Biomedical Imaging

From this we can deduce the variance of the Poisson distribution

$\langle n^2\rangle -\langle n^2\rangle =\mu =\langle n\rangle ,\left(3.131\right)$

which turns out to be identical to the mean, a well known characteristic property of this distribution. The standard deviation σ is hence

$\sigma =\sqrt{\langle n^2\rangle -{\langle n\rangle }^2}=\sqrt{\mu }.\left(3.132\right)$

This results in a relative uncertainty of a measurement of

$\frac{\sqrt{\langle n^2\rangle -{\langle n\rangle }^2}}{\langle n\rangle }=\frac{1}{\sqrt{\mu }}.\left(3.133\right)$

In other words, the higher the photon count (i.e., μ), the higher the absolute uncertainty, but the lower the relative uncertainty. This almost trivial statement has significant consequences, since it sets a limit to the possible reduction in the x-ray dose. The smaller the dose, the smaller the information for diagnostics. Therefore, modern developments ...