 (2002).
    [Crossref]
  17. J. C. Li, P. Tankam, Z. J. Peng, and P. Picart, “Digital holographic reconstruction of large objects using a convolution approach and adjustable magnification,” Opt. Lett. 34(5), 572–574 (2009).
    [Crossref] [PubMed]
  18. S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36(2), 103–126 (2001).
    [Crossref]
  19. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29(14), 1668–1670 (2004).
    [Crossref] [PubMed]
  20. J. F. Restrepo and J. Garcia-Sucerquia, “Magnified reconstruction of digitally recorded holograms by Fresnel-Bluestein transform,” Appl. Opt. 49(33), 6430–6435 (2010).
    [Crossref] [PubMed]
  21. P. Picart and P. Tankam, “Analysis and adaptation of convolution algorithms to reconstruct extended objects in digital holography,” Appl. Opt. 52(1), A240–A253 (2013).
    [Crossref] [PubMed]
  22. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25(7), 1744–1761 (2008).
    [Crossref] [PubMed]
  23. J. Mundt and T. Kreis, “Digital holographic recording and reconstruction of large scale objects for metrology and display,” Opt. Eng. 49(12), 125801 (2010).
    [Crossref]
  24. F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed., (McGraw-Hill, Inc., 1981).
  25. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. 29(8), 854–856 (2004).
    [Crossref] [PubMed]
  26. P. Tankam and P. Picart, “Use of digital color holography for crack investigation in electronic components,” Opt. Lasers Eng. 49(11), 1335–1342 (2011).
    [Crossref]
  27. P. V. C. Hough, Method and means for recognizing complex patterns, US Patent 3069654 (1960).

2013 (3)

2012 (2)

J. Garcia-Sucerquia, “Color lensless digital holographic microscopy with micrometer resolution,” Opt. Lett. 37(10), 1724–1726 (2012).
[Crossref] [PubMed]

Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
[Crossref]

2011 (2)

2010 (3)

2009 (2)

2008 (3)

2007 (1)

2006 (1)

2005 (1)

2004 (2)

2002 (1)

T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41(4), 771–778 (2002).
[Crossref]

2001 (1)

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36(2), 103–126 (2001).
[Crossref]

2000 (1)

1996 (1)

U. Schnars, T. M. Kreis, and W. O. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35(4), 977–982 (1996).
[Crossref]

Alfieri, D.

Aspert, N.

Awatsuji, Y.

Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
[Crossref]

P. Xia, Y. Shimozato, Y. Ito, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Improvement of color reproduction in color digital holography by using spectral estimation technique,” Appl. Opt. 50(34), H177–H182 (2011).
[Crossref] [PubMed]

Baumbach, T.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36(2), 103–126 (2001).
[Crossref]

Bieda, M.

Bingham, P. R.

Bourquin, S.

Charrière, F.

Colomb, T.

Coppola, G.

Cuche, E.

De Nicola, S.

Denicola, S.

Depeursinge, C.

Desse, J.-M.

Ferraro, P.

Finizio, A.

Garcia-Sucerquia, J.

Grilli, S.

P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
[Crossref]

S. De Nicola, A. Finizio, G. Pierattini, D. Alfieri, S. Grilli, L. Sansone, and P. Ferraro, “Recovering correct phase information in multiwavelength digital holographic microscopy by compensation for chromatic aberrations,” Opt. Lett. 30(20), 2706–2708 (2005).
[Crossref] [PubMed]

Ito, Y.

Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
[Crossref]

P. Xia, Y. Shimozato, Y. Ito, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Improvement of color reproduction in color digital holography by using spectral estimation technique,” Appl. Opt. 50(34), H177–H182 (2011).
[Crossref] [PubMed]

Javidi, B.

P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
[Crossref]

S. Yeom, B. Javidi, P. Ferraro, D. Alfieri, S. Denicola, and A. Finizio, “Three-dimensional color object visualization and recognition using multi-wavelength computational holography,” Opt. Express 15(15), 9394–9402 (2007).
[Crossref] [PubMed]

Juptner, W.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36(2), 103–126 (2001).
[Crossref]

Jüptner, W. O.

U. Schnars, T. M. Kreis, and W. O. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35(4), 977–982 (1996).
[Crossref]

Kakue, T.

Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
[Crossref]

P. Xia, Y. Shimozato, Y. Ito, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Improvement of color reproduction in color digital holography by using spectral estimation technique,” Appl. Opt. 50(34), H177–H182 (2011).
[Crossref] [PubMed]

Karray, M.

Kim, M. K.

Kolodziejczyk, A.

Kowalczyk, A.

Kreis, T.

J. Mundt and T. Kreis, “Digital holographic recording and reconstruction of large scale objects for metrology and display,” Opt. Eng. 49(12), 125801 (2010).
[Crossref]

Kreis, T. M.

T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41(4), 771–778 (2002).
[Crossref]

U. Schnars, T. M. Kreis, and W. O. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35(4), 977–982 (1996).
[Crossref]

Kubota, T.

Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
[Crossref]

P. Xia, Y. Shimozato, Y. Ito, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Improvement of color reproduction in color digital holography by using spectral estimation technique,” Appl. Opt. 50(34), H177–H182 (2011).
[Crossref] [PubMed]

Kühn, J.

Leval, J.

Li, J. C.

Makowski, M.

Mann, C. J.

Marian, A.

Marquet, P.

Massig, J. H.

Matoba, O.

Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
[Crossref]

P. Xia, Y. Shimozato, Y. Ito, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Improvement of color reproduction in color digital holography by using spectral estimation technique,” Appl. Opt. 50(34), H177–H182 (2011).
[Crossref] [PubMed]

Miccio, L.

P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
[Crossref]

Montfort, F.

Mounier, D.

Mundt, J.

J. Mundt and T. Kreis, “Digital holographic recording and reconstruction of large scale objects for metrology and display,” Opt. Eng. 49(12), 125801 (2010).
[Crossref]

Nishio, K.

Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
[Crossref]

P. Xia, Y. Shimozato, Y. Ito, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Improvement of color reproduction in color digital holography by using spectral estimation technique,” Appl. Opt. 50(34), H177–H182 (2011).
[Crossref] [PubMed]

Osten, W.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36(2), 103–126 (2001).
[Crossref]

Paquit, V. C.

Peng, Z. J.

Picart, P.

Pierattini, G.

Restrepo, J. F.

Sansone, L.

Schnars, U.

U. Schnars, T. M. Kreis, and W. O. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35(4), 977–982 (1996).
[Crossref]

Seebacher, S.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36(2), 103–126 (2001).
[Crossref]

Shimozato, Y.

Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
[Crossref]

P. Xia, Y. Shimozato, Y. Ito, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Improvement of color reproduction in color digital holography by using spectral estimation technique,” Appl. Opt. 50(34), H177–H182 (2011).
[Crossref] [PubMed]

Song, Q.

Stadelmaier, A.

Sypek, M.

Tahara, T.

Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
[Crossref]

P. Xia, Y. Shimozato, Y. Ito, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Improvement of color reproduction in color digital holography by using spectral estimation technique,” Appl. Opt. 50(34), H177–H182 (2011).
[Crossref] [PubMed]

Tankam, P.

Tobin, K. W.

Ura, S.

Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
[Crossref]

P. Xia, Y. Shimozato, Y. Ito, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Improvement of color reproduction in color digital holography by using spectral estimation technique,” Appl. Opt. 50(34), H177–H182 (2011).
[Crossref] [PubMed]

Xia, P.

Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
[Crossref]

P. Xia, Y. Shimozato, Y. Ito, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Improvement of color reproduction in color digital holography by using spectral estimation technique,” Appl. Opt. 50(34), H177–H182 (2011).
[Crossref] [PubMed]

Yamaguchi, I.

Yaroslavsky, L. P.

Yeom, S.

Zhang, F.

Appl. Opt. (4)

J. Disp. Technol. (2)

Y. Ito, Y. Shimozato, P. Xia, T. Tahara, T. Kakue, Y. Awatsuji, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Four-wavelength color digital holography,” J. Disp. Technol. 8(10), 570–576 (2012).
[Crossref]

P. Ferraro, S. Grilli, L. Miccio, D. Alfieri, S. De Nicola, A. Finizio, and B. Javidi, “Full color 3-D imaging by digital holography and removal of chromatic aberrations,” J. Disp. Technol. 4(1), 97–100 (2008).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Eng. (3)

J. Mundt and T. Kreis, “Digital holographic recording and reconstruction of large scale objects for metrology and display,” Opt. Eng. 49(12), 125801 (2010).
[Crossref]

U. Schnars, T. M. Kreis, and W. O. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35(4), 977–982 (1996).
[Crossref]

T. M. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41(4), 771–778 (2002).
[Crossref]

Opt. Express (4)

Opt. Lasers Eng. (2)

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36(2), 103–126 (2001).
[Crossref]

P. Tankam and P. Picart, “Use of digital color holography for crack investigation in electronic components,” Opt. Lasers Eng. 49(11), 1335–1342 (2011).
[Crossref]

Opt. Lett. (7)

P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. 29(8), 854–856 (2004).
[Crossref] [PubMed]

F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29(14), 1668–1670 (2004).
[Crossref] [PubMed]

J. C. Li, P. Tankam, Z. J. Peng, and P. Picart, “Digital holographic reconstruction of large objects using a convolution approach and adjustable magnification,” Opt. Lett. 34(5), 572–574 (2009).
[Crossref] [PubMed]

A. Stadelmaier and J. H. Massig, “Compensation of lens aberrations in digital holography,” Opt. Lett. 25(22), 1630–1632 (2000).
[Crossref] [PubMed]

S. De Nicola, A. Finizio, G. Pierattini, D. Alfieri, S. Grilli, L. Sansone, and P. Ferraro, “Recovering correct phase information in multiwavelength digital holographic microscopy by compensation for chromatic aberrations,” Opt. Lett. 30(20), 2706–2708 (2005).
[Crossref] [PubMed]

J. Garcia-Sucerquia, “Color lensless digital holographic microscopy with micrometer resolution,” Opt. Lett. 37(10), 1724–1726 (2012).
[Crossref] [PubMed]

P. Tankam, Q. Song, M. Karray, J. C. Li, J.-M. Desse, and P. Picart, “Real-time three-sensitivity measurements based on three-color digital Fresnel holographic interferometry,” Opt. Lett. 35(12), 2055–2057 (2010).
[Crossref] [PubMed]

Other (3)

R. Kingslake, Lens Design Fundamentals, (Academic Pr, 1978).

P. V. C. Hough, Method and means for recognizing complex patterns, US Patent 3069654 (1960).

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed., (McGraw-Hill, Inc., 1981).

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Figures (9)

Fig. 1
Fig. 1

Basic scheme for digital color holography using 3 wavelengths

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Fig. 2
Fig. 2

Origins of chromatism in a digital color holographic set-up, (a) the negative lens in front of the sensor brings a shift in the positions and sizes of the virtual images, (b) a small misalignment of the three color-beams in the reference wave brings a spatial shift in the reconstruction of the images

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Fig. 3
Fig. 3

Three reconstructed images, from left to right: λ = 457, 532 and 660nm respectively

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Fig. 4
Fig. 4

Three reconstructed images superposed without any correction

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Fig. 5
Fig. 5

Three reconstructed images superposed using the convolution algorithm with correction of the lateral shift only

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Fig. 6
Fig. 6

Hough Transform of the reconstructed green image of the grid in the polar coordinates; the white squares circle the intersection points, indicating all the real lines

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Fig. 7
Fig. 7

Three images superposed after the adjustable magnification and the lateral shift corrections

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Fig. 8
Fig. 8

From left to right: color photography, the superposition of the non-corrected reconstructed images, and the superposition of the corrected reconstructed images of a yellow and pink mask

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Fig. 9
Fig. 9

From left to right: color photography, the superposition of the non-corrected reconstructed images, and the superposition of the corrected reconstructed images of a green and purple mask

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Tables (2)

Tables Icon

Table 1 Experimental results for the parameters used in the modified zero-padding algorithm

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Table 2 Size and magnification parameters estimated using the Hough transform

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Equations (16)

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

A r ( x,y,d )= −id λ c ∫ −∞ +∞ ∫ −∞ +∞ H( X,Y )w( X,Y, λ c , R c ) × exp[ 2iπ/ λ c d 2 + ( X−x ) 2 + ( Y−y ) 2 ] d 2 + ( X−x ) 2 + ( Y−y ) 2 dXdY
A r ( x,y,d )= −iexp( 2iπd/λ ) λd exp[ iπ λd ( x 2 + y 2 ) ]× ∑ k=−K/2 k=K/2−1 ∑ l=−L/2 l=L/2−1 H(l p x, k p y )exp[ iπ λd ( l 2 p x 2 + k 2 p y 2 ) ]× exp[ − 2iπ λd ( l p x x+k p y y ) ]
A r =F T −1 [ FT[ H×w ]×G ]
G( u,v,d )={ exp[ 2iπ d/λ  1−λ   2   ( u− u 0 λ ) 2 −λ   2   ( v− v 0 λ ) 2 ]   if | u− u 0 λ |≤L p x /2λd and | v− v 0 λ |≤K p y /2λd 0  elsewhere
γ=− d r d 0
Δd= p ' R 2 ν×f'
Δy'=y ' R Δ γ opt γ opt Δd p ' R
{ u 0 λ = sin θ x λ λ v 0 λ = sin θ y λ λ
Δη( λ )= λ d r λ K λ p x =constant
K λ 1 = λ 1 d r λ 1 λ 2 d r λ 2 K λ 2 , K λ ∈Ν
{ d r λ 1 d r λ 2 = d 0 λ 1 d 0 λ 2 d r λ 1 + d r λ 2 = d 0 λ 1 + d 0 λ 2
K λ 1 = λ 1 d 0 λ 1 λ 2 d 0 λ 2 K λ 2 , K λ ∈Ν
{ d r λ 1 = d 0 λ 1 + d 0 λ 2 1+ K λ 2 * λ 1 K λ1 * λ 2 d r λ 2 = d 0 λ 1 + d 0 λ 2 − d r λ 1
{ K λ 3 * = λ 3 d 0 λ 3 λ 2 d 0 λ 2 K λ 2 * d r λ 3 = λ 2 K λ 3 * λ 3 K λ 2 * d r λ 2
d r 'λ =Γ(λ) d 0 λ
{ u 0 'λ = u 0 λ + Δ X λ λ d 0 λ v 0 'λ = v 0 λ + Δ Y λ λ d 0 λ

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