Anscombe transform

An Anscombe transform is a transformation operation that converts a random variable with Poisson distribution into one with a Normal distribution.

• See: Poisson Distribution, Normal distribution, Transformation Operation.

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References

2016

• (Wikipedia, 2016) ⇒ https://www.wikiwand.com/en/Anscombe_transform Retrieved 2016-07-31
  • 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 [math]\displaystyle{ m }[/math] and variance [math]\displaystyle{ v }[/math] are not independent: [math]\displaystyle{ m = v }[/math]. The Anscombe transform

[math]\displaystyle{ A:x \mapsto 2\sqrt{x+\tfrac{3}{8}} \, }[/math]

aims at transforming the data so that the variance is set approximately 1 whatever the mean. It transforms Poissonian data [math]\displaystyle{ x }[/math] (with mean [math]\displaystyle{ m }[/math]) to approximately Gaussian data of mean [math]\displaystyle{ 2\sqrt{m + 3/8} - 1/(4\sqrt{m}) }[/math] and standard deviation 1. This approximation is valid provided that [math]\displaystyle{ m }[/math] is larger than 4.

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