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Дата изменения: Mon Apr 18 22:44:17 1994
Дата индексирования: Sun Dec 23 19:49:10 2007
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Prior Models <A NAME=prior><IMG ALIGN=MIDDLE SRC="/icons/invis_anchor.xbm"></A>



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Prior Models

Consider an image with no stars but only with regions of smoothly varying luminosity. We then expect and to be spatially smooth. Probably the simplest probability models that can be used to model smoothness are spatial autoregressions (Ripley 1981).

The conditional autoregression (CAR) model is defined by

where is the unknown hyper-parameter, matrix is such that if cells and are spatial neighbors (pixels at distance one) and zero otherwise, and scalar is just less than 0.25. The term represents in matrix notation the sum of squares of the values minus times the sum of for neighboring pixels and .

The parameters can be interpreted by the following expressions describing the conditional distribution

where the suffix `' denotes the four neighbor pixels at distance one from pixel . The parameter measures the smoothness of the `true' image.

Assuming a toroidal edge correction, the eigenvalues of the matrix are . So, the density of f has the form

where , .

This prior model can be easily modified to work at log scale, which is the right scale for the deconvolution of galaxies (Molina et al. 1992a). Furthermore, this model can also take into account the existence of different objects in the image (Molina et al. 1992b).

Let us now examine the prior model used in the R-L restoration method. This method aims at maximizing when this conditional distribution is Poissonian. Under the Bayesian framework this is the same as maximizing the posterior distribution for the prior model , together with the Poissonian noise model. The meaning of this prior is simple; all possible restorations have the same probability.


rlw@sundog.stsci.edu
Mon Apr 18 14:28:26 EDT 1994