# Find the noise settings to reproduce it

I work on images that contain shot noise (Poisson) acquired from a microscop. On few images, I have a "flat" zone that is supposed to have the exact same intensity, but it's not because of the noise. I was thinking that I could use this zone to estimate the quantity of noise, and then estimate the parameters K and Lambda of the Poisson equation.

Is there a method to do it?

What you'd often be looking for would the variance – the expectation of the squared difference to the mean value in that region. If the noise you add to the actual image is zero-mean, then the mean of the flat region is (in expectation) the actual intensity, and subtracting it from all pixels and squaring the result would give you the noise power.

However, shot noise is not zero-mean:

each flat-zone image pixel $$v_i$$ follows an actual-intensity-"offset" poisson distribution

$$v_i = m + n_i,\quad n_i\sim\text{Pois}(\lambda)$$

with $$m$$ being the actual intensity of the object, $$n_i$$ being shot noise samples, and $$\lambda$$ the intensity of the Poisson variable.

Now, since shot noise and actual image are independent,

$$\mathbb E(v_i) = \mathbb E(m + n_i) = \mathbb E(m) + \mathbb E(n_i) = m + \lambda\text.$$

Sadly, since we don't know the actual $$m$$ a priori, we'll have to dig one moment deeper:

\begin{align} \DeclareMathOperator{\Var}{Var} \Var (v_i) &= \mathbb E\left((m + n_i- \mathbb E(v_i))^2\right)\\ & = \mathbb E\left((m + n_i- m - \lambda)^2\right) \\ & = \mathbb E\left(( n_i - \lambda)^2\right) \\ &=: \Var(n_i)\\ &=\lambda \end{align}

So, with the Variance of your observation sample as estimator for the $$\lambda$$ of your Poisson variable, you get the sole statistical property of your shot noise for free. If you want to know the actual flat color, you'd also subtract that from the average of the flat region.

Poisson noise estimation can be complicated. I remember two papers, doing that in a wavelet domain. I still have some Matlab code for the oldest one, if needed:

In this paper, we present the estimation of Poisson intensity based on hypothesis testing in the wavelet domain for any dimensional data. The testing framework for wavelet-based Poisson intensity estimation was first introduced by Kolaczyk, where a thresholding estimator, which realizes the hypothesis testing, is derived for Haar wavelet coefficients. Here we propose for the same wavelet a new thresholding estimator which is based on Fisher's normal approximation. Furthermore, we have demonstrated that non-normalized biorthogonal Haar coefficients converge in distribution to non-normalized Haar coefficients as the scale increases. This allows us to directly apply the threshold in the biorthogonal Haar domain. Therefore we gain, by using this more regular wavelet, a reconstruction with less artifacts. Simulations show that on a wide range of intensity types, the proposed threshold combined with undecimated biorthogonal Haar transform gives one of the best estimation result compared with existing estimators of various kinds. Finally, potential applicability of our approach is illustrated on astronomical data.

The ubiquity of integrating detectors in imaging and other applications implies that a variety of real-world data are well modeled as Poisson random variables whose means are in turn proportional to an underlying vector-valued signal of interest. In this article, we first show how the so-called Skellam distribution arises from the fact that Haar wavelet and filterbank transform coefficients corresponding to measurements of this type are distributed as sums and differences of Poisson counts. We then provide two main theorems on Skellam shrinkage, one showing the near-optimality of shrinkage in the Bayesian setting and the other providing for unbiased risk estimation in a frequentist context. These results serve to yield new estimators in the Haar transform domain, including an unbiased risk estimate for shrinkage of Haar-Fisz variance-stabilized data, along with accompanying low-complexity algorithms for inference. We conclude with a simulation study demonstrating the efficacy of our Skellam shrinkage estimators both for the standard univariate wavelet test functions as well as a variety of test images taken from the image processing literature, confirming that they offer some performance improvements over existing alternatives.