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 for large enough mean; for mean zero, the variance is still zero.

It transforms Poissonian data $x$ (with mean $m$ ) to approximately Gaussian data of mean $2{\sqrt {m+{\tfrac {3}{8}}}}-{\tfrac {1}{4\,m^{1/2}}}+O\left({\tfrac {1}{m^{3/2}}}\right)$ and standard deviation $1+O\left({\tfrac {1}{m^{2}}}\right)$ . This approximation is good provided that $m$ is larger than 4. For a transformed variable of the form $2{\sqrt {x+c}}$ , the expression for the variance has an additional term ${\frac {{\tfrac {3}{8}}-c}{m}}$ ; it is reduced to zero at $c={\tfrac {3}{8}}$ , which is exactly the reason why this value was picked.

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

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. Indeed, from the delta method,

$V[2{\sqrt {x}}]\approx \left({\frac {d(2{\sqrt {m}})}{dm}}\right)^{2}V[x]=\left({\frac {1}{\sqrt {m}}}\right)^{2}m=1$ .

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 2 Anscombe, F. J. (1948), "The transformation of Poisson, binomial and negative-binomial data", Biometrika, 35 (3–4), pp. 246–254, JSTOR 2332343, doi:10.1093/biomet/35.3-4.246
↑ Mäkitalo, M.; Foi, A. (2011), "Optimal inversion of the Anscombe transformation in low-count Poisson image denoising", IEEE Transactions on Image Processing, 20 (1), pp. 99–109, doi:10.1109/TIP.2010.2056693
↑ 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), pp. 2697–2698, doi:10.1109/TIP.2011.2121085
↑ 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), pp. 803–804, doi:10.1093/biomet/75.4.803
↑ Freeman, M. F.; Tukey, J. W. (1950), "Transformations related to the angular and the square root", The Annals of Mathematical Statistics, 21 (4), pp. 607–611, JSTOR 2236611, doi:10.1214/aoms/1177729756
↑ Starck, J.L.; Murtagh, F.; Bijaoui, A. (1998). Image Processing and Data Analysis. Cambridge University Press. ISBN 9780521599146.
↑ Mäkitalo, M.; Foi, A. (2013), "Optimal inversion of the generalized Anscombe transformation for Poisson-Gaussian noise", IEEE Transactions on Image Processing, 22 (1), pp. 91–103, doi:10.1109/TIP.2012.2202675