Anscombe transform explained

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 transformA: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\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 (\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 neededin order to return the variance-stabilized and denoised data y to the original range.Applying the algebraic inverseA ^{-1} :y \mapsto \left( (y)/(2) \right) ^{2} - (3)/(8)

usually introduces undesired bias to the estimate of the mean

m , because the forward square-roottransform is not linear. Sometimes using the asymptotically unbiased inversey \mapsto \left( (y)/(2) \right) ^{2} - (1)/(8)

mitigates the issue of bias, but this is not the case in photon-limited imaging, for whichthe exact unbiased inverse given by the implicit mapping

\operatorname{E} \left[ 2 \sqrt{x + \tfrac{ 3 }{ 8 }} \mid m \right] = 2 \sum ^{+inf ty} _{x =0} \left( \sqrt{x + \tfrac{ 3 }{ 8 }} ⋅ (m ^{x} e ^{-m})/(x!) \right) \mapsto m

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

y \mapsto (1)/(4) y ^{2} - (1)/(8) + (1)/(\sqrt{ 4) (3)/(2) } y^ - \frac y^ + \frac \sqrt y^.

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}] ≈ \left( (d ( 2 \sqrt{m)/()}{d) m} \right) ^{2} V[x] = \left( (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^[1] and its asymptotically unbiased or exact unbiased inverses.

See also

• Variance-stabilizing transformation
• Box–Cox transformation

Notes and References

1. Book: Starck . J.L. . Murtagh . F. . Bijaoui . A. . 1998 . Image Processing and Data Analysis . registration . 9780521599146 . Cambridge University Press.