On the Distribution of Photon Counts with Censoring in Two-Photon Laser Scanning Microscopy

  • Burcin Simsek
  • Satish IyengarEmail author


We address the problem of counting emitted photons in two-photon laser scanning microscopy. Following a laser pulse, photons are emitted after exponentially distributed waiting times. Modeling the counting process is of interest because photon detectors have a dead period after a photon is detected that leads to an underestimate of the count of emitted photons. We describe a model which has a Poisson \((\alpha )\) number N of photons emitted, and a dead period \(\Delta \) that is standardized by the fluorescence time constant \(\tau (\delta = \Delta /\tau )\), and an observed count D. The estimate of \(\alpha \) determines the intensity of a single pixel in an image. We first derive the distribution of D and study its properties. We then use it to estimate \(\alpha \) and \(\delta \) simultaneously by maximum likelihood. We show that our results improve the signal-to-noise ratio, hence the quality of actual images.


Counter Dead period Exponential waiting times Grouping Inhomogeneous Poisson process Loss of information Maximum likelihood 

Mathematics Subject Classification

60G55 60K40 62E15 



We thank David Kleinfeld for his guidance on the physics background, especially in contrasting our work with those of Isbaner and his colleagues. We thank David Kleinfeld and Philbert Tsai (UC San Diego) for generating the imaging data. We also thank the reviewers for their helpful comments. Dr. Simsek was supported by an Andrew Mellon Predoctoral Fellowship at the University of Pittsburgh.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of PittsburghPittsburghUSA

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