New Approach: Space Variant Hyperparameter

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.