Leo360 wrote:

Joofa wrote:

bobn2 wrote:

The random process is occuring in the light not the pixel.

Actually, both. Even if we knew the exact number of photons impinging on a pixel, and consider only the QE among any other effects, the number of photoelectrons generated is random (see bullet #3 below). The mean value of photoelectrons can be estimated as the mean number of photons multiplied by the QE.

The key thing to remember is:

- the number of photons being impinged is Poisson.
- the number of photoelectrons generated is Poisson.
- the number of photoelectrons generated for a given number of photons impinged upon a pixel is not Poisson, rather binomial.
- the number of photons impinged upon a pixel given the number of photoelectrons generated is not exactly Poisson, rather a variant of it.

No, it gets catered for simply by multiplying the photon count by the QE.

The interesting thing to note regarding the above is that the estimate of the number of photons impinged obtained by dividing the photoelectron count by QE (something that is done many times due to lack of other information) is a relatively poor estimater, with its mean square error being 1 / QE of the mean square error of the best (L2) estimater.

Hello Joofa,

Thanks for a concise, yet, precise outline. It was fun to derive #2 through #4 from scratch as well as to play with the estimators. And indeed, as you said, the naive n_e/QE estimator has a std.dev that is 1/QE worse that the std.dev of the conditional mean E{n_photon|n_e}.

Leo

Hi Leo,

Thanks for the effort. Glad that you found it interesting.

An interesting thing to note is that it is an established fact that the mean number of photoelectrons can be estimated as the mean number of photons x QE, therefore, (mean photons) = (mean photoelectrons) / QE. However, our confidence in a particular estimate of photons impinged on a pixel, given as (number of photonelectrons collected in the pixel) / QE, for a particular measurement gets increasingly worse with lower QE values. The 'naive' estimator = (number of photoelectrons) / QE is actually kind of a maximum-likelihood estimator. What makes the L2 estimator better is that it brings some sort of 'prior knowledge' embedded through the estimate of mean number of incident photons.

Sincerely,

Joofa