Anscombe transform

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Standard deviation of the transformed Poisson random variable as a function of the mean m {\displaystyle m} {\displaystyle 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.

Anscombe transform animated. Here μ {\displaystyle \mu } {\displaystyle \mu } is the mean of the Anscombe-transformed Poisson distribution, normalized by subtracting by 2 m + 3 8 − 1 4 m 1 / 2 {\displaystyle 2{\sqrt {m+{\tfrac {3}{8}}}}-{\tfrac {1}{4\,m^{1/2}}}} {\displaystyle 2{\sqrt {m+{\tfrac {3}{8}}}}-{\tfrac {1}{4\,m^{1/2}}}}, and σ {\displaystyle \sigma } {\displaystyle \sigma } is its standard deviation (estimated empirically). We notice that m 3 / 2 μ {\displaystyle m^{3/2}\mu } {\displaystyle m^{3/2}\mu } and m 2 ( σ − 1 ) {\displaystyle m^{2}(\sigma -1)} {\displaystyle m^{2}(\sigma -1)} remains roughly in the range of [ 0 , 10 ] {\displaystyle [0,10]} {\displaystyle [0,10]} over the period, giving empirical support for μ = O ( m − 3 / 2 ) , σ = 1 + O ( m − 2 ) {\displaystyle \mu =O(m^{-3/2}),\sigma =1+O(m^{-2})} {\displaystyle \mu =O(m^{-3/2}),\sigma =1+O(m^{-2})}

Definition

For the Poisson distribution the mean m {\displaystyle m} {\displaystyle m} and variance v {\displaystyle v} {\displaystyle v} are not independent: m = v {\displaystyle m=v} {\displaystyle m=v}. The Anscombe transform[1]

A : x ↦ 2 x + 3 8 {\displaystyle A:x\mapsto 2{\sqrt {x+{\tfrac {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} {\displaystyle x} (with mean m {\displaystyle 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)} {\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)} {\displaystyle 1+O\left({\tfrac {1}{m^{2}}}\right)}. This approximation gets more accurate for larger m {\displaystyle 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}}} {\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}}} {\displaystyle {\frac {{\tfrac {3}{8}}-c}{m}}}; it is reduced to zero at c = 3 8 {\displaystyle c={\tfrac {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} {\displaystyle x} an estimate of m {\displaystyle m} {\displaystyle m}), its inverse transform is also needed in order to return the variance-stabilized and denoised data y {\displaystyle 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}}} {\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} {\displaystyle m}, because the forward square-root transform is not linear. Sometimes using the asymptotically unbiased inverse[1]

y ↦ ( y 2 ) 2 − 1 8 {\displaystyle y\mapsto \left({\frac {y}{2}}\right)^{2}-{\frac {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 mapping[3]

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} {\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 is[4]

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}.} {\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[2] a family of such transformations which includes the Anscombe transform. Another member of the family is the Freeman-Tukey transformation[5]

A : x ↦ x + 1 + x . {\displaystyle A:x\mapsto {\sqrt {x+1}}+{\sqrt {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}}\,} {\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} {\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[6] and its asymptotically unbiased or exact unbiased inverses.[7]

See also

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
  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
  3. 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, doi:10.1109/TIP.2010.2056693, PMID 20615809, S2CID 10229455
  4. 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
  5. 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
  6. Starck, J.L.; Murtagh, F.; Bijaoui, A. (1998). Image Processing and Data Analysis. Cambridge University Press. ISBN 978-0-521-59914-6.
  7. 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

Further reading