Abstract

This paper addresses parameter estimation for an optical transient signal when the received data has been right-censored. We develop an expectation-maximization (EM) algorithm to estimate the amplitude of a Poisson intensity with a known shape in the presence of additive background counts, where the measurements are subject to saturation effects. We compare the results of our algorithm with those of an EM algorithm that is unaware of the censoring.

© 2018 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Object recognition based on impulse restoration with use of the expectation-maximization algorithm

Ahmad Abu-Naser, Nikolas P. Galatsanos, Miles N. Wernick, and Dan Schonfeld
J. Opt. Soc. Am. A 15(9) 2327-2340 (1998)

Amplitude and phase retrieval with simultaneous diversity estimation using expectation maximization

Joyce Fang and Dmitry Savransky
J. Opt. Soc. Am. A 35(2) 293-300 (2018)

Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm

Nannan Cao, Arye Nehorai, and Mathews Jacob
Opt. Express 15(21) 13695-13708 (2007)

References

  • View by:
  • |
  • |
  • |

  1. H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
    [Crossref]
  2. D. L. Snyder, R. L. White, and A. M. Hammoud, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
    [Crossref]
  3. A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1–38 (1977).
  4. S. B. Lee and C. Park, “Parameter estimation from censored samples using the expectation-maximization algorithm,” (Department of Mathematical Sciences, Clemson University, 2003).
  5. R. D. Palkki and A. D. Lanterman, “Minimum description length approach to detecting chemicals via their Raman spectra,” Opt. Eng. 50, 083601 (2011).
    [Crossref]
  6. C. Viwatwongkasem, “EM algorithm for truncated and censored Poisson likelihoods,” Procedia Comput. Sci. 86, 240–243 (2016).
    [Crossref]
  7. M. J. Kagie, “Time-of-arrival estimation for saturated optical transients using censored probabilistic models,” Master’s thesis (Georgia Institute of Technology, 2016).
  8. D. L. Snyder, C. W. Helstrom, A. D. Lanterman, R. L. White, and M. Faisal, “Compensation for readout noise in CCD images,” J. Opt. Soc. Am. A 12, 272–283 (1995).
    [Crossref]

2016 (1)

C. Viwatwongkasem, “EM algorithm for truncated and censored Poisson likelihoods,” Procedia Comput. Sci. 86, 240–243 (2016).
[Crossref]

2011 (1)

R. D. Palkki and A. D. Lanterman, “Minimum description length approach to detecting chemicals via their Raman spectra,” Opt. Eng. 50, 083601 (2011).
[Crossref]

1995 (1)

1993 (1)

1977 (1)

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

Blakeslee, R. J.

H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
[Crossref]

Boccippio, D. J.

H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
[Crossref]

Boeck, W. L.

H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
[Crossref]

Buechler, D. E.

H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
[Crossref]

Christian, H. J.

H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
[Crossref]

Dempster, A. P.

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

Driscoll, K. T.

H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
[Crossref]

Faisal, M.

Goodman, S. J.

H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
[Crossref]

Hall, J. M.

H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
[Crossref]

Hammoud, A. M.

Helstrom, C. W.

Kagie, M. J.

M. J. Kagie, “Time-of-arrival estimation for saturated optical transients using censored probabilistic models,” Master’s thesis (Georgia Institute of Technology, 2016).

Koshak, W. J.

H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
[Crossref]

Laird, N. M.

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

Lanterman, A. D.

R. D. Palkki and A. D. Lanterman, “Minimum description length approach to detecting chemicals via their Raman spectra,” Opt. Eng. 50, 083601 (2011).
[Crossref]

D. L. Snyder, C. W. Helstrom, A. D. Lanterman, R. L. White, and M. Faisal, “Compensation for readout noise in CCD images,” J. Opt. Soc. Am. A 12, 272–283 (1995).
[Crossref]

Lee, S. B.

S. B. Lee and C. Park, “Parameter estimation from censored samples using the expectation-maximization algorithm,” (Department of Mathematical Sciences, Clemson University, 2003).

Mach, D. M.

H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
[Crossref]

Palkki, R. D.

R. D. Palkki and A. D. Lanterman, “Minimum description length approach to detecting chemicals via their Raman spectra,” Opt. Eng. 50, 083601 (2011).
[Crossref]

Park, C.

S. B. Lee and C. Park, “Parameter estimation from censored samples using the expectation-maximization algorithm,” (Department of Mathematical Sciences, Clemson University, 2003).

Rubin, D. B.

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

Snyder, D. L.

Stewart, M. F.

H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
[Crossref]

Viwatwongkasem, C.

C. Viwatwongkasem, “EM algorithm for truncated and censored Poisson likelihoods,” Procedia Comput. Sci. 86, 240–243 (2016).
[Crossref]

White, R. L.

J. Opt. Soc. Am. A (2)

J. R. Stat. Soc. B (1)

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

Opt. Eng. (1)

R. D. Palkki and A. D. Lanterman, “Minimum description length approach to detecting chemicals via their Raman spectra,” Opt. Eng. 50, 083601 (2011).
[Crossref]

Procedia Comput. Sci. (1)

C. Viwatwongkasem, “EM algorithm for truncated and censored Poisson likelihoods,” Procedia Comput. Sci. 86, 240–243 (2016).
[Crossref]

Other (3)

M. J. Kagie, “Time-of-arrival estimation for saturated optical transients using censored probabilistic models,” Master’s thesis (Georgia Institute of Technology, 2016).

S. B. Lee and C. Park, “Parameter estimation from censored samples using the expectation-maximization algorithm,” (Department of Mathematical Sciences, Clemson University, 2003).

H. J. Christian, R. J. Blakeslee, D. J. Boccippio, W. L. Boeck, D. E. Buechler, K. T. Driscoll, S. J. Goodman, J. M. Hall, W. J. Koshak, D. M. Mach, and M. F. Stewart, “Global frequency and distribution of lightning as observed from space by the optical transient detector,” J. Geophys. Res.108, ACL 4-1–ACL 4-15 (2003).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1. Summands for (top) a low-noise, partially saturated, low-photon-count scenario and (bottom) a high-noise, partially saturated, high-photon-count scenario.
Fig. 2.
Fig. 2. Template of a lightning strike (μ).
Fig. 3.
Fig. 3. Example reconstructions for (top) an unsaturated scenario and (bottom) a high-noise, partially saturated, low-photon-count scenario.
Fig. 4.
Fig. 4. Reconstruction examples for (top) a low-noise, mostly saturated, high-photon-count scenario and (bottom) a moderate-noise, partially saturated, low-photon-count scenario.
Fig. 5.
Fig. 5. MC simulation comparing the standard Poisson EM algorithm and the censored Poisson EM algorithm.

Tables (1)

Tables Icon

Table 1. Table of Scenarios for Conditional Expectation Evaluation

Equations (15)

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

xi=min(yi,T),1iN,
λi=αμi+λ0,
YiPoisson(λi),
I(α|x)=i=1nhi,where  λi=αμi+λ0andhi={xiln(λi)λiln(xi!)forxi<T,ln(1eλij=0Tλijj!)forxiT,
x=xS+xB,
yi=yiS+yiB,for  iC,
C(α)=i=1Nαμi+iNXiSln(αμi)+iCYiSln(αμi).
Q(α|α(j))=αi=1Nμi+Cln(α),
C=iNE[XiS|xi;α(j)]+iCE[YiS|xi;α(j)].
α(j+1)=Ci=1Nμi.
fXiS(xiS|xi;α(j))=Pr[XiS=xisXiB=xixiS;α(j)]Pr[Xi=xi;α(j)]=(xixis)(α(j)μiα(j)μi+λ0)xiS(1α(j)μiα(j)μi+λ0)xixiS.
E[XiS|xi;α(j)]=xiα(j)μiα(j)μi+λ0.
fYiS(yiS|xi;α(j))=Pr[YiS=yisYiBTyiS;α(j)]Pr[YiT;α(j)]=[eα(j)μi(α(j)μi)yiS/yiS][1F(TyiS1;λ0)]1F(T1;α(j)μi+λ0),
E[YiS|xi;α(j)]=yiS=0yiSfYiS(yiS|xi;α(j)).
α(j+1)=1i=1nμi(i=1nxiμiα(j)α(j)μi+λ0).