The philosophy within statistics known as Bayesian inference has a very long history. It is distinguished from the perhaps more familiar classical statistical ideas by using prior information about the images being studied.
Bayesian methods start with a prior distribution, a probability distribution
over images ,
, (it is here that we incorporate
information on the expected structure within an image), it is also necessary to
specify the probability distribution
, of observed images
g if f were the true image. The Bayesian paradigm dictates that
inference about the true
should be based on
given by
To show just one restoration, it is common to choose the mode of this
posterior distribution, that is to display the image
which satisfies
This is known as the maximum a posteriori (MAP) estimate of .
Equivalently, we can choose to minimize
The first term in (1) is the familiar log likelihood of .
The second term can be thought of as a roughness penalty, as images
which do not correspond to our prior conceptions will be
assigned a small
and hence a large penalty.
In statistical physics it is common to define probabilities by the
energy of a system, so that
where is
,
being the temperature and
Boltzmann's
constant. If we adopt this notation,
minimizes
We can recognize this as a Lagrangian form, so its solution is equivalent to solving
and to
which correspond to the regularization approach to image restoration.
Many other deconvolution principles fit into one of these forms; in particular - as we will see later - the R-L restoration method. Maximum entropy methods also fit into this framework (Molina et al. 1992a).
Having described the Bayesian paradigm let us move on to examine the
two ingredients of this paradigm, the observation process,
, and the prior model or image model,
.