Abstract

Compressive sampling enables signal reconstruction using less than one measurement per reconstructed signal value. Compressive measurement is particularly useful in generating multidimensional images from lower dimensional data. We demonstrate single frame 3D tomography from 2D holographic data.

© 2009 Optical Society of America

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References

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  1. C. E. Shannon, "Communications in the presence of noise," Proc. IREv 37, 10-21 (1949).
    [CrossRef]
  2. M. Golay, "Multislit spectroscopy," J. Opt. Soc. Am. 39,437-444 (1949).
    [CrossRef] [PubMed]
  3. E. J. Candes, J. K. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math. 59,1207-1223 (2006).
    [CrossRef]
  4. E. J. Candes and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?," IEEE Transactions on Information Theory 52(12), 5406-5425 (2006).
    [CrossRef]
  5. D. L. Donoho, "Compressed sensing," IEEE Trans. Info. Theory 52(4), 1289-1306 (2006).
    [CrossRef]
  6. D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera architecture using optical-domain compression," in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.
  7. M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, "Single-shot compressive spectral imaging with a dual-disperser architecture," Opt. Express 15(21), 14013-14027 (2007).
    [CrossRef]
  8. A. Wagadarikar, R. John, R. Willett, and D. J. Brady, "Single disperser design for coded aperture snapshot spectral imaging," Appl. Opt. 47(10), B44-B51 (2008).
    [CrossRef]
  9. D. Gabor, "A new microscopic principle," Nature 161,777-778 (1948).
    [CrossRef] [PubMed]
  10. E. N. Leith and J. Upatnieks, "Reconstructed wavefronts and communication theory," J. Opt. Soc. Am. 52,1123-1130 (1962).
    [CrossRef]
  11. S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, "Lensfree holographic imaging for on-chip cytometry and diagnostics," Lab on a Chip 9(6), 777-787 (2009).
    [CrossRef]
  12. W. Jueptner and U. Schnars, Digital Holography, (New York, Springer-Verlag, Berlin Heidelberg, 2005).
  13. T. C. Poon, Digital holography and three-dimensional display, (New York; London: Springer, 2006).
    [CrossRef]
  14. M. E. Brezinski, Optical coherence tomography. (Amsterdam; Boston: Academic Press, 2006).
  15. D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, "Visible cone-beam tomography with a lensless interferometric camera," Science 284(5423), 2164-2166 (1999).
    [CrossRef]
  16. J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, and J. Hecksher-Sorensen, R. Baldock, and D. Davidson, "Optical projection tomography as a tool for 3d microscopy and gene expression studies," Science 296(5567), 541-545 (2002).
    [CrossRef]
  17. A. J. Devaney, "Nonuniqueness in the inverse scattering problem," J. Math. Phys. 19(7), 1526-1531 (1978).
    [CrossRef]
  18. A. J. Devaney, "Geophysical diffraction tomography," IEEE Trans. Geoscie. Remote Sens. GE-22, 3-13 (1984).
    [CrossRef]
  19. W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, "Terahertz imaging with compressed sensing and phase retrieval," Opt. Lett. 33(9), 974-976 (2008).
    [CrossRef]
  20. I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, "Image reconstruction only by phase data in phaseshifting digital holography," Appl. Opt. 45(5), 975-983 (2006).
    [CrossRef]
  21. R. Baraniuk and P. Steeghs, "Compressive radar imaging," IEEE Radar Conference, pp. 128-133, April, (2007).
  22. L. Li, W. Zhang, F. Li, "Compressive diffraction tomography for Weakly Scattering," Submitted to IEEE Trans. on Geoscie. Remote Sens. (2009).
  23. D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via _1 minimization," Proc. Nat. Acad. Scie. U. S. A. 100(5), 2197-2202 (2003).
    [CrossRef]
  24. R. E. Blahut, Theory of Remote Image Formation (Cambridge University Press, 2004).
    [CrossRef]
  25. J. W. Goodman, Introduction to Fourier optics, 3rd Ed., (Roberts and Company Publishers, 2005).
  26. D. M. Paganin, Coherent X-ray Optics (Oxford Science Publications, 2006).
    [CrossRef]
  27. A. C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001).
    [CrossRef]
  28. D. J. Brady, Optical Imaging and Spectroscopy (Wiley, 2009).
    [CrossRef]
  29. L. I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60(1-4), 259-268 (1992).
    [CrossRef]
  30. J. M. Bioucas-Dias and M. A. T. Figueiredo, "A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration," IEEE Trans. Image Proc. 16, 2992-3004 (2007).
    [CrossRef] [PubMed]
  31. V. Mic’o, J. Garc’ıa, Z. Zalevsky, and B. Javidi, "Phase-shifting Gabor holography," Opt. Express 17, 1492-1494 (2009).
  32. T. Latychevskaia and H. W. Fink, "Solution to the Twin Image Problem in Holography," Phys. Rev. Lett. 98, 233901 (2007).
    [CrossRef] [PubMed]

2009 (2)

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, "Lensfree holographic imaging for on-chip cytometry and diagnostics," Lab on a Chip 9(6), 777-787 (2009).
[CrossRef]

V. Mic’o, J. Garc’ıa, Z. Zalevsky, and B. Javidi, "Phase-shifting Gabor holography," Opt. Express 17, 1492-1494 (2009).

2008 (2)

2007 (3)

M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, "Single-shot compressive spectral imaging with a dual-disperser architecture," Opt. Express 15(21), 14013-14027 (2007).
[CrossRef]

T. Latychevskaia and H. W. Fink, "Solution to the Twin Image Problem in Holography," Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

J. M. Bioucas-Dias and M. A. T. Figueiredo, "A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration," IEEE Trans. Image Proc. 16, 2992-3004 (2007).
[CrossRef] [PubMed]

2006 (4)

E. J. Candes, J. K. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math. 59,1207-1223 (2006).
[CrossRef]

E. J. Candes and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?," IEEE Transactions on Information Theory 52(12), 5406-5425 (2006).
[CrossRef]

D. L. Donoho, "Compressed sensing," IEEE Trans. Info. Theory 52(4), 1289-1306 (2006).
[CrossRef]

I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, "Image reconstruction only by phase data in phaseshifting digital holography," Appl. Opt. 45(5), 975-983 (2006).
[CrossRef]

2003 (1)

D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via _1 minimization," Proc. Nat. Acad. Scie. U. S. A. 100(5), 2197-2202 (2003).
[CrossRef]

2002 (1)

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, and J. Hecksher-Sorensen, R. Baldock, and D. Davidson, "Optical projection tomography as a tool for 3d microscopy and gene expression studies," Science 296(5567), 541-545 (2002).
[CrossRef]

1999 (1)

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, "Visible cone-beam tomography with a lensless interferometric camera," Science 284(5423), 2164-2166 (1999).
[CrossRef]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60(1-4), 259-268 (1992).
[CrossRef]

1984 (1)

A. J. Devaney, "Geophysical diffraction tomography," IEEE Trans. Geoscie. Remote Sens. GE-22, 3-13 (1984).
[CrossRef]

1978 (1)

A. J. Devaney, "Nonuniqueness in the inverse scattering problem," J. Math. Phys. 19(7), 1526-1531 (1978).
[CrossRef]

1962 (1)

1949 (1)

1948 (1)

D. Gabor, "A new microscopic principle," Nature 161,777-778 (1948).
[CrossRef] [PubMed]

Ahlgren, U.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, and J. Hecksher-Sorensen, R. Baldock, and D. Davidson, "Optical projection tomography as a tool for 3d microscopy and gene expression studies," Science 296(5567), 541-545 (2002).
[CrossRef]

Baldock, R.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, and J. Hecksher-Sorensen, R. Baldock, and D. Davidson, "Optical projection tomography as a tool for 3d microscopy and gene expression studies," Science 296(5567), 541-545 (2002).
[CrossRef]

Baraniuk, R. G.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, "A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration," IEEE Trans. Image Proc. 16, 2992-3004 (2007).
[CrossRef] [PubMed]

Brady, D. J.

Brady, R. B.

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, "Visible cone-beam tomography with a lensless interferometric camera," Science 284(5423), 2164-2166 (1999).
[CrossRef]

Candes, E. J.

E. J. Candes, J. K. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math. 59,1207-1223 (2006).
[CrossRef]

E. J. Candes and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?," IEEE Transactions on Information Theory 52(12), 5406-5425 (2006).
[CrossRef]

Chan, W. L.

Davidson, D.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, and J. Hecksher-Sorensen, R. Baldock, and D. Davidson, "Optical projection tomography as a tool for 3d microscopy and gene expression studies," Science 296(5567), 541-545 (2002).
[CrossRef]

Devaney, A. J.

A. J. Devaney, "Geophysical diffraction tomography," IEEE Trans. Geoscie. Remote Sens. GE-22, 3-13 (1984).
[CrossRef]

A. J. Devaney, "Nonuniqueness in the inverse scattering problem," J. Math. Phys. 19(7), 1526-1531 (1978).
[CrossRef]

Donoho, D. L.

D. L. Donoho, "Compressed sensing," IEEE Trans. Info. Theory 52(4), 1289-1306 (2006).
[CrossRef]

D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via _1 minimization," Proc. Nat. Acad. Scie. U. S. A. 100(5), 2197-2202 (2003).
[CrossRef]

Elad, M.

D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via _1 minimization," Proc. Nat. Acad. Scie. U. S. A. 100(5), 2197-2202 (2003).
[CrossRef]

Erlinger, A.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, "Lensfree holographic imaging for on-chip cytometry and diagnostics," Lab on a Chip 9(6), 777-787 (2009).
[CrossRef]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60(1-4), 259-268 (1992).
[CrossRef]

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, "A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration," IEEE Trans. Image Proc. 16, 2992-3004 (2007).
[CrossRef] [PubMed]

Fink, H. W.

T. Latychevskaia and H. W. Fink, "Solution to the Twin Image Problem in Holography," Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Gabor, D.

D. Gabor, "A new microscopic principle," Nature 161,777-778 (1948).
[CrossRef] [PubMed]

Gehm, M. E.

Golay, M.

Hecksher-Sorensen, J.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, and J. Hecksher-Sorensen, R. Baldock, and D. Davidson, "Optical projection tomography as a tool for 3d microscopy and gene expression studies," Science 296(5567), 541-545 (2002).
[CrossRef]

Hill, B.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, and J. Hecksher-Sorensen, R. Baldock, and D. Davidson, "Optical projection tomography as a tool for 3d microscopy and gene expression studies," Science 296(5567), 541-545 (2002).
[CrossRef]

John, R.

Latychevskaia, T.

T. Latychevskaia and H. W. Fink, "Solution to the Twin Image Problem in Holography," Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Leith, E. N.

Marks, D. L.

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, "Visible cone-beam tomography with a lensless interferometric camera," Science 284(5423), 2164-2166 (1999).
[CrossRef]

Mills, G. A.

Mittleman, D. M.

Moravec, M. L.

Munson, D. C.

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, "Visible cone-beam tomography with a lensless interferometric camera," Science 284(5423), 2164-2166 (1999).
[CrossRef]

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60(1-4), 259-268 (1992).
[CrossRef]

Ozcan, A.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, "Lensfree holographic imaging for on-chip cytometry and diagnostics," Lab on a Chip 9(6), 777-787 (2009).
[CrossRef]

Perry, P.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, and J. Hecksher-Sorensen, R. Baldock, and D. Davidson, "Optical projection tomography as a tool for 3d microscopy and gene expression studies," Science 296(5567), 541-545 (2002).
[CrossRef]

Romberg, J. K.

E. J. Candes, J. K. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math. 59,1207-1223 (2006).
[CrossRef]

Ross, A.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, and J. Hecksher-Sorensen, R. Baldock, and D. Davidson, "Optical projection tomography as a tool for 3d microscopy and gene expression studies," Science 296(5567), 541-545 (2002).
[CrossRef]

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60(1-4), 259-268 (1992).
[CrossRef]

Schulz, T. J.

Seo, S.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, "Lensfree holographic imaging for on-chip cytometry and diagnostics," Lab on a Chip 9(6), 777-787 (2009).
[CrossRef]

Sharpe, J.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, and J. Hecksher-Sorensen, R. Baldock, and D. Davidson, "Optical projection tomography as a tool for 3d microscopy and gene expression studies," Science 296(5567), 541-545 (2002).
[CrossRef]

Stack, R. A.

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, "Visible cone-beam tomography with a lensless interferometric camera," Science 284(5423), 2164-2166 (1999).
[CrossRef]

Su, T. W.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, "Lensfree holographic imaging for on-chip cytometry and diagnostics," Lab on a Chip 9(6), 777-787 (2009).
[CrossRef]

Tao, T.

E. J. Candes, J. K. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math. 59,1207-1223 (2006).
[CrossRef]

E. J. Candes and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?," IEEE Transactions on Information Theory 52(12), 5406-5425 (2006).
[CrossRef]

Tseng, D. K.

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, "Lensfree holographic imaging for on-chip cytometry and diagnostics," Lab on a Chip 9(6), 777-787 (2009).
[CrossRef]

Upatnieks, J.

Wagadarikar, A.

Willett, R.

Willett, R. M.

Yamaguchi, I.

Yamamoto, K.

Yokota, M.

Appl. Opt. (2)

Commun. Pure Appl. Math. (1)

E. J. Candes, J. K. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math. 59,1207-1223 (2006).
[CrossRef]

IEEE Trans. Geoscie. Remote Sens. (1)

A. J. Devaney, "Geophysical diffraction tomography," IEEE Trans. Geoscie. Remote Sens. GE-22, 3-13 (1984).
[CrossRef]

IEEE Trans. Image Proc. (1)

J. M. Bioucas-Dias and M. A. T. Figueiredo, "A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration," IEEE Trans. Image Proc. 16, 2992-3004 (2007).
[CrossRef] [PubMed]

IEEE Trans. Info. Theory (1)

D. L. Donoho, "Compressed sensing," IEEE Trans. Info. Theory 52(4), 1289-1306 (2006).
[CrossRef]

IEEE Transactions on Information Theory (1)

E. J. Candes and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?," IEEE Transactions on Information Theory 52(12), 5406-5425 (2006).
[CrossRef]

J. Math. Phys. (1)

A. J. Devaney, "Nonuniqueness in the inverse scattering problem," J. Math. Phys. 19(7), 1526-1531 (1978).
[CrossRef]

J. Opt. Soc. Am. (2)

Lab on a Chip (1)

S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, "Lensfree holographic imaging for on-chip cytometry and diagnostics," Lab on a Chip 9(6), 777-787 (2009).
[CrossRef]

Nature (1)

D. Gabor, "A new microscopic principle," Nature 161,777-778 (1948).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

T. Latychevskaia and H. W. Fink, "Solution to the Twin Image Problem in Holography," Phys. Rev. Lett. 98, 233901 (2007).
[CrossRef] [PubMed]

Physica D (1)

L. I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60(1-4), 259-268 (1992).
[CrossRef]

Proc. Nat. Acad. Scie. U. S. A. (1)

D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via _1 minimization," Proc. Nat. Acad. Scie. U. S. A. 100(5), 2197-2202 (2003).
[CrossRef]

Science (2)

D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, "Visible cone-beam tomography with a lensless interferometric camera," Science 284(5423), 2164-2166 (1999).
[CrossRef]

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, and J. Hecksher-Sorensen, R. Baldock, and D. Davidson, "Optical projection tomography as a tool for 3d microscopy and gene expression studies," Science 296(5567), 541-545 (2002).
[CrossRef]

Other (12)

W. Jueptner and U. Schnars, Digital Holography, (New York, Springer-Verlag, Berlin Heidelberg, 2005).

T. C. Poon, Digital holography and three-dimensional display, (New York; London: Springer, 2006).
[CrossRef]

M. E. Brezinski, Optical coherence tomography. (Amsterdam; Boston: Academic Press, 2006).

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera architecture using optical-domain compression," in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.

C. E. Shannon, "Communications in the presence of noise," Proc. IREv 37, 10-21 (1949).
[CrossRef]

R. E. Blahut, Theory of Remote Image Formation (Cambridge University Press, 2004).
[CrossRef]

J. W. Goodman, Introduction to Fourier optics, 3rd Ed., (Roberts and Company Publishers, 2005).

D. M. Paganin, Coherent X-ray Optics (Oxford Science Publications, 2006).
[CrossRef]

A. C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

D. J. Brady, Optical Imaging and Spectroscopy (Wiley, 2009).
[CrossRef]

R. Baraniuk and P. Steeghs, "Compressive radar imaging," IEEE Radar Conference, pp. 128-133, April, (2007).

L. Li, W. Zhang, F. Li, "Compressive diffraction tomography for Weakly Scattering," Submitted to IEEE Trans. on Geoscie. Remote Sens. (2009).

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Figures (4)

Fig. 1.
Fig. 1.

Gabor hologram geometry.

Fig. 2.
Fig. 2.

Experimental apparatus.

Fig. 3.
Fig. 3.

(a) Raw Gabor hologram for seed parachutes of taraxacum arranged as in Fig. 2, (b) and (c) photographs of the individual objects, (d) transverse slices at various ranges of the tomographic reconstruction of the 3D data volume containing both objects, and (e) transverse slices at various ranges of the backpropagated (numerical refocusing) field.

Fig. 4.
Fig. 4.

Simulations showing the effects of squared field term e: the 3D datacube estimates from the squared field with no diffraction (a) and the squared field with diffraction (b).

Equations (21)

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

(1δS)fT22HTfT22(1+δS)fT22,
μ1(H,Ψ)=Nmaxhm,ψn,
1mM
1nN
MC μ12 (H,Ψ)SlogN,
θe=argminθθ1suchthatg=Hfe=HΨθe,
μ2(D)=maxij,1i,jN{diTdjdi2dj2},
S<12(1+1μ2(D)),
I(x,y)=A+E(x,y)2
=A2+E(x,y)2+A*E(x,y)+AE*(x,y),
E(x,y)=dxdydzη(x,y,z)h(xx,yy,zz),
En1n2=E (n1Δ,n2Δ)
=1(2π)2dzdxdydkxdkydxdyη(x,y,z)eikzei(kxx+kyy)Σm1Σm2δ(xm1Δ)
δ (ym2Δ)eizk2kx2ky2δ(zz)ei(kxx+kyy)δ(xn1Δ)δ(yn2Δ)Σm1Σm2δ(kxm1Δk)
δ (kym2Δk)Σlδ(zlΔz)
=1N2ΣlΣm1Σm2[Σm'2Σm2ηm1m2lei2πm1m1+m2m2N]eiklΔzeilΔzk2m12Δk2m22Δk2ei2πn1m1+n2m2N,
En1n2=𝓕2D1 {Σlη̂m1m2leiklΔzeilΔzk2m12Δk2m22Δk2} ,
gˉ=G2DQBf,
g=2Re{gˉ}=2Re{G2DQBf}=2 Re {Hf}+e+n,
f̂=argminffTVsuchthatg=Hf,
fkTV=ΣkΣn1Σn2(fk)n1,n2,

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