Estimation in Poisson Noise: Properties of the Conditional Mean\n Estimator

This paper considers estimation of a random variable in Poisson noise with\nsignal scaling coefficient and dark current as explicit parameters of the noise\nmodel. Specifically, the paper focuses on properties of the conditional mean\nestimator as a function of the scaling coefficient, the dark current parameter,\nthe distribution of the input random variable and channel realizations. With\nrespect to the scaling coefficient and the dark current, several identities in\nterms of derivatives are established. For example, it is shown that the\ngradient of the conditional mean estimator with respect to the scaling\ncoefficient and dark current parameter is proportional to the conditional\nvariance. Moreover, a score function is proposed and a Tweedie-like formula for\nthe conditional expectation is recovered. With respect to the distribution,\nseveral regularity conditions are shown. For instance, it is shown that the\nconditional mean estimator uniquely determines the input distribution.\nMoreover, it is shown that if the conditional expectation is close to a linear\nfunction in terms of mean squared error, then the input distribution is\napproximately gamma in the L\\'evy distance. Furthermore, sufficient and\nnecessary conditions for linearity are found. Interestingly, it is shown that\nthe conditional mean estimator cannot be linear when the dark current parameter\nof the Poisson noise is non-zero.\n