Anscombe transform

File:Anscombe stabilized stdev.svg

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|thumb|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

{{Citation

| last = Anscombe

| first = F. J.

| authorlink = Frank Anscombe

| year = 1948

| title = The transformation of Poisson, binomial and negative-binomial data

| periodical = Biometrika

| volume = 35

| issue = 3–4

| pages = 246–254

| doi = 10.1093/biomet/35.3-4.246

| jstor = 2332343

| publisher = [Oxford University Press, Biometrika Trust]

}}

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

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 $m$ ,{{Citation |last1=Bar-Lev |first1=S. K. |last2=Enis |first2=P. |year=1988 |title=On the classical choice of variance stabilizing transformations and an application for a Poisson variate |periodical=Biometrika |volume=75 |issue=4 |pages=803–804 |doi=10.1093/biomet/75.4.803

}} 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{{Citation

| last1 = Mäkitalo

| first1 = M.

| last2 = Foi

| first2 = A.

| year = 2011

| title = Optimal inversion of the Anscombe transformation in low-count Poisson image denoising

| periodical = IEEE Transactions on Image Processing

| volume = 20

| issue = 1

| pages = 99–109

| doi = 10.1109/TIP.2010.2056693

| pmid = 20615809

| bibcode = 2011ITIP...20...99M

| citeseerx = 10.1.1.219.6735

| s2cid = 10229455

}}

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

should be used. A closed-form approximation of this exact unbiased inverse is{{Citation |last1=Mäkitalo |first1=M. |last2=Foi |first2=A. |year=2011 |title=A closed-form approximation of the exact unbiased inverse of the Anscombe variance-stabilizing transformation |periodical=IEEE Transactions on Image Processing |volume=20 |issue=9 |pages=2697–2698 |doi=10.1109/TIP.2011.2121085|pmid=21356615 |bibcode=2011ITIP...20.2697M |s2cid=7937596 }}

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

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{{Citation |last1=Freeman |first1=M. F. |last2=Tukey |first2=J. W. |authorlink2=John Tukey |year=1950 |title=Transformations related to the angular and the square root |periodical=The Annals of Mathematical Statistics |volume=21 |issue=4 |pages=607–611 |jstor=2236611 |doi=10.1214/aoms/1177729756|doi-access=free }}

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

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,

$V[2\sqrt{x}] \approx \left(\frac{d (2\sqrt{m})}{d m} \right)^2 V[x] = \left(\frac{1}{\sqrt{m}} \right)^2 m = 1$ .

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

{{cite book

| last1 = Starck

| first1 = J.L.

| last2 = Murtagh

| first2 = F.

| last3 = Bijaoui

| first3 = A.

| year = 1998

| title = Image Processing and Data Analysis

| url = https://archive.org/details/imageprocessingd0000star

| url-access = registration

| isbn = 9780521599146

| publisher = Cambridge University Press

}} and its asymptotically unbiased or exact unbiased inverses.{{Citation

| last1 = Mäkitalo

| first1 = M.

| last2 = Foi

| first2 = A.

| year = 2013

| title = Optimal inversion of the generalized Anscombe transformation for Poisson-Gaussian noise

| periodical = IEEE Transactions on Image Processing

| volume = 22

| issue = 1

| pages = 91–103

| doi = 10.1109/TIP.2012.2202675

| pmid = 22692910

| bibcode = 2013ITIP...22...91M

| s2cid = 206724566

}}

Anscombe transform

Definition

Inversion

Alternatives

Generalization

See also

References

Further reading