# Filtering performance on Poisson noise with quadratic data-fidelity

We recently performed a work on signal filtering/component separation (sparse signal/trend/noise). The cost function contains:

1. A quadratic data fidelity term,
2. Some smoothed $\ell_1$ terms for sparsity and positivity promotion.

This was initially designed for Gaussian noise, and performed satisfactorily. Reviewers asked to test the algorithm on the same signals with Poisson noise, and it performed well also.

In many works concerned with Poisson-Gaussian noise mixtures, authors usually use more involved penalty functions and/or use variance stabilization transforms (such as Anscombe's).

Presenting the work to different signal processing folks, several colleagues (signal and image persons equally) said they were "not surprised" that a simple quadratic term would work well with Poisson noise too.

Since I am less a theoretician than a frequentist, and a bayesian (much) less than a frequentist, I do not understand the reason behind the relatively good behavior with both Gaussian and Poisson perturbations.

May a reader offer practical or theorical hints behind this "non surprise"?

• It's nice to see how your problem evolves! Wishing you the best, though I can not guarantee my answer is going to be of overly high value. – Marcus Müller Sep 18 '15 at 13:09

## 1 Answer

For large intensities / large "bins", i.e. "areas for which events are counted and accumulated", Poisson processes lead to nearly Gaussian distributed individual values -- basically, without trying to derive this, I think that's the application of the CLT on a lot of realization of a point process.

EDIT: Shameless plug: I really really like the wikipedia page on "Relationships among probability distributions; notice the arrow between the Poisson and the normal distribution:

• Poisson noise is also relatively well-behaved (unlike,e.g. Cauchy). As you say, almost any summation of welll-behaved random variables tends to start looking like a Gaussian random variable because of the CLT. That's why working with FFT data is so nice: almost regardless of the time-domain noise type, FFT bin (frequency-domain) noise can be regarded as almost exclusively Gaussian. – Peter K. Sep 18 '15 at 14:27
• I was aware of the possibility of Going from the Poisson distribution to the Gaussian, in the contexte of probability laws. I am still wondering about an interpretation in terms of cost functions (squared error) or transformations (Anscombe and related). – Laurent Duval Sep 19 '15 at 13:02