The observed image differs from the true brightness distribution in having been blurred and encountering statistical noise in the recording process.
If is the size of the image, the blurring process is described by where is a vector and is the point spread matrix defining the systematic blur and assumed to be known.
Let us describe the noise models. For each component , of the observed vector we could use a Poissonian model, thus obtaining . This model can be approximated, at least for high brightness values, by the Gaussian distribution .
An alternative model would be to assume where . For this model we could use the following Gaussian approximation:
Substituting by , we would have
Finally, a model like
and the corresponding can be approximated by normal distributions having the form for appropriate constants and .
If we use the Gaussian approximation we have
with , being the standard deviation, , in the noise model just described.
This Gaussian approximation allows the easy incorporation of robust statistics concepts to deal with detector errors (Molina and Ripley 1989). The idea is to down-weight observations which are far away from their means. Such values are given too much weight in (4). The squared term in represents the number of standard deviations that is away from its mean. In robust statistics is replaced by for a function which penalizes extreme values less severely. A typical function is the Huber's `Proposal 2' function defined by
This is quadratic in the center, but penalizes large deviations linearly rather than quadratically. Equivalently, observations are down-weighted if exceeds 2. In practice is chosen at about 2, which down-weights only those observations more than two standard deviations away from their means.
The noise model we will use in this paper is Poissonian; see Molina and Ripley (1989) for the use of robust Gaussian noise models.