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


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

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,

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

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 and its asymptotically unbiased or exact unbiased inverses.