# Anscombe transform

Standard deviation of the transformed Poisson random variable as a function of the mean $m$.

In statistics, the Anscombe transform, named after Francis Anscombe, is a variance-stabilizing transformation that transforms a random variable with a Poisson distribution into one with an approximately standard Gaussian distribution. The Anscombe transform is widely used in photon-limited imaging (astronomy, X-ray) where images naturally follow the Poisson law. The Anscombe transform is usually used to pre-process the data in order to make the standard deviation approximately constant. Then denoising algorithms designed for the framework of additive white Gaussian noise are used; the final estimate is then obtained by applying an inverse Anscombe transformation to the denoised data.

## Definition

For the Poisson distribution the mean $m$ and variance $v$ are not independent: $m = v$. The Anscombe transform[1]

$A:x \mapsto 2\sqrt{x+\tfrac{3}{8}} \,$

aims at transforming the data so that the variance is set approximately 1 whatever the mean. It transforms Poissonian data $x$ (with mean $m$) to approximately Gaussian data of mean $2\sqrt{m + 3/8} - 1/(4\sqrt{m})$ and standard deviation 1. This approximation is valid provided that $m$ is larger than 4.[citation needed]

## Inversion

When the Anscombe transform is used in denoising (i.e. when the goal is to obtain from $x$ an estimate of $m$), its inverse transform is also needed in order to return the variance-stabilized and denoised data $y$ to the original range. Applying the algebraic inverse

$A^{-1}:y \mapsto \left( \frac{y}{2} \right)^2 - \frac{3}{8}$

usually introduces undesired bias to the estimate of the mean $m$, because the forward square-root transform is not linear. Sometimes using the asymptotically unbiased inverse[1]

$y \mapsto \left( \frac{y}{2} \right)^2 - \frac{1}{8}$

mitigates the issue of bias, but this is not the case in photon-limited imaging, for which the exact unbiased inverse given by the implicit mapping[2]

$\operatorname{E} \left[ 2\sqrt{x+\tfrac{3}{8}} \mid m \right] = 2 \sum_{x=0}^{+\infty} \left( \sqrt{x+\tfrac{3}{8}} \cdot \frac{m^x e^{-m}}{x!} \right) \mapsto m$

should be used. A closed-form approximation of this exact unbiased inverse is[3]

$y \mapsto \frac{1}{4} y^2 + \frac{1}{4} \sqrt{\frac{3}{2}} y^{-1} - \frac{11}{8} y^{-2} + \frac{5}{8} \sqrt{\frac{3}{2}} y^{-3} - \frac{1}{8}.$

## Alternatives

There are many other possible variance-stabilizing transformations for the Poisson distribution. Bar-Lev and Enis report[4] a family of such transformations which includes the Anscombe transform. Another member of the family is the Freeman-Tukey transformation[5]

$A:x \mapsto \sqrt{x+1}+\sqrt{x}. \,$

A simplified transformation, obtained as the primitive of the reciprocal of the standard deviation of the data, is

$A:x \mapsto 2\sqrt{x} \,$

which, while it is not quite so good at stabilizing the variance, has the advantage of being more easily understood.

## Generalization

While the Anscombe transform is appropriate for pure Poisson data, in many applications the data presents also an additive Gaussian component. These cases are treated by a Generalized Anscombe transform[6] and its asymptotically unbiased or exact unbiased inverses.[7]

## References

1. ^ a b Anscombe, F. J. (1948), "The transformation of Poisson, binomial and negative-binomial data", Biometrika 35 (3–4): 246–254, doi:10.1093/biomet/35.3-4.246, JSTOR 2332343
2. ^ Mäkitalo, M.; Foi, A. (2011), "Optimal inversion of the Anscombe transformation in low-count Poisson image denoising", IEEE Transactions on Image Processing 20 (1): 99–109, doi:10.1109/TIP.2010.2056693
3. ^ Mäkitalo, M.; Foi, A. (2011), "A closed-form approximation of the exact unbiased inverse of the Anscombe variance-stabilizing transformation", IEEE Transactions on Image Processing 20 (9): 2697–2698, doi:10.1109/TIP.2011.2121085
4. ^ Bar-Lev, S. K.; Enis, P. (1988), "On the classical choice of variance stabilizing transformations and an application for a Poisson variate", Biometrika 75 (4): 803–804, doi:10.1093/biomet/75.4.803
5. ^ Freeman, M. F.; Tukey, J. W. (1950), "Transformations related to the angular and the square root", The Annals of Mathematical Statistics 21 (4): 607–611, doi:10.1214/aoms/1177729756, JSTOR 2236611
6. ^ Starck, J.L.; Murtagh, F.; Bijaoui, A. (1998). Image Processing and Data Analysis. Cambridge University Press. ISBN 9780521599146.
7. ^ Mäkitalo, M.; Foi, A. (2013), "Optimal inversion of the generalized Anscombe transformation for Poisson-Gaussian noise", IEEE Transactions on Image Processing 22 (1): 91–103, doi:10.1109/TIP.2012.2202675