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 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 and variance are not independent:. 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 to approximately Gaussian data of mean
and standard deviation.
This approximation is good provided that is larger than 4.
For a transformed variable of the form, the expression for the variance has an additional term ; it is reduced to zero at, which is exactly the reason why this value was picked.

Inversion

When the Anscombe transform is used in denoising, its inverse transform is also needed
in order to return the variance-stabilized and denoised data to the original range.
Applying the algebraic inverse
usually introduces undesired bias to the estimate of the mean, 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
should be used. A closed-form approximation of this exact unbiased inverse is

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

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.