Anscombe transform

Inversion

When the Anscombe transform is used in denoising (i.e. when the goal is to obtain from ${\displaystyle x}$ an estimate of ${\displaystyle m}$), its inverse transform is also needed in order to return the variance-stabilized and denoised data ${\displaystyle y}$ to the original range. Applying the algebraic inverse

${\displaystyle A^{-1}:y\mapsto {\left(\frac{y}{2}\right)}^2-\frac{3}{8}}$

usually introduces undesired bias to the estimate of the mean ${\displaystyle m}$, because the forward square-root transform is not linear. Sometimes using the asymptotically unbiased inverse

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

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

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

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

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

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

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

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

${\displaystyle A:x\mapsto 2\sqrt{x}}$

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,

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

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.