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.

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

Definition

For the Poisson distribution the mean $m$ and variance $v$ are not independent: $m=v$. The Anscombe transform

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

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 gets more accurate for larger $m$, as can be also seen in the figure.

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

$y\mapsto \left({\frac {y}{2}}\right)^{2}-{\frac {1}{8}}$

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

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

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

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

Alternatives

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

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

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 and its asymptotically unbiased or exact unbiased inverses.

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