# Anscombe transform

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Anscombe transform

In statistics, the Anscombe transform (1948) is a variance-stabilizing transformation that transforms a random variable with a Poisson distribution into one with an approximately 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 have the noise of constant standard deviation so as to apply denoising algorithms in the extensively studied framework of Gaussian additive noise.

Definition

For the Poisson distribution the mean "m", and variance "v", are not independent: "m = v". The Anscombe transform

: $A:x o 2sqrt\left\{x+frac\left\{3\right\}\left\{8$

aims at transforming the data so that the variance is set approximately 1 whatever the mean. It transforms Poissonian data to approximately Gaussian data of standard deviation 1 and is valid provided that the mean value of the Poissonian data is more than 20.

Alternatives

There are many other possible variance stabilising transformations for the Poisson distribution. Bar-Lev and Enis (1988) report a family of such transformations which includes the Anscombe transform. Another member of the family is (Freeman & Tukey, 1950)

: $A:x o sqrt\left\{x+1\right\}+sqrt\left\{x\right\}.$

A simplified transformation is

: $A:x o 2sqrt\left\{x\right\}$

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

ee also

*Box-Cox transformation

References

*F.J. Anscombe, The transformation of Poisson, binomial and negative-binomial data. Biometrika, vol. 35, pp. 246-254, 1948.
* [http://citeseer.ist.psu.edu/cache/papers/cs/22670/http:zSzzSzjstarck.free.frzSzIEEE_SP_01.pdf/starck01astronomical.pdf An article about astronomical image and signal processing]

*Bar-Lev S.K., Enis P. (1988). On the classical choice of variance stabilising transformations and an application for a Poisson variate. Biometrika, 75 (4), 803-4.

*Freeman, M.F., Tukey, J.W. (1950). Transformations related to the angular and the square root. Ann. Math. Statist., 21, 607-11.

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