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.

For the Poisson distribution the mean ${\displaystyle m}$ and variance ${\displaystyle v}$ are not independent: ${\displaystyle m=v}$. The Anscombe transform

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 ${\displaystyle x}$ (with mean ${\displaystyle m}$) to approximately Gaussian data of mean ${\displaystyle 2{\sqrt {m+{\tfrac {3}{8}}}}-{\tfrac {1}{4\,m^{1/2}}}+O\left({\tfrac {1}{m^{3/2}}}\right)}$ and standard deviation ${\displaystyle 1+O\left({\tfrac {1}{m^{2}}}\right)}$. This approximation gets more accurate for larger ${\displaystyle m}$, as can be also seen in the figure.

For a transformed variable of the form ${\displaystyle 2{\sqrt {x+c}}}$, the expression for the variance has an additional term ${\displaystyle {\frac {{\tfrac {3}{8}}-c}{m}}}$; it is reduced to zero at ${\displaystyle c={\tfrac {3}{8}}}$, which is exactly the reason why this value was picked.

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

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

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