## Abstract

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.

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## Acknowledgements

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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## Appendix: Proofs

### Appendix: Proofs

*Proof of Lemma 1:* Suppose that \(d = 2c\) is even; we omit the details for the case of odd *d*, which are similar. To have a concise presentation, let \(\theta = e^{-\delta }\) so that \(A_{k}= A_{k}(\theta )\), and equation (1) becomes

Multiply both sides of (5) by the least common denominator

to get the polynomials \(\{f_{i}: 1 \le i \le d+1\}\) in the first column of Table 1. The cumulative sums \(F_{3} = f_{1}+f_{2}\) and \(\{F_{k} = F_{k-1} + f_{k-1}: 4 \le k \le d\}\) are in the second column there. Using the fact that for \(i < j\), \(A_{i} - A_{j} = -\theta ^{i} A_{j-i}\), we start with

the other cumulative sums given in Table 1 follow similarly, ending with the \(F_{d+1} = -f_{d+1}\) term.

*Proof of Theorem 1:* Define the functions

and note that

Thus, in equation (2), integrate out \(t_{d}\) to get

Next, integrate out \(t_{d-1}\) in (2) to get

The last expression is obtained by using Lemma 1, to combine the coefficients of the \(U_{1}\) terms. Next, integrate out \(t_{d-2}\) down to \(t_{2}\) — each time applying Lemma 1 — to get

Finally, integrate out \(t_{1}\) to get

*Proof of Corollary 1:* First we define the vectors \(\varvec{p}_{d} = (p_{0},\ldots ,p_{d})'\) and \(\varvec{\zeta }_{d} = (\zeta _{0},\ldots ,\zeta _{d})'\). We can restate the result in Theorem 1 as

and the result in the Corollary as

thus, we must show that \(Q_{d}R_{d}=I\). Note that for \(0 \le a,b \le d\),

and

Therefore, their product \(S_{d} = (s_{ab})\), where

By inspection, \(s_{ab} =0\) for \(b >a\), and \(s_{aa}=1\). Next, let \(j=k-b\) in this expression and use Lemma 1 with \(d=a-b\) to get \(s_{ab} = 0\) for \(b<a\).

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Simsek, B., Iyengar, S. On the Distribution of Photon Counts with Censoring in Two-Photon Laser Scanning Microscopy.
*J Math Imaging Vis* **58, **47–56 (2017). https://doi.org/10.1007/s10851-016-0690-4

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### Keywords

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

### Mathematics Subject Classification

- 60G55
- 60K40
- 62E15