Abstract

The inverse problem involving the determination of a three-dimensional biological structure from images obtained by means of optical-sectioning microscopy is ill posed. Although the linear least-squares solution can be obtained rapidly by inverse filtering, we show here that it is unstable because of the inversion of small eigenvalues of the microscope’s point-spread-function operator. We have regularized the problem by application of the linear-precision-gauge formalism of Joyce and Root [ J. Opt. Soc. Am. A 1, 149 ( 1984)]. In our method the solution is regularized by being constrained to lie in a subspace spanned by the eigenvectors corresponding to a selected number of large eigenvalues. The trade-off between the variance and the regularization error determines the number of eigenvalues inverted in the estimation. The resulting linear method is a one-step algorithm that yields, in a few seconds, solutions that are optimal in the mean-square sense when the correct number of eigenvalues are inverted. Results from sensitivity studies show that the proposed method is robust to noise and to underestimation of the width of the point-spread function. The method proposed here is particularly useful for applications in which processing speed is critical, such as studies of living specimens and time-lapse analyses. For these applications existing iterative methods are impractical without expensive and/or specially designed hardware.

© 1992 Optical Society of America

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References

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  1. F. S. Gibson, F. Lanni, “Diffraction by a circular aperture as a model for three-dimensional optical microscopy,” J. Opt. Soc. Am. A 6, 1357–1367 (1989).
    [CrossRef] [PubMed]
  2. M. Weinstein, K. R. Castleman, “Reconstructing 3-D specimens from 2-D section images,” in Quantitative Imagery in the Biomedical Sciences I, R. E. Herron, ed., Proc. Soc. Photo-Opt. Instrum. Eng.26, 131–138 (1971).
    [CrossRef]
  3. D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
    [CrossRef] [PubMed]
  4. A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Opt. 24, 194–200 (1985).
    [CrossRef] [PubMed]
  5. R. S. Fay, K. E. Fogarty, J. M. Coggins, “Analysis of molecular distribution in single cells using a digital imaging microscope,” in Optical Methods in Cell Physiology, P. DeWeer, B. M. Salzberg, eds. (Society of General Physiologists and Wiley-Interscience, New York, 1986), Vol. 40, pp. 51–63.
  6. W. Carrington, K. Fogarty, “3-D molecular distribution in living cells by deconvolution of optical sections using light microscopy,” in Proceedings of the Thirteenth Annual Northeast Bioengineering Conference, K. R. Foster, ed. (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 108–111.
  7. P. A. Jansson, R. H. Hunt, E. K. Plyler, “Resolution enhancement of spectra,”J. Opt. Soc. Am. 60, 596–599 (1970).
    [CrossRef]
  8. L. S. Joyce, W. L. Root, “Precision bounds in superresolution processing,” J. Opt. Soc. Am. A 1, 149–168 (1984).
    [CrossRef]
  9. N. Streibl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
    [CrossRef]
  10. C. Preza, “A regularized linear reconstruction method for optical sectioning microscopy,” master’s thesis (Sever Institute of Technology, Washington University, St. Louis, Mo., 1990).
  11. B. Noble, Applied Linear Algebra (Prentice-Hall, Englewood Cliffs, N.J., 1969).
  12. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
    [CrossRef]
  13. W. L. Root, “Ill-posedness and precision in object-field reconstruction problems,” J. Opt. Soc. Am. A 4, 171–179 (1987).
    [CrossRef]
  14. L. L. Tella, “The determination of a microscope’s three-dimensional transfer function for use in image restoration,” master’s thesis (Worcester Polytechnic Institute, Worcester, Mass., 1985).

1989 (2)

F. S. Gibson, F. Lanni, “Diffraction by a circular aperture as a model for three-dimensional optical microscopy,” J. Opt. Soc. Am. A 6, 1357–1367 (1989).
[CrossRef] [PubMed]

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[CrossRef] [PubMed]

1987 (1)

1985 (1)

1984 (2)

1970 (1)

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Agard, D. A.

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[CrossRef] [PubMed]

Bille, J.

Carrington, W.

W. Carrington, K. Fogarty, “3-D molecular distribution in living cells by deconvolution of optical sections using light microscopy,” in Proceedings of the Thirteenth Annual Northeast Bioengineering Conference, K. R. Foster, ed. (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 108–111.

Castleman, K. R.

M. Weinstein, K. R. Castleman, “Reconstructing 3-D specimens from 2-D section images,” in Quantitative Imagery in the Biomedical Sciences I, R. E. Herron, ed., Proc. Soc. Photo-Opt. Instrum. Eng.26, 131–138 (1971).
[CrossRef]

Coggins, J. M.

R. S. Fay, K. E. Fogarty, J. M. Coggins, “Analysis of molecular distribution in single cells using a digital imaging microscope,” in Optical Methods in Cell Physiology, P. DeWeer, B. M. Salzberg, eds. (Society of General Physiologists and Wiley-Interscience, New York, 1986), Vol. 40, pp. 51–63.

Erhardt, A.

Fay, R. S.

R. S. Fay, K. E. Fogarty, J. M. Coggins, “Analysis of molecular distribution in single cells using a digital imaging microscope,” in Optical Methods in Cell Physiology, P. DeWeer, B. M. Salzberg, eds. (Society of General Physiologists and Wiley-Interscience, New York, 1986), Vol. 40, pp. 51–63.

Fogarty, K.

W. Carrington, K. Fogarty, “3-D molecular distribution in living cells by deconvolution of optical sections using light microscopy,” in Proceedings of the Thirteenth Annual Northeast Bioengineering Conference, K. R. Foster, ed. (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 108–111.

Fogarty, K. E.

R. S. Fay, K. E. Fogarty, J. M. Coggins, “Analysis of molecular distribution in single cells using a digital imaging microscope,” in Optical Methods in Cell Physiology, P. DeWeer, B. M. Salzberg, eds. (Society of General Physiologists and Wiley-Interscience, New York, 1986), Vol. 40, pp. 51–63.

Gibson, F. S.

Hiraoka, Y.

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[CrossRef] [PubMed]

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Hunt, R. H.

Jansson, P. A.

Joyce, L. S.

Komitowski, D.

Lanni, F.

Noble, B.

B. Noble, Applied Linear Algebra (Prentice-Hall, Englewood Cliffs, N.J., 1969).

Plyler, E. K.

Preza, C.

C. Preza, “A regularized linear reconstruction method for optical sectioning microscopy,” master’s thesis (Sever Institute of Technology, Washington University, St. Louis, Mo., 1990).

Root, W. L.

Sedat, J. W.

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[CrossRef] [PubMed]

Shaw, P.

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[CrossRef] [PubMed]

Streibl, N.

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

Tella, L. L.

L. L. Tella, “The determination of a microscope’s three-dimensional transfer function for use in image restoration,” master’s thesis (Worcester Polytechnic Institute, Worcester, Mass., 1985).

Weinstein, M.

M. Weinstein, K. R. Castleman, “Reconstructing 3-D specimens from 2-D section images,” in Quantitative Imagery in the Biomedical Sciences I, R. E. Herron, ed., Proc. Soc. Photo-Opt. Instrum. Eng.26, 131–138 (1971).
[CrossRef]

Zinser, G.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Methods Cell Biol. (1)

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[CrossRef] [PubMed]

Opt. Acta (1)

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

Proc. R. Soc. London Ser. A (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Other (6)

L. L. Tella, “The determination of a microscope’s three-dimensional transfer function for use in image restoration,” master’s thesis (Worcester Polytechnic Institute, Worcester, Mass., 1985).

C. Preza, “A regularized linear reconstruction method for optical sectioning microscopy,” master’s thesis (Sever Institute of Technology, Washington University, St. Louis, Mo., 1990).

B. Noble, Applied Linear Algebra (Prentice-Hall, Englewood Cliffs, N.J., 1969).

M. Weinstein, K. R. Castleman, “Reconstructing 3-D specimens from 2-D section images,” in Quantitative Imagery in the Biomedical Sciences I, R. E. Herron, ed., Proc. Soc. Photo-Opt. Instrum. Eng.26, 131–138 (1971).
[CrossRef]

R. S. Fay, K. E. Fogarty, J. M. Coggins, “Analysis of molecular distribution in single cells using a digital imaging microscope,” in Optical Methods in Cell Physiology, P. DeWeer, B. M. Salzberg, eds. (Society of General Physiologists and Wiley-Interscience, New York, 1986), Vol. 40, pp. 51–63.

W. Carrington, K. Fogarty, “3-D molecular distribution in living cells by deconvolution of optical sections using light microscopy,” in Proceedings of the Thirteenth Annual Northeast Bioengineering Conference, K. R. Foster, ed. (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 108–111.

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Figures (10)

Fig. 1
Fig. 1

PSF’s of a diffraction-limited, aberration-free 40 × 1.0 N.A. oil lens at defocus distance Zd equal to (a) 3.0 μm, (b) 2.0 μm, (c) 1.0 μm, and (d) 0.0 μm (in-focus PSF). The height corresponds to intensity values in the x, y plane calculated from the inverse Fourier transform of the optical-transfer function, as discussed in Ref. 10.

Fig. 2
Fig. 2

Planar sections at several axial frequencies of the 3-D optical-transfer function, |T(u,v,η)|, of an aberration-free 40 × 1.0 N.A. oil lens. The plotted intensity (calculated from theoretical formulations that are due to Hopkins,12 as discussed in Ref. 10) is in the u,v frequency plane at different frequencies η: (a) 0.24 μm−1, (b) 0.18 μm−1,(c) 0.12 μm−1, (d) 0.0 μm−1. The cutoff frequencies are ηmax = 1.0 μm−1 and umax = vmax = 4.0 μm−1. (e) Radial profiles through the planar sections.

Fig. 3
Fig. 3

Phantom consisting of seven structures with dimensions as tabulated. Along the z axis the structures are separated by 2 μm, while the separation between structures in the x, y plane varies between 0.375 and 1.375 μm. The intensity at each voxel is either 0 or 255.

Fig. 4
Fig. 4

(a) Section images (64 × 64) of the intensity of the phantom (Fig. 3) at 2.0-μm z-axis spacings. (b) Corresponding images from the simulated microscopic data. Focus is stepped from the top image on the left to the bottom image on the right.

Fig. 5
Fig. 5

Eigenspace dimension used for each estimation as a function of the estimate number. The estimate number indicates the number of distinct eigenvalues used.

Fig. 6
Fig. 6

Statistical behavior of the estimation sequence shown in terms of the variation of (a) the average MSE, (b) the absolute SNR, and (c) the bias and the STD with respect to the eigenspace dimension. The largest eigenspace dimension shown is 30.5% of the maximum one.

Fig. 7
Fig. 7

x, z views of three different estimates compared with corresponding x, z views of a, the simulated data and b, the phantom. The eigenspace dimension κ, the experimental average MSE, and the absolute SNR for each estimate are as follows: c, κ = 19112 (14.6% of κmax), MSE = 925, SNR = 7.5 dB; d, κ = 37106 (28.3% of κmax), MSE = 3837, SNR = 1.3 dB; e, κ = 40602 (34.1% of κmax), MSE = 10094, SNR = −2.9 dB. The bars on the right-hand side indicate the range of intensity values in each image.

Fig. 8
Fig. 8

x, y section images obtained from the reconstructed 3-D data at a fixed z coordinate for the three different cases of Fig. 7 compared with corresponding images obtained from a, the simulated microscopic data and b, the phantom. The x, y plane is a cut from structures E and G of the phantom (see Fig. 3). The algorithm was able to remove the out-of-focus light (two lower structures) in the simulated data (a), as the images from the reconstructions (d and e) show. Note that the out-of-focus light is due to structures A, B, C, and D in the phantom (Fig. 3). The bars on the right-hand side indicate the range of intensity values in each image.

Fig. 9
Fig. 9

Effect of noise level on the performance of the regularized linear method shown in terms of the variation of (a) subeigenspace dimension, (b) average bias, and (c) output SNR with respect to the noise level.

Fig. 10
Fig. 10

Results of a set of simulations to determine the sensitivity of the regularized linear method to error in the FWHM of a Gaussian PSF. The two curves plot the SNR of estimates with STD’s equal to 6 and 35 at different percent errors in the FWHMZ.

Equations (25)

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i k ( x , y ) = j = 0 J - 1 [ o j ( x , y ) * s k - j ( x , y ) ] + n ( x , y ) ,
χ = S τ + ω ,
S = [ S 0 x y S K - 1 x y S K - 2 x y S 1 x y S 1 x y S 0 x y S K - 1 x y S z x y S K - 1 x y S K - 2 x y S K - 3 x y S 0 x y ] .
τ ^ = ( B * K - 1 B ) - 1 B * K - 1 χ ,
τ ^ = ( B * B ) - 1 B * χ ,
E { τ - τ ^ 2 } = Tr [ ( B * K - 1 B ) - 1 ]
var ( τ ^ ) = σ 2 Tr [ ( B * B ) - 1 ] .
var ( τ ^ ) = σ 2 k = 0 m - 1 λ k - 1 ,             λ k 0.
τ ^ κ = arg min τ ˜ κ χ - S τ ˜ κ 2 , subject to τ ˜ κ { τ ˜ κ τ ˜ κ = i = 0 κ - 1 a i e i } ,
min a i χ - S i = 0 κ - 1 a i e i 2 ,
χ - i = 0 κ - 1 a i μ i e i , e j = 0 ,             j = 0 , , κ - 1.
τ ^ κ = i = 0 κ - 1 a i e i ,             κ m ,
var ( τ ^ κ ) = σ 2 i = 0 κ - 1 λ i - 1 ,             κ m ,
Q k = [ e 0 , e 1 , , e κ - 1 ] ,             P κ = Q κ * .
τ κ = Q κ α κ ,             α κ P κ τ .
χ κ P κ χ = P κ S τ + P κ ω ,
χ κ = P κ S [ τ κ + ( τ - τ κ ) ] + ω κ ,
χ κ = P κ S Q κ α κ + ω κ = S κ α κ + ω κ ,
α ^ κ = S k - 1 χ κ = α κ + S κ - 1 ω κ .
min α κ χ κ - S κ α κ 2 ,
min α κ χ κ - P κ S Q κ α κ 2 ,
τ ^ κ = Q κ S κ - 1 χ κ ,             κ m ,
bias κ ( τ ^ κ ) E { τ - τ ^ κ } = τ - Q κ P κ τ ,
MSE κ ( τ ^ κ ) = var κ ( τ ^ κ ) + bias κ ( τ ^ κ ) * bias κ ( τ ^ κ ) = σ 2 i = 0 κ - 1 λ i - 1 + τ * τ - τ * Q κ P κ τ ,
SNR = 10 log 10 [ x , y , z o ( x , y , z ) 2 x , y , z MSE ( x , y , z ) ] .

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