Abstract

Computational optical-sectioning microscopy with a nonconfocal microscope is fundamentally limited because the optical transfer function, the Fourier transform of the point-spread function, is exactly zero over a conic region of the spatial-frequency domain. Because of this missing cone of optical information, images are potentially artifactual. To overcome this limitation, superresolution, in the sense of band extrapolation, is necessary. I present a frequency-domain analysis of the expectation-maximization algorithm for maximum-likelihood image estimation that shows how the algorithm achieves this band extrapolation. This analysis gives the theoretical absolute bandwidth of the restored image; however, this absolute value may not be realistic in many cases. Then a second analysis is presented that assumes a Gaussian point-spread function and a specimen function and shows more realistic behavior of the algorithm and demonstrates some of its properties. Experimental results on the superresolving capability of the algorithm are also presented.

© 1998 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [PubMed]
  37. D. L. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
    [CrossRef] [PubMed]
  38. S. Joshi, M. I. Miller, “Maximum a posteriori estimation with Good’s roughness for optical sectioning microscopy,” J. Opt. Soc. Am. A 10, 1078–1085 (1993).
    [CrossRef] [PubMed]
  39. D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. MI-6, 228–238 (1987).
    [CrossRef]

1997

1995

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

1994

1993

1992

1991

1990

J. A. Conchello, E. W. Hansen, “Enhanced 3-D reconstruction from confocal scanning microscope images. 1: Deterministic and maximum likelihood reconstructions,” Appl. Opt. 29, 3795–3804 (1990).
[CrossRef] [PubMed]

J. Biemond, R. L. Lagendijk, R. M. Meresereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856–883 (1990).
[CrossRef]

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and super-resolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

1989

1988

1987

B. Bertero, D. De Mol, E. R. Pike, “Analytic inversion formula for confocal scanning microscopy,” J. Opt. Soc. Am. A 4, 1748–1750 (1987).
[CrossRef]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. MI-6, 228–238 (1987).
[CrossRef]

1986

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging II,” Optik (Stuttgart) 74, 128–129 (1986).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Stuttgart) 72, 131–133 (1986).

1985

1984

N. Streibl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik (Stuttgart) 66, 341–354 (1984).

1976

H. C. Andrews, C. L. Patterson, “Singular value decompositions and digital image processing,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-24, 26–53 (1976).
[CrossRef]

1969

1968

1967

Agard, D. A.

R. S. Aikens, D. A. Agard, J. W. Sedat, “Solid-state imagers for microscopy,” in Methods in Cell Biology, Y.-L. Wang, D. L. Taylor, eds. (Academic, New York, 1989), Vol. 29, pp. 291–313.
[PubMed]

Aikens, R. S.

R. S. Aikens, D. A. Agard, J. W. Sedat, “Solid-state imagers for microscopy,” in Methods in Cell Biology, Y.-L. Wang, D. L. Taylor, eds. (Academic, New York, 1989), Vol. 29, pp. 291–313.
[PubMed]

Andrews, H. C.

H. C. Andrews, C. L. Patterson, “Singular value decompositions and digital image processing,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-24, 26–53 (1976).
[CrossRef]

Bertero, B.

Bertero, M.

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and super-resolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

Biemond, J.

J. Biemond, R. L. Lagendijk, R. M. Meresereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856–883 (1990).
[CrossRef]

Bille, J.

Boccacci, P.

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and super-resolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications 2e (McGraw-Hill, New York, 1978), pp. 148–156.

R. N. Bracewell, The Fourier Transform and Its Applications 2e (McGraw-Hill, New York, 1978), pp. 160–163.

R. N. Bracewell, The Fourier Transform and Its Applications 2e (McGraw-Hill, New York, 1978), p. 157.

Brakenhoff, G. J.

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and super-resolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

Brown, J. W.

R. V. Churchill, J. W. Brown, Complex Variables and Applications 4e (McGraw-Hill, New York, 1984), Sect. 103.

Carrington, W. A.

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

Churchill, R. V.

R. V. Churchill, J. W. Brown, Complex Variables and Applications 4e (McGraw-Hill, New York, 1984), Sect. 103.

Conchello, J. A.

J. A. Conchello, J. J. Kim, E. W. Hansen, “Enhanced three-dimensional reconstruction from confocal scanning microscope images. 2: depth discrimination versus signal-to-noise ratio in partially confocal images,” Appl. Opt. 33, 3740–3750 (1994).
[CrossRef] [PubMed]

J. G. McNally, C. Preza, J. A. Conchello, L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A 11, 1056–1067 (1994).
[CrossRef]

J. A. Conchello, E. W. Hansen, “Enhanced 3-D reconstruction from confocal scanning microscope images. 1: Deterministic and maximum likelihood reconstructions,” Appl. Opt. 29, 3795–3804 (1990).
[CrossRef] [PubMed]

J. A. Conchello, “Three-dimensional reconstruction of noisy images from partially confocal scanning microscope,” Ph.D. dissertation (Dartmouth College, Hanover, N.H., 1990).

J. A. Conchello, “Super-resolution and point spread function sensitivity analysis of the expectation-maximization algorithm for computational optical sectioning microscopy,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 369–378 (1994).
[CrossRef]

De Mol, D.

Erhardt, A.

Fay, F. S.

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

Fogarty, K.

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

Frieden, B. R.

Gibson, F. S.

Hammoud, A. M.

Hansen, E. W.

Holmes, T. J.

Hunt, B. R.

Isenberg, G.

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

Joshi, S.

Jovin, T. M.

Kim, J. J.

Komitowski, D.

Lagendijk, R. L.

J. Biemond, R. L. Lagendijk, R. M. Meresereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856–883 (1990).
[CrossRef]

Lanni, F.

Liu, Y.-H.

Lynch, R. M.

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

Malfanti, F.

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and super-resolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

Mao, X. Q.

McNally, J. G.

Meresereau, R. M.

J. Biemond, R. L. Lagendijk, R. M. Meresereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856–883 (1990).
[CrossRef]

Miller, M. I.

Moore, E. D. W.

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

Nadar, M. S.

Nielsen-Delaney, P. A.

O’Sullivan, J. A.

D. L. Snyder, T. J. Schultz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Sig. Proc. 40, 1143–1150 (1992).
[CrossRef]

Patterson, C. L.

H. C. Andrews, C. L. Patterson, “Singular value decompositions and digital image processing,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-24, 26–53 (1976).
[CrossRef]

Pike, E. R.

Politte, D. G.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. MI-6, 228–238 (1987).
[CrossRef]

Preza, C.

Scalas, E.

Schultz, T. J.

D. L. Snyder, T. J. Schultz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Sig. Proc. 40, 1143–1150 (1992).
[CrossRef]

Sedat, J. W.

R. S. Aikens, D. A. Agard, J. W. Sedat, “Solid-state imagers for microscopy,” in Methods in Cell Biology, Y.-L. Wang, D. L. Taylor, eds. (Academic, New York, 1989), Vol. 29, pp. 291–313.
[PubMed]

Sementilli, P. J.

Sheppard, C. J. R.

C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260–1269 (1989).
[CrossRef]

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging II,” Optik (Stuttgart) 74, 128–129 (1986).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Stuttgart) 72, 131–133 (1986).

Snyder, D. L.

D. L. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
[CrossRef] [PubMed]

D. L. Snyder, T. J. Schultz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Sig. Proc. 40, 1143–1150 (1992).
[CrossRef]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. MI-6, 228–238 (1987).
[CrossRef]

Stokseth, P. A.

Streibl, N.

N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
[CrossRef]

N. Streibl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik (Stuttgart) 66, 341–354 (1984).

Thomas, L. J.

Toraldo di Francia, G.

van der Voort, H. T. M.

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and super-resolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

Verveer, P. J.

Viano, G. A.

Walsh, D. O.

White, R. L.

Zinser, G.

Appl. Opt.

IEEE Trans. Acoust., Speech, Signal Process.

H. C. Andrews, C. L. Patterson, “Singular value decompositions and digital image processing,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-24, 26–53 (1976).
[CrossRef]

IEEE Trans. Med. Imag.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. MI-6, 228–238 (1987).
[CrossRef]

IEEE Trans. Sig. Proc.

D. L. Snyder, T. J. Schultz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Sig. Proc. 40, 1143–1150 (1992).
[CrossRef]

J. Microsc.

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and super-resolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
[CrossRef]

C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260–1269 (1989).
[CrossRef]

T. J. Holmes, Y.-H. Liu, “Acceleration of maximum-likelihood image restoration for fluorescence microscopy and other noncoherent imagery,” J. Opt. Soc. Am. A 8, 893–907 (1991).
[CrossRef]

F. S. Gibson, F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A 8, 1601–1613 (1991).
[CrossRef]

B. Bertero, D. De Mol, E. R. Pike, “Analytic inversion formula for confocal scanning microscopy,” J. Opt. Soc. Am. A 4, 1748–1750 (1987).
[CrossRef]

J. G. McNally, C. Preza, J. A. Conchello, L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A 11, 1056–1067 (1994).
[CrossRef]

D. O. Walsh, P. A. Nielsen-Delaney, “Direct method for superresolution,” J. Opt. Soc. Am. A 11, 572–579 (1994).
[CrossRef]

C. Preza, M. I. Miller, L. J. Thomas, J. G. McNally, “Regularized method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A 9, 219–228 (1992).
[CrossRef] [PubMed]

E. Scalas, G. A. Viano, “Resolving power and information theory in signal recovery,” J. Opt. Soc. Am. A 10, 991–996 (1993).
[CrossRef]

P. J. Verveer, T. M. Jovin, “Efficient superresolutionrestoration algorithms using maximum a posteriori estimations with applications to fluorescence microscopy,” J. Opt. Soc. Am. A 14, 1696–1706 (1997).
[CrossRef]

T. J. Holmes, “Maximum-likelihood image restoration adapted for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
[CrossRef]

T. J. Holmes, “Expectation-maximization restoration of band-limited, truncated point-process intensities with application in microscopy,” J. Opt. Soc. Am. A 6, 1006–1014 (1989).
[CrossRef]

P. J. Sementilli, B. R. Hunt, M. S. Nadar, “Analysis of the limit to superresolution in incoherent imaging,” J. Opt. Soc. Am. A 10, 2265–2276 (1993).
[CrossRef]

D. L. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
[CrossRef] [PubMed]

S. Joshi, M. I. Miller, “Maximum a posteriori estimation with Good’s roughness for optical sectioning microscopy,” J. Opt. Soc. Am. A 10, 1078–1085 (1993).
[CrossRef] [PubMed]

Opt. Acta

N. Streibl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

Optik (Stuttgart)

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik (Stuttgart) 66, 341–354 (1984).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging II,” Optik (Stuttgart) 74, 128–129 (1986).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Stuttgart) 72, 131–133 (1986).

Proc. IEEE

J. Biemond, R. L. Lagendijk, R. M. Meresereau, “Iterative methods for image deblurring,” Proc. IEEE 78, 856–883 (1990).
[CrossRef]

Science

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

Other

J. A. Conchello, “Three-dimensional reconstruction of noisy images from partially confocal scanning microscope,” Ph.D. dissertation (Dartmouth College, Hanover, N.H., 1990).

J. A. Conchello, “Super-resolution and point spread function sensitivity analysis of the expectation-maximization algorithm for computational optical sectioning microscopy,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 369–378 (1994).
[CrossRef]

R. V. Churchill, J. W. Brown, Complex Variables and Applications 4e (McGraw-Hill, New York, 1984), Sect. 103.

R. S. Aikens, D. A. Agard, J. W. Sedat, “Solid-state imagers for microscopy,” in Methods in Cell Biology, Y.-L. Wang, D. L. Taylor, eds. (Academic, New York, 1989), Vol. 29, pp. 291–313.
[PubMed]

R. N. Bracewell, The Fourier Transform and Its Applications 2e (McGraw-Hill, New York, 1978), pp. 148–156.

R. N. Bracewell, The Fourier Transform and Its Applications 2e (McGraw-Hill, New York, 1978), pp. 160–163.

R. N. Bracewell, The Fourier Transform and Its Applications 2e (McGraw-Hill, New York, 1978), p. 157.

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Figures (8)

Fig. 1
Fig. 1

Frequency-domain support of the OTF of the microscope [(a) nonconfocal (b) confocal] equal to the frequency support of the estimated object after two iterations.

Fig. 2
Fig. 2

Width σk of the estimated specimen function relative to the width Σ of the true specimen function plotted as a function of iteration for the case in which the PSF and the specimen function have the same width. Solid curve, values obtained during the deconvolution; dots, values calculated from Eq. (15).

Fig. 3
Fig. 3

Width σk of the estimated specimen function relative to the width Σ of the true specimen function plotted as a function of iteration for different values of the width H of the PSF.

Fig. 4
Fig. 4

Recorded image of a 10-µm diameter fluorescent microsphere. Left, horizontal or xy medial section. Right, vertical or xz medial section. For display purposes the xz section was interpolated in z to achieve a 1:1 aspect ratio.

Fig. 5
Fig. 5

EM specimen-function estimates for a 10-µm diameter fluorescent microsphere. Left column, horizontal or xy medial sections; Right column, vertical or xz medial sections. For display purposes the xz section was interpolated in z to achieve a 1:1 aspect ratio.

Fig. 6
Fig. 6

Frequency support of the OTF (top), of the recorded image (top image, left column), and of the estimated specimen function from 16 to 4096 iterations with doubling of number of iterations. The regions shown are where the corresponding function is greater than 1% (0.01) of its peak value. Numbers indicate number of iterations. “Raw” indicates the unprocessed data. The horizontal axis is the radial spatial frequency ρ. The vertical axis is the axial spatial frequency ζ. For display purposes the figures were scaled in ρ to achieve a 1:1 aspect ratio.

Fig. 7
Fig. 7

Simulated image (top row) and estimated specimen function for a spherical shell of uniform fluorescence with outer and inner diameters of 10 µm and 8 µm, respectively. Left column, lateral or xy sections through the center of the shell; right column, xz or axial sections through the center of the shell.

Fig. 8
Fig. 8

Axial frequency components of the estimated specimen function for a synthetic image of a fluorescent spherical shell for different numbers of iterations. Also shown are the frequency components of the image (long-dashed curve) and of the true specimen function (solid curve).

Equations (47)

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g(x)=Oh(x-x)s(x)dx=h(x)s(x),
gˆ(k)(x)=h(x)sˆ(k)(x)
d(k)(x)=g(x)/gˆ(k)(x).
r(k)(x)=[1/H(0)]h(-x)d(k)(x),
H(0)=h(x)dx;
sˆ(k+1)(x)=sˆ(k)(x)r(k)(x).
Sˆ(k)(ν)=F{sˆ(k)(x)}=-sˆ(k)(x)exp(-j2πν·x)dx,
H(ν)=0forνΩH.
Sˆ(k)(ν)=0forνΩk.
Sˆ(k)(ν)0forallνΩk;
sˆ(1)(x)=h(x)g(x),
Sˆ(k+1)(ν)=Sˆ(k)(ν)R(k)(ν).
d(k)(x)=g(x)/gˆ(k)(x)1
g1(x)=h(x)s1(x),g2(x)=h(x)s2(x),
WS(k)=1Sˆ(k)(0)Sˆ(k)(ν)dν,
k2=ν2|Sˆ(k)(ν)|2dν|Sˆ(k)(ν)|2dν-ν|Sˆ(k)(ν)|2dν|Sˆ(k)(ν)|2dν2.
σk2k21/4π,
rect(x)=1for|x|1/20otherwise
h(x)=exp(-πx2/H2),H(ν)=H exp(-πν2H2)
s(x)=exp(-πx2/2),S(ν)= exp(-πν22)
g(x)=exp(-πx2/Γ2),G(ν)=Γ exp(-πν2Γ2)
sˆ(k)(x)=exp(-πx2/σk2),Sˆ(k)(ν)=σk exp(-πσk2ν2)
g(x)=s(x)h(x).
σk+12=σk2 1+σk2/H2+(σk2/H2)(1+2/H2)(1+σk2/H2)2.
σ0>,
σk>σk+1>.
(σk2/H2)(1+σk2/H2)+2σk2/H2
>(k2/H2)(1+σk2/H2)+2σk2/H2.
2<σk[1+σk2/H2+(σk2/H2)(1+2/H2)]/(1+σk2/H2)2.
1+2σk2/H2+2σk2/H4<1+2σk2/H2+σk4/H4,
Fk=(1+2σk2/H2+2σk2/H4)/(1+σk2/H2)2<1.
Fk=1-[σk2(σk2-2)/H4]/(1+σk2/H2)2<1.
Ii(z)=-h(x, y, z)dxdy
sˆ(k)(x, y, Nz-z)=sˆ(k)(x, y, z)
forz=0, 1,, Nz-1.
sˆ(k)(x, y, Nz-z)=Nz-znzsˆ(k)(x, y, 0)+znzsˆ(k)(x, y, Nz-1) forz=0, 1,, Nz-1.
|Sˆ(k)(ξ, η, ζ)/Sˆ(k)(0, 0, 0)|>0.01.
G(ν)=K exp[-πν2(2+H2)],
g(x)=K exp[-πx/(2+H2)].
gˆ(k)(x)=K exp{-(πx)/(σk2+H2)].
d(k)(x)=g(x)/gˆ(k)(x)=K exp{-πx2(σk2-2)/[(2+H2)(σk2+H2)]}
D(k)(ν)=K exp[-πν2(2+H2)×(σk2+H2)/(σk2-2)].
R(k)(ν)=K exp[-πν2(2σk2+2σk2H2+H4)/(σk2-2)],
r(k)(x)=K exp[-πx(σk2-2)/(2σk2+2σk2H2+H4)].
sˆ(k+1)(x)=K exp-πxσk2-22σk2+2σk2H2+H4-1σk2.
1σk+12=σk2-2(H2+σk2)2-σk4+2σk2-1σk2
σk+12=σk2 H4+2H2σk2+2H2(H2+σk2)2.

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