- Anscombe transform
In

statistics , the**Anscombe transform**(1948) is a variance-stabilizing transformation that transforms arandom variable with aPoisson distribution into one with an approximately Gaussian distribution. The Anscombe transform is widely used inphoton-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: $A:x\; o\; 2sqrt\{x+frac\{3\}\{8$

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)

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

A simplified transformation is

: $A:x\; o\; 2sqrt\{x\}$

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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