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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]

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

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 $${\displaystyle 2{\sqrt {m+{\tfrac {3}{8}}}}-{\tfrac {1}{4\,m^{1/2}}}+O\left({\tfrac {1}{m^{3/2}}}\right)}$$ and standard deviation $${\displaystyle 1+O\left({\tfrac {1}{m^{2}}}\right)}$$. This approximation is good provided that m is larger than 4.[2]

For a transformed variable of the form $${\displaystyle 2{\sqrt {x+c}}}$$, the expression for the variance has an additional term $${\displaystyle {\frac {{\tfrac {3}{8}}-c}{m}}};$$ it is reduced to zero at $${\displaystyle c={\tfrac {3}{8}}}$$, 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[3]

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

should be used. A closed-form approximation of this exact unbiased inverse is[4]

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

Alternatives

There are many other possible variance-stabilizing transformations for the Poisson distribution. Bar-Lev and Enis report[5] a family of such transformations which includes the Anscombe transform. Another member of the family is the Freeman-Tukey transformation[6]

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

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,

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

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[7] and its asymptotically unbiased or exact unbiased inverses.[8]

Variance-stabilizing transformation
Box–Cox transformation

References

Anscombe, F. J. (1948), "The transformation of Poisson, binomial and negative-binomial data", Biometrika, [Oxford University Press, Biometrika Trust], 35 (3–4), pp. 246–254, doi:10.1093/biomet/35.3-4.246, JSTOR 2332343
"POISSON NOISE REMOVAL FROM FLUORESCENCE IMAGES" (PDF).
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, Bibcode:2011ITIP...20...99M, CiteSeerX 10.1.1.219.6735, doi:10.1109/TIP.2010.2056693, PMID 20615809
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, Bibcode:2011ITIP...20.2697M, 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, doi:10.1214/aoms/1177729756, JSTOR 2236611
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, Bibcode:2013ITIP...22...91M, doi:10.1109/TIP.2012.2202675, PMID 22692910

Starck, J.-L.; Murtagh, F. (2001), "Astronomical image and signal processing: looking at noise, information and scale", Signal Processing Magazine, IEEE, 18 (2), pp. 30–40, Bibcode:2001ISPM...18...30S, doi:10.1109/79.916319