• Images
• Videos
• Documents
• Books
• Articles

Definition

For the Poisson distribution the mean m {\displaystyle m} and variance v {\displaystyle v} are not independent: m = v {\displaystyle m=v} . The Anscombe transform¹

A : x ↦ 2 x + 3 8 {\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 {\displaystyle x} (with mean m {\displaystyle m} ) to approximately Gaussian data of mean 2 m + 3 8 − 1 4 m 1 / 2 + O ( 1 m 3 / 2 ) {\displaystyle 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 ( 1 m 2 ) {\displaystyle 1+O\left({\tfrac {1}{m^{2}}}\right)} . This approximation gets more accurate for larger m {\displaystyle m} ,² as can be also seen in the figure.

For a transformed variable of the form 2 x + c {\displaystyle 2{\sqrt {x+c}}} , the expression for the variance has an additional term 3 8 − c m {\displaystyle {\frac {{\tfrac {3}{8}}-c}{m}}} ; it is reduced to zero at c = 3 8 {\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 {\displaystyle x} an estimate of m {\displaystyle m} ), its inverse transform is also needed in order to return the variance-stabilized and denoised data y {\displaystyle y} to the original range. Applying the algebraic inverse

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

usually introduces undesired bias to the estimate of the mean m {\displaystyle m} , because the forward square-root transform is not linear. Sometimes using the asymptotically unbiased inverse³

y ↦ ( y 2 ) 2 − 1 8 {\displaystyle 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 mapping⁴

E ⁡ [ 2 x + 3 8 ∣ m ] = 2 ∑ x = 0 + ∞ ( x + 3 8 ⋅ m x e − m x ! ) ↦ m {\displaystyle \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 is⁵

y ↦ 1 4 y 2 − 1 8 + 1 4 3 2 y − 1 − 11 8 y − 2 + 5 8 3 2 y − 3 . {\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 report⁶ a family of such transformations which includes the Anscombe transform. Another member of the family is the Freeman-Tukey transformation⁷

A : x ↦ x + 1 + x . {\displaystyle 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 ↦ 2 x {\displaystyle 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 x ] ≈ ( d ( 2 m ) d m ) 2 V [ x ] = ( 1 m ) 2 m = 1 {\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 transform⁸ and its asymptotically unbiased or exact unbiased inverses.⁹

See also

• Variance-stabilizing transformation
• Box–Cox transformation

Further reading

• Starck, J.-L.; Murtagh, F. (2001), "Astronomical image and signal processing: looking at noise, information and scale", Signal Processing Magazine, IEEE, vol. 18, no. 2, pp. 30–40, Bibcode:2001ISPM...18...30S, doi:10.1109/79.916319, S2CID 13210703

References

1. Anscombe, F. J. (1948), "The transformation of Poisson, binomial and negative-binomial data", Biometrika, vol. 35, no. 3–4, [Oxford University Press, Biometrika Trust], pp. 246–254, doi:10.1093/biomet/35.3-4.246, JSTOR 2332343 /wiki/Frank_Anscombe ↩

2. Bar-Lev, S. K.; Enis, P. (1988), "On the classical choice of variance stabilizing transformations and an application for a Poisson variate", Biometrika, vol. 75, no. 4, pp. 803–804, doi:10.1093/biomet/75.4.803 /wiki/Doi_(identifier) ↩

3. Anscombe, F. J. (1948), "The transformation of Poisson, binomial and negative-binomial data", Biometrika, vol. 35, no. 3–4, [Oxford University Press, Biometrika Trust], pp. 246–254, doi:10.1093/biomet/35.3-4.246, JSTOR 2332343 /wiki/Frank_Anscombe ↩

4. Mäkitalo, M.; Foi, A. (2011), "Optimal inversion of the Anscombe transformation in low-count Poisson image denoising", IEEE Transactions on Image Processing, vol. 20, no. 1, pp. 99–109, Bibcode:2011ITIP...20...99M, CiteSeerX 10.1.1.219.6735, doi:10.1109/TIP.2010.2056693, PMID 20615809, S2CID 10229455 /wiki/Bibcode_(identifier) ↩

5. 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, vol. 20, no. 9, pp. 2697–2698, Bibcode:2011ITIP...20.2697M, doi:10.1109/TIP.2011.2121085, PMID 21356615, S2CID 7937596 /wiki/Bibcode_(identifier) ↩

6. Bar-Lev, S. K.; Enis, P. (1988), "On the classical choice of variance stabilizing transformations and an application for a Poisson variate", Biometrika, vol. 75, no. 4, pp. 803–804, doi:10.1093/biomet/75.4.803 /wiki/Doi_(identifier) ↩

7. Freeman, M. F.; Tukey, J. W. (1950), "Transformations related to the angular and the square root", The Annals of Mathematical Statistics, vol. 21, no. 4, pp. 607–611, doi:10.1214/aoms/1177729756, JSTOR 2236611 /wiki/John_Tukey ↩

8. Starck, J.L.; Murtagh, F.; Bijaoui, A. (1998). Image Processing and Data Analysis. Cambridge University Press. ISBN 9780521599146. 9780521599146 ↩

9. Mäkitalo, M.; Foi, A. (2013), "Optimal inversion of the generalized Anscombe transformation for Poisson-Gaussian noise", IEEE Transactions on Image Processing, vol. 22, no. 1, pp. 91–103, Bibcode:2013ITIP...22...91M, doi:10.1109/TIP.2012.2202675, PMID 22692910, S2CID 206724566 /wiki/Bibcode_(identifier) ↩