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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.