Anscombe transform

File:Anscombe stabilized stdev.svg

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.

File:Anscombe transform animated.gif

Anscombe transform animated. Here $\mu$ is the mean of the Anscombe-transformed Poisson distribution, normalized by subtracting by $2\sqrt{m+\frac{3}{8}}-\frac{1}{4m^{1/2}}$, 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

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

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 $2\sqrt{m+\frac{3}{8}}-\frac{1}{4m^{1/2}}+O\left(\frac{1}{m^{3/2}}\right)$ and standard deviation $1+O\left(\frac{1}{m^2}\right)$. 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 $\frac{\frac{3}{8}-c}{m}$; it is reduced to zero at $c=\frac{3}{8}$, 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

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

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