Abstract

The enhanced depth discrimination of a confocal scanning optical microscope is produced by a pinhole aperture placed in front of the detector to reject out-of-focus light. Strictly confocal behavior is impractical because an infinitesimally small aperture would collect very little light and would result in images with a poor signal-to-noise ratio (SNR), while a finite-sized partially confocal aperture provides a better SNR but reduced depth discrimination. Reconstruction algorithms, such as the expectation-maximization algorithm for maximum likelihood, can be applied to partially confocal images in order to achieve better resolution, but because they are sensitive to noise in the data, there is a practical trade-off involved. With a small aperture, fewer iterations of the reconstruction algorithm are necessary to achieve the desired resolution, but the low a priori SNR will result in a noisy reconstruction, at least when no regularization is used. With a larger aperture the a priori SNR is larger but the resolution is lower, and more iterations of the algorithm are necessary to reach the desired resolution; at some point the a posteriori SNR is lower than the a priori value. We present a theoretical analysis of the SNR-to-resolution trade-off partially confocal imaging, and we present two studies that use the expectation-maximization algorithm as a postprocessor; these studies show that a for a given task there is an optimum aperture size, departures from which result in a lower a posteriori SNR.

© 1994 Optical Society of America

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1992 (1)

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

1991 (1)

M. I. Miller, B. Roysam, “Bayesian image reconstruction for emission tomography incorporating Good’s roughness prior on massively parallel processors,” Proc. Natl. Acad. Sci. (USA) 88, 3223–3227 (1991).
[CrossRef]

1990 (2)

1989 (1)

1988 (3)

1987 (2)

T. Wilson, A. R. Carlini, “Size of the detector in confocal imaging systems,” Opt. Lett. 12, 227–229 (1987).
[CrossRef] [PubMed]

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

1986 (1)

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

1985 (5)

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. (Oxford) 138, 29–34 (1985).
[CrossRef]

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–20 (1985).
[CrossRef]

A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopy images by digital image processing,” Appl. Opt. 24, 194–200 (1985).
[CrossRef] [PubMed]

K. Carlsson, P. E. Danielsson, R. Lenz, A. Liljeborg, L. Majlöf, N. Åslund, “Three-dimensional microscopy using a confocal laser scanning microscope,” Opt. Lett. 10, 53–58 (1985).
[CrossRef] [PubMed]

1983 (2)

F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of α-actinin in single isolated smooth muscle cells,” J. Cell Biol. 96, 783–795 (1983).
[CrossRef] [PubMed]

D. A. Agard, J. W. Sedat, “Three-dimensional structure of a polytene nucleus,” Nature (London) 302, 676–681 (1983).
[CrossRef]

1980 (1)

I. J. Good, R. A. Gaskins, “Density estimation and bump-hunting by the penalized likelihood method exemplified by scattering and meteorite data,” J. Am. Stat. Assoc. 75, 42–73 (1980).
[CrossRef]

1979 (1)

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. (Oxford) 117, 219–232 (1979).
[CrossRef]

1977 (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1–38 (1977).

1971 (1)

I. J. Good, R. A. Gaskins, “Nonparametric roughness penalties for probability densities,” Biometrika 58, 255–277 (1971).
[CrossRef]

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, J. W. Sedat, “Three-dimensional structure of a polytene nucleus,” Nature (London) 302, 676–681 (1983).
[CrossRef]

Åslund, N.

Barends, P.

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. (Oxford) 117, 219–232 (1979).
[CrossRef]

Bille, J.

Blom, P.

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. (Oxford) 117, 219–232 (1979).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 333.

Brakenhoff, G. J.

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. (Oxford) 117, 219–232 (1979).
[CrossRef]

Carlini, A. R.

T. Wilson, A. R. Carlini, “Three-dimensional imaging in confocal imaging systems with finite sized detectors,” J. Microsc. (Oxford) 149, 51–66 (1988).
[CrossRef]

T. Wilson, A. R. Carlini, “Size of the detector in confocal imaging systems,” Opt. Lett. 12, 227–229 (1987).
[CrossRef] [PubMed]

Carlsson, K.

Concehllo, J.-A.

J.-A. Concehllo, “Three-dimensional reconstruction of noisy images from partially confocal scanning microscope,” Ph.D. dissertation (Dartmouth College, Hanover, N.H., 1990), pp. 145–146.

Conchello, J.-A.

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

J.-A. Conchello, E. W. Hansen, “Resolution and signal-to-noise trade-offs in confocal microscopy,” in Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FI2.

J.-A. Conchello, J. J. Kim, E. W. Hansen, “Digital postprocessing of partially confocal images: signal-to-noise ratio and depth discrimination,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 49.

J.-A. Conchello, E. W. Hansen, “Three-dimensional reconstruction of noisy confocal scanning microscope images,” in New Methods in Microscopy and Low-Light Imaging, J. E. Wampler, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1161, 279–285 (1989).

J.-A. Conchello, “Three-dimensional reconstruction of noisy images from partially confocal scanning microscope,” Ph.D. dissertation (Dartmouth College, Hanover, N.H., 1990), pp. 147–150.

Danielsson, P. E.

Davoust, J.

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. (Oxford) 138, 29–34 (1985).
[CrossRef]

DeGroot, M. H.

M. H. DeGroot, Probability and Statistics (Addison-Wesley, Reading, Mass., 1987), pp. 412–413.

Dempster, A. P.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1–38 (1977).

Erhardt, A.

Fay, F. S.

F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of α-actinin in single isolated smooth muscle cells,” J. Cell Biol. 96, 783–795 (1983).
[CrossRef] [PubMed]

Fogarty, K. E.

F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of α-actinin in single isolated smooth muscle cells,” J. Cell Biol. 96, 783–795 (1983).
[CrossRef] [PubMed]

Fujiwara, K.

F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of α-actinin in single isolated smooth muscle cells,” J. Cell Biol. 96, 783–795 (1983).
[CrossRef] [PubMed]

Gaskins, R. A.

I. J. Good, R. A. Gaskins, “Density estimation and bump-hunting by the penalized likelihood method exemplified by scattering and meteorite data,” J. Am. Stat. Assoc. 75, 42–73 (1980).
[CrossRef]

I. J. Good, R. A. Gaskins, “Nonparametric roughness penalties for probability densities,” Biometrika 58, 255–277 (1971).
[CrossRef]

Good, I. J.

I. J. Good, R. A. Gaskins, “Density estimation and bump-hunting by the penalized likelihood method exemplified by scattering and meteorite data,” J. Am. Stat. Assoc. 75, 42–73 (1980).
[CrossRef]

I. J. Good, R. A. Gaskins, “Nonparametric roughness penalties for probability densities,” Biometrika 58, 255–277 (1971).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 112.

Hansen, E. W.

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

J.-A. Conchello, E. W. Hansen, “Resolution and signal-to-noise trade-offs in confocal microscopy,” in Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FI2.

J.-A. Conchello, J. J. Kim, E. W. Hansen, “Digital postprocessing of partially confocal images: signal-to-noise ratio and depth discrimination,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 49.

J.-A. Conchello, E. W. Hansen, “Three-dimensional reconstruction of noisy confocal scanning microscope images,” in New Methods in Microscopy and Low-Light Imaging, J. E. Wampler, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1161, 279–285 (1989).

J. J. Kim, E. W. Hansen, “Psychophysical study of resolution in confocal scanning microscopy,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992) p. 220.

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), p. 354.

Holmes, T. J.

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]

Kaplan, R.

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. (Oxford) 138, 29–34 (1985).
[CrossRef]

Kaufman, L.

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–20 (1985).
[CrossRef]

Kim, J. J.

J. J. Kim, E. W. Hansen, “Psychophysical study of resolution in confocal scanning microscopy,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992) p. 220.

J.-A. Conchello, J. J. Kim, E. W. Hansen, “Digital postprocessing of partially confocal images: signal-to-noise ratio and depth discrimination,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 49.

Kimura, S.

Komitowski, D.

Laird, N. M.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1–38 (1977).

Lenz, R.

Liang, Z.

Liljeborg, A.

Majlöf, L.

Marsman, H. K. B.

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. (Oxford) 138, 29–34 (1985).
[CrossRef]

McNally, J. G.

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

Miller, M. I.

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

M. I. Miller, B. Roysam, “Bayesian image reconstruction for emission tomography incorporating Good’s roughness prior on massively parallel processors,” Proc. Natl. Acad. Sci. (USA) 88, 3223–3227 (1991).
[CrossRef]

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

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

Munakata, C.

Munakta, C.

Politte, D. G.

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

Preza, C.

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

Rees, D. D.

F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of α-actinin in single isolated smooth muscle cells,” J. Cell Biol. 96, 783–795 (1983).
[CrossRef] [PubMed]

Roysam, B.

M. I. Miller, B. Roysam, “Bayesian image reconstruction for emission tomography incorporating Good’s roughness prior on massively parallel processors,” Proc. Natl. Acad. Sci. (USA) 88, 3223–3227 (1991).
[CrossRef]

Rubin, D. B.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1–38 (1977).

Sedat, J. W.

D. A. Agard, J. W. Sedat, “Three-dimensional structure of a polytene nucleus,” Nature (London) 302, 676–681 (1983).
[CrossRef]

Shepp, L. A.

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–20 (1985).
[CrossRef]

Sheppard, C. J. R.

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

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), pp. 48–49.

Snyder, D. L.

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

Steltzer, E. H. K.

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. (Oxford) 138, 29–34 (1985).
[CrossRef]

Striker, R.

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. (Oxford) 138, 29–34 (1985).
[CrossRef]

Synder, D. L.

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

Thomas, L. J.

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

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

Vardi, Y.

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–20 (1985).
[CrossRef]

Wijnaendts van Resandt, R. W.

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. (Oxford) 138, 29–34 (1985).
[CrossRef]

Wilson, T.

T. Wilson, A. R. Carlini, “Three-dimensional imaging in confocal imaging systems with finite sized detectors,” J. Microsc. (Oxford) 149, 51–66 (1988).
[CrossRef]

T. Wilson, A. R. Carlini, “Size of the detector in confocal imaging systems,” Opt. Lett. 12, 227–229 (1987).
[CrossRef] [PubMed]

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), pp. 48–49.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 333.

Zajac, A.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), p. 354.

Zinser, G.

Appl. Opt. (3)

Biometrika (1)

I. J. Good, R. A. Gaskins, “Nonparametric roughness penalties for probability densities,” Biometrika 58, 255–277 (1971).
[CrossRef]

IEEE Trans. Med. Imaging (1)

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

IEEE Trans. Nucl. Sci. (1)

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

J. Am. Stat. Assoc. (2)

I. J. Good, R. A. Gaskins, “Density estimation and bump-hunting by the penalized likelihood method exemplified by scattering and meteorite data,” J. Am. Stat. Assoc. 75, 42–73 (1980).
[CrossRef]

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–20 (1985).
[CrossRef]

J. Cell Biol. (1)

F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of α-actinin in single isolated smooth muscle cells,” J. Cell Biol. 96, 783–795 (1983).
[CrossRef] [PubMed]

J. Microsc. (Oxford) (3)

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. (Oxford) 138, 29–34 (1985).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Image formation in the partially confocal scanning microscope.

Fig. 2
Fig. 2

Approximate support (footprint) of the partially confocal OTF H(ρ, ζ) for various detector apertures. Cutoff frequencies for the nonconfocal case are ζ o = NA2/2nλ (longitudinal) and 2ρ o = 2NA/λ (lateral). A missing cone about the longitudinal axis appears for detector apertures larger than 0.61λ/NA.

Fig. 3
Fig. 3

Depth discrimination and SNR as functions of detector aperature for a partially confocal microscope. Longitudinal resolution (dashed curve) is characterized by the HWHM of |H(0, ζ)|, i.e., the MTF along the longitudinal spatial frequency axis normalized by the confocal HWHM. The SNR (solid curve) is characterized by the integrated intensity at the in-focus plane [Eq. (5)], normalized by its nonconfocal value.

Fig. 4
Fig. 4

Cross plot of SNR (normalized by the nonconfocal value) and HWHM of the axial MTF (normalized by the confocal value) for different sizes of the detector aperture. The knee in the curve occurs at r d ~ 0.5.

Fig. 5
Fig. 5

Simulated images for a partially confocal microscope subject to Poisson noise. From left to right the columns show the following: practically confocal (r d = 0.1), r d = 0.305, r d = 0.61, r d = 1.22. From top to bottom each row shows the following: the optical slice immediately above the cross, the slice at which the cross is in focus, the slice at which the circle is in focus, and the slice immediately below the circle.

Fig. 6
Fig. 6

Reconstruction of a noisy partially confocal image after 10 iterations of the expectation-maximization algorithm. The columns and the rows have the same parameters as in Fig. 5.

Fig. 7
Fig. 7

Reconstruction of a noisy partially confocal image after 20 iterations of the expectation-maximization algorithm. The columns and the rows have the same parameters as in Fig. 5.

Fig. 8
Fig. 8

Reconstruction of a noisy partially confocal image after 40 iterations of the expectation-maximization algorithm. The columns and the rows have the same parameters as in Fig. 5.

Fig. 9
Fig. 9

Average behavior of postprocessing SNR and visibility versus iteration for a 20-realization ensemble of images formed from two parallel lines separated longitudinally by 400 nm. Ensemble averages for visibility (dashed curve) and SNR (solid curve) for an aperture with normalized radius r d = 0.61. The point of zero visibility (Sparrow resolution) occurs at ~ 350 iterations, with a posterior SNR of 10.5 dB. The SNR decreases approximately linearly on this log-log scale, with an average slope of roughly −5 dB/decade.

Fig. 10
Fig. 10

Average postprocessed SNR versus detector aperture radius for weak (solid curves) and strong (dashed curves) sources at Rayleigh (circles) and Sparrow (squares) resolution. The preprocessing SNR (triangles) is shown for comparison. The SNR degrades for detector apertures larger than 0.43. The SNR is linearly proportional to illumination intensity.

Fig. 11
Fig. 11

Mean number of iterations necessary to achieve Rayleigh (circles) and Sparrow (squares) resolution for the weak (solid curves) and the strong (dashed curves) sources. The amount of computation grows rapidly for apertures larger than 0.43.

Fig. 12
Fig. 12

Mean number of iterations versus mean SNR at Sparrow resolution (zero visibility) for the strong source. The SNR decreases and the computation increases rapidly for apertures larger than 0.43.

Fig. 13
Fig. 13

Visibility versus iteration for different aperture sizes (strong source). The visibility initially decreases, and the number of iterations for zero visibility (Sparrow) crossover increases with aperture size. The rate of visibility improvement slows as the aperture is enlarged.

Equations (17)

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I o ( x , x o ) = P l A o h o ( x o ) 2 s ( x - x o ) ,
I i ( x i , x o , x ) = P l A c A o h o ( x o ) 2 s ( x - x o ) h c ( x o - x i ) 2 ,
I d ( x ) = P l A c A o d x o h o ( x o ) 2 s ( x - x o ) d x i × h c ( x o - x i ) 2 p d ( x i , y i ) δ ( z i ) ,
I d ( x ) = P l A c A o ( s * * * h f ) ( x ) , h f ( x ) = h o ( x ) 2 ( p d * * h c 2 ) ( x ) ,
P ( ρ ; z ) = exp [ j 2 π z z o ( ρ ρ 0 ) ] circ ( ρ ρ 0 ) ,
circ ( ρ ) = { 1 for ρ 1 0 otherwise } .
g = γ A c A o P l s o d x i d y i p d ( x i , y i ) d x o d y o × h o ( x o , y o , 0 ) 2 h c ( x o - x i , y o - y i , 0 ) 2 .
g = 128 γ A c A o P l s o r o 2 r c 2 T π 3 [ A ( r d / r c ; Λ ) ] , A ( r ; Λ ) = 2 r 0 1 J 1 ( 4 π r ρ ) [ cos - 1 ρ - ρ ( 1 - ρ 2 ) 1 / 2 ] × { cos - 1 ρ Λ - ρ Λ [ 1 - ( ρ Λ ) 2 ] 1 / 2 } d ρ ,
H f ( ρ , ζ ) = H o ( ρ , ζ ) * * * [ P d ( ρ ) H c ( ρ , ζ ) ] ,
s ^ i ( k + 1 ) = s ^ i ( k ) ( H T d ( k ) ) i j ( H ) i j + ξ ( k ) z i ( k ) , d i ( k ) = g i ( H s ^ ( k ) ) i , z i ( k ) = [ - s i ln f ( s ) ] s = s ^ ( k ) ,
s ^ ( x ) = s ( x ) * * h f ( x , y , 0 ) ,
r d < pixel size λ / NA = 5 μ m / 128 514 nm / 1.4 = 0.106 ;
s ^ m ( k ) ( y ) = s ^ ( k ) ( 0 , y , z u ) + s ^ ( k ) ( 0 , y , z l ) 2 ,
V ( k ) ( y ) = s ^ m ( k ) ( y ) - s ^ ( k ) ( 0 , y , z m ) s ^ m ( k ) ( y ) + s ^ ( k ) ( 0 , y , z m ) ,
V ( k ) = 1 N y y V ( k ) ( y ) ,
SNR ( k ) = s ¯ ( k ) σ s ( k ) ,
s ¯ ( k ) = 1 N y y s ^ m ( k ) ( y ) = 1 2 N y y s ^ ( k ) ( 0 , y , z u ) + s ^ ( k ) ( 0 , y , z l ) , [ σ s ( k ) ] 2 = 2 2 N y y { [ s ^ ( k ) ( 0 , y , z u ) ] 2 + [ s ^ ( k ) ( 0 , y , z l ) ] 2 } - [ s ¯ ( k ) ] 2 .

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