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The hyperparameter 
is a key parameter in the Bayesian framework. It
determines the relative weight between the prior information and the likelihood
in the Bayesian function and therefore determines the chosen solution between
the default image and the maximum likelihood solution. Ultimately, it defines
the degree of smoothness of the solution. Up to now, we have considered
to be an adjustable parameter belonging to object space that is
constant for a given image. The value of can be adjusted by
cross-validation (Núñez and Llacer 1991; Núñez 1993).
The use of a single in Bayesian restoration produces a good global
fit of the data, but usually the part of the image in which the light level is
usually noisy (over-restored) while the bright objects (stars) are too smooth
(under-restored). This effect is also present in the Richardson-Lucy algorithm
at the stopping point. Furthermore, Bayesian methods with entropy and MEM
present photometric linearity problems.
The effect can be observed in the image of residuals:
If the data are Poisson, the 
values should be randomly distributed around
unity. However, for low values of the hyperparameter the residuals are
strongly correlated with the data. For a very high value of the hyperparameter,
the residuals are not correlated with the data but they are very near zero. In
this case the data are overfitted and the resulting image very is noisy with
high peaks and low valleys. In general, after obtaining the optimum value of
by cross-validation, the residuals for the background part of the
image are random but too low, resulting in a noisy restoration. On the other
hand, the residuals at the bright stars are large and still correlated with the
data, giving a too smooth restoration for the bright objects. As stated above,
this effect is also present in the Richardson-Lucy algorithm for a given
stopping point.
To solve this problem several approaches have been presented in this conference
(Katsaggelos 1994, Lucy 1994, Puetter 1994, Stark et al. 1994, White 1994). Our
approach consists in the introduction of a space variant
hyperparameter in the Bayesian framework. A variable allows
different degrees of smoothing in different regions, ranging from very smooth
(or very similar to the prior) regions 
to MLE
regions 
. We have used this variable
hyperparameter in connection with known default images for medical tomography
(Llacer, Veklerov, and Núñez 1991).
Assume that the intensity increment described in § 3 is
space variant in the object space. Let 
be the
variable increment. Now, in place of 
units we distribute
units. The number of ways in which an
object can occur is then
A natural choice for the prior energy distribution is
Using the Stirling approximation and taking the prior probability
to be proportional to the multiplicity, its logarithm is
where
With the modifications described, the algorithm FMAPEVAR is
The modified algorithm has the same desirable characteristics as the FMAPE
algorithm, but now it is possible to change the values of 
in order
to determine the degree of smoothing in different regions or, in connection
with known default images, to adjust the proximity between the restoration and
the default image 
.
To compute the for the different regions, we can use segmentation
techniques in conjunction with local cross-validation to define several levels
of data fitting, depending on the signal-to-noise ratio. We are currently
studying the suitability of the different segmentation algorithms to this
problem.
Next: Results
Up: HST Image Restoration with
Previous: The FMAPE Algorithm