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

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 $$(\alpha )$$(α) number N of photons emitted, and a dead period $$\Delta $$Δ that is standardized by the fluorescence time constant $$\tau (\delta = \Delta /\tau )$$τ(δ=Δ/τ), and an observed count D. The estimate of $$\alpha $$α 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 $$\alpha $$α and $$\delta $$δ simultaneously by maximum likelihood. We show that our results improve the signal-to-noise ratio, hence the quality of actual images.