Anscombe transform

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

: $A:x \mapsto 2 \sqrt \,$

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 $2\sqrt - \tfrac + O\left(\tfrac\right)$
and standard deviation $1 + O\left(\tfrac\right)$.
This approximation gets more accurate for larger $m$, as can be also seen in the figure.

For a transformed variable of the form $2 \sqrt$, the expression for the variance has an additional term $\frac$; it is reduced to zero at $c = \tfrac$, 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

: $A^:y \mapsto \left( \frac \right)^2 - \frac$

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

: $y \mapsto \left( \frac \right)^2 - \frac$

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 \left 2\sqrt \mid m \right= 2 \sum_^ \left( \sqrt \cdot \frac \right) \mapsto m

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

: $y \mapsto \frac y^2 - \frac + \frac \sqrt y^ - \frac y^ + \frac \sqrt y^.$

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

: $A:x \mapsto \sqrt+\sqrt. \,$

A simplified transformation, obtained as the primitive of the reciprocal of the standard deviation of the data, is

: $A:x \mapsto 2\sqrt \,$

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 \sqrt\approx \left(\frac \right)^2 V = \left(\frac \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

References

Further reading

*{{Citation , last1=Starck , first1=J.-L. , last2=Murtagh , first2=F. , year=2001 , title=Astronomical image and signal processing: looking at noise, information and scale , periodical=Signal Processing Magazine, IEEE , volume=18 , issue=2 , pages=30–40 , doi=10.1109/79.916319, bibcode=2001ISPM...18...30S , s2cid=13210703

Poisson distribution
Normal distribution
Statistical data transformation