Abstract

We present in this article the use of probabilistic background constraints in astronomical image deconvolution to approach a solution as an interval estimate. We elaborate our objective—the interval estimate of the unknown object from observed data and our approach—Monte Carlo experiment and analysis of marginal distributions of image values. One-dimensional observation and deconvolution using the proposed approach are simulated. Confidence intervals revealing the uncertainties due to the background constraint are calculated and significance levels for sources retrieved from restored images are provided.

© 2012 Optical Society of America

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References

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  1. A. Meister, Deconvolution Problems in Nonparametric Statistics (Springer-Verlag, 2009), p. 2.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: A review,” Publ. Astron. Soc. Pac. 114, 1051–1069 (2002).
    [CrossRef]
  12. D. N. Esch, A. Connors, M. Karovska, and D. A. van Dyk, “An image restoration technique with error estimates,” Astrophys. J. 610, 1213–1227 (2004).
    [CrossRef]
  13. C. Byrne, “Iterative algorithms for deblurring and deconvolution with constraints,” Inverse Probl. 14, 1455 (1998).
    [CrossRef]
  14. A. Lannes, S. Roques, and M. J. Casanove, “Stabilized reconstruction in signal and image processing—i. partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
    [CrossRef]
  15. P. Hansen, “Regularization tools,” Numer. Algorithms 6, 1–35 (1994).
    [CrossRef]
  16. T.-P. Li and M. Wu, “A direct restoration method for spectral and image analysis,” Astrophys. Space Sci. 206, 91–102 (1993).
    [CrossRef]
  17. N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, “Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech. 69, 260–266 (2006).
    [CrossRef]
  18. G. M. P. van Kempen and L. J. van Vliet, “Background estimation in nonlinear image restoration,” J. Opt. Soc. Am. A 17, 425–433 (2000).
    [CrossRef]
  19. G. M. Wing, A Primer on Integral Equations of the First Kind: The Problem of Deconvolution and Unfolding (Society for Industrial Mathematics, 1991), pp. 92–103.
  20. I. S. McLean, Electronic Imaging in Astronomy: Detectors and Instrumentation, 2nd ed. (Springer, 2008), pp. 241–313.
  21. J.-L. Starck and F. Murtagh, Astronomical Image and Data Analysis (Springer-Verlag, 2006), pp. 40–41.

2007

2006

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, “Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech. 69, 260–266 (2006).
[CrossRef]

2005

R. Puetter, T. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139–194 (2005).
[CrossRef]

2004

D. N. Esch, A. Connors, M. Karovska, and D. A. van Dyk, “An image restoration technique with error estimates,” Astrophys. J. 610, 1213–1227 (2004).
[CrossRef]

2002

J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: A review,” Publ. Astron. Soc. Pac. 114, 1051–1069 (2002).
[CrossRef]

2000

1998

C. Byrne, “Iterative algorithms for deblurring and deconvolution with constraints,” Inverse Probl. 14, 1455 (1998).
[CrossRef]

1995

S. Zaroubi, Y. Hoffman, K. B. Fisher, and O. Lahav, “Wiener reconstruction of the large-scale structure,” Astrophys. J. 449, 446 (1995).
[CrossRef]

1994

P. Hansen, “Regularization tools,” Numer. Algorithms 6, 1–35 (1994).
[CrossRef]

1993

T.-P. Li and M. Wu, “A direct restoration method for spectral and image analysis,” Astrophys. Space Sci. 206, 91–102 (1993).
[CrossRef]

1988

1987

A. Lannes, S. Roques, and M. J. Casanove, “Stabilized reconstruction in signal and image processing—i. partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

1974

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 417–426 (1974).

1972

1966

1951

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
[CrossRef]

Ayers, G. R.

Blanc-Feraud, L.

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, “Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech. 69, 260–266 (2006).
[CrossRef]

Byrne, C.

C. Byrne, “Iterative algorithms for deblurring and deconvolution with constraints,” Inverse Probl. 14, 1455 (1998).
[CrossRef]

Casanove, M. J.

A. Lannes, S. Roques, and M. J. Casanove, “Stabilized reconstruction in signal and image processing—i. partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

Connors, A.

D. N. Esch, A. Connors, M. Karovska, and D. A. van Dyk, “An image restoration technique with error estimates,” Astrophys. J. 610, 1213–1227 (2004).
[CrossRef]

Dainty, J. C.

Dey, N.

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, “Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech. 69, 260–266 (2006).
[CrossRef]

Esch, D. N.

D. N. Esch, A. Connors, M. Karovska, and D. A. van Dyk, “An image restoration technique with error estimates,” Astrophys. J. 610, 1213–1227 (2004).
[CrossRef]

Fisher, K. B.

S. Zaroubi, Y. Hoffman, K. B. Fisher, and O. Lahav, “Wiener reconstruction of the large-scale structure,” Astrophys. J. 449, 446 (1995).
[CrossRef]

Gosnell, T.

R. Puetter, T. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139–194 (2005).
[CrossRef]

Hansen, P.

P. Hansen, “Regularization tools,” Numer. Algorithms 6, 1–35 (1994).
[CrossRef]

Harris, L.

Hoffman, Y.

S. Zaroubi, Y. Hoffman, K. B. Fisher, and O. Lahav, “Wiener reconstruction of the large-scale structure,” Astrophys. J. 449, 446 (1995).
[CrossRef]

Högbom, J. A.

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 417–426 (1974).

James, S.

Kam, Z.

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, “Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech. 69, 260–266 (2006).
[CrossRef]

Karovska, M.

D. N. Esch, A. Connors, M. Karovska, and D. A. van Dyk, “An image restoration technique with error estimates,” Astrophys. J. 610, 1213–1227 (2004).
[CrossRef]

Lahav, O.

S. Zaroubi, Y. Hoffman, K. B. Fisher, and O. Lahav, “Wiener reconstruction of the large-scale structure,” Astrophys. J. 449, 446 (1995).
[CrossRef]

Landweber, L.

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
[CrossRef]

Lannes, A.

A. Lannes, S. Roques, and M. J. Casanove, “Stabilized reconstruction in signal and image processing—i. partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

Li, T.-P.

T.-P. Li and M. Wu, “A direct restoration method for spectral and image analysis,” Astrophys. Space Sci. 206, 91–102 (1993).
[CrossRef]

Löfdahl, M. G.

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

McLean, I. S.

I. S. McLean, Electronic Imaging in Astronomy: Detectors and Instrumentation, 2nd ed. (Springer, 2008), pp. 241–313.

Meister, A.

A. Meister, Deconvolution Problems in Nonparametric Statistics (Springer-Verlag, 2009), p. 2.

Murtagh, F.

J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: A review,” Publ. Astron. Soc. Pac. 114, 1051–1069 (2002).
[CrossRef]

J.-L. Starck and F. Murtagh, Astronomical Image and Data Analysis (Springer-Verlag, 2006), pp. 40–41.

Olivo-Marin, J.-C.

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, “Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech. 69, 260–266 (2006).
[CrossRef]

Pantin, E.

J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: A review,” Publ. Astron. Soc. Pac. 114, 1051–1069 (2002).
[CrossRef]

Puetter, R.

R. Puetter, T. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139–194 (2005).
[CrossRef]

Richardson, W. H.

Roques, S.

A. Lannes, S. Roques, and M. J. Casanove, “Stabilized reconstruction in signal and image processing—i. partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

Roux, P.

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, “Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech. 69, 260–266 (2006).
[CrossRef]

Starck, J. L.

J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: A review,” Publ. Astron. Soc. Pac. 114, 1051–1069 (2002).
[CrossRef]

Starck, J.-L.

J.-L. Starck and F. Murtagh, Astronomical Image and Data Analysis (Springer-Verlag, 2006), pp. 40–41.

van Dyk, D. A.

D. N. Esch, A. Connors, M. Karovska, and D. A. van Dyk, “An image restoration technique with error estimates,” Astrophys. J. 610, 1213–1227 (2004).
[CrossRef]

van Kempen, G. M. P.

van Vliet, L. J.

Wing, G. M.

G. M. Wing, A Primer on Integral Equations of the First Kind: The Problem of Deconvolution and Unfolding (Society for Industrial Mathematics, 1991), pp. 92–103.

Wu, M.

T.-P. Li and M. Wu, “A direct restoration method for spectral and image analysis,” Astrophys. Space Sci. 206, 91–102 (1993).
[CrossRef]

Yahil, A.

R. Puetter, T. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139–194 (2005).
[CrossRef]

Zaroubi, S.

S. Zaroubi, Y. Hoffman, K. B. Fisher, and O. Lahav, “Wiener reconstruction of the large-scale structure,” Astrophys. J. 449, 446 (1995).
[CrossRef]

Zerubia, J.

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, “Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech. 69, 260–266 (2006).
[CrossRef]

Zimmer, C.

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, “Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech. 69, 260–266 (2006).
[CrossRef]

Am. J. Math.

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
[CrossRef]

Annu. Rev. Astron. Astrophys.

R. Puetter, T. Gosnell, and A. Yahil, “Digital image reconstruction: deblurring and denoising,” Annu. Rev. Astron. Astrophys. 43, 139–194 (2005).
[CrossRef]

Appl. Opt.

Astron. Astrophys.

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 417–426 (1974).

Astron. J.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Astrophys. J.

D. N. Esch, A. Connors, M. Karovska, and D. A. van Dyk, “An image restoration technique with error estimates,” Astrophys. J. 610, 1213–1227 (2004).
[CrossRef]

S. Zaroubi, Y. Hoffman, K. B. Fisher, and O. Lahav, “Wiener reconstruction of the large-scale structure,” Astrophys. J. 449, 446 (1995).
[CrossRef]

Astrophys. Space Sci.

T.-P. Li and M. Wu, “A direct restoration method for spectral and image analysis,” Astrophys. Space Sci. 206, 91–102 (1993).
[CrossRef]

Inverse Probl.

C. Byrne, “Iterative algorithms for deblurring and deconvolution with constraints,” Inverse Probl. 14, 1455 (1998).
[CrossRef]

J. Mod. Opt.

A. Lannes, S. Roques, and M. J. Casanove, “Stabilized reconstruction in signal and image processing—i. partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Microsc. Res. Tech.

N. Dey, L. Blanc-Feraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, “Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution,” Microsc. Res. Tech. 69, 260–266 (2006).
[CrossRef]

Numer. Algorithms

P. Hansen, “Regularization tools,” Numer. Algorithms 6, 1–35 (1994).
[CrossRef]

Opt. Lett.

Publ. Astron. Soc. Pac.

J. L. Starck, E. Pantin, and F. Murtagh, “Deconvolution in astronomy: A review,” Publ. Astron. Soc. Pac. 114, 1051–1069 (2002).
[CrossRef]

Other

A. Meister, Deconvolution Problems in Nonparametric Statistics (Springer-Verlag, 2009), p. 2.

G. M. Wing, A Primer on Integral Equations of the First Kind: The Problem of Deconvolution and Unfolding (Society for Industrial Mathematics, 1991), pp. 92–103.

I. S. McLean, Electronic Imaging in Astronomy: Detectors and Instrumentation, 2nd ed. (Springer, 2008), pp. 241–313.

J.-L. Starck and F. Murtagh, Astronomical Image and Data Analysis (Springer-Verlag, 2006), pp. 40–41.

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Figures (11)

Fig. 1.
Fig. 1.

Simulated object O(x). Sources: 1000photons/s at 200 pixel, 4000photons/s at 400 pixel, 10,000photons/s at 600 pixel, and 5000photons/s at 800 pixel. The intensity of the background is 1000photons/s.

Fig. 2.
Fig. 2.

Simulated PSF P(x).

Fig. 3.
Fig. 3.

Simulated observed data.

Fig. 4.
Fig. 4.

Background data IR(x).

Fig. 5.
Fig. 5.

Object estimate O^MAP. Deconvolving with 2500 MAP iterations, using B^=IR(x)¯ as the background constraint.

Fig. 6.
Fig. 6.

Convergence of mean square error (MSE). It shows the MSE between two adjacent iteration steps.

Fig. 7.
Fig. 7.

Convergence of posterior probability density function (PDF). It shows the logarithm to base 10 of the posterior PDF of the estimate in each iteration step.

Fig. 8.
Fig. 8.

Normalized occurrence frequency of image values (backgrounds are truncated) on x=310pixel (top), x=400pixel (middle), and x=600pixel (bottom).

Fig. 9.
Fig. 9.

Medians of marginal distributions of image values for all the pixels as well as their 99.9% (about 3.3σ) CIs in images. Medians, upper limits, and lower limits are indicated by solid line, dashes, and dots, respectively. Data series are shifted, smoothed, and plotted with logarithmic scaled y axis to highlight differences between different series especially on faint structures.

Fig. 10.
Fig. 10.

Null distribution of flux restored from random background data sets with the same means and variances as the background data extracted from the observed data. The dashed line with a circle, the dotted line with a cross, dash-and-dotted line with a diamond, and the dashed line with a square denote the 95%, 99%, 99.5%, and 99.9% cumulative probabilities, respectively.

Fig. 11.
Fig. 11.

Fluxes (y coordinates) and positions (x coordinates) of retrieved sources, 99.9% CIs of both fluxes and positions, as well as the 0.01% significance level for fluxes of retrieved sources. Blue crosses indicate the fluxes and of sources retrieved from Monte Carlo experiment using probabilistic background constraints, while error bars along x axis indicate central 99.9% CIs of positions and error bars along y axis indicate central 99.9% CIs of fluxes. Green solid lines with circles indicate sources in the simulated object. Red dashed lines with triangles indicate sources retrieved from a single run of MAP deconvolution. Black dotted line indicates the 0.01% significance level. Here the significance level is 142counts/s.

Equations (4)

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

I(x)=O(x1)P(xx1)dx1+N(x)=(O*P)(x)+N(x),
O^(x)={O^(x)O^(x)BBotherwise,
ϵ(k)=(OMAP(k)(x)OMAP(k1)(x))2¯,
p(OMAP(k)|I)=Πx(OMAP(k)*P)I(x)(x)e(OMAP(k)*P)(x)I(x)!

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