1. DESCRIPTION
1.1. Introduction
1.2. Image Statistics
1.3. GSM
1.4. Plus Noise
1.5. BLS-GSM
1.6. Results
1.7. References
2. EXAMPLES
3. TEST IMAGES
4. SOFTWARE
1.5. Bayes Least-Squares solution: the BLS-GSM method
Let's assume that we have a model, an estimate or a reasonable guess for the mixing density p(z). Then it is easy to demonstrate [Portilla 2002, Portilla 2003b] that the LS solution for estimating the central coefficient of the GSM x vector can be written as:
where x_c is the central or reference coefficient of the neighborhood, and E{x_c | y, z} is the central element of the vector Wiener solution obtained for a particular conditionally Gaussian observation y for a given scale z, assuming the observed sample covariance Cy and zero-mean noise of known covariance Cw:
This solution is computed, in practice, for a finite (and relatively small - 10, for instance) number of possible z values, converting in practice the continuous GSM into a discrete scale mixture. For every chosen z value we also compute numerically the posteriors p(z | y), for every observed noisy vector y. The latter computation is easy by applying the Bayes rule, given that we know p(y | z), and that we have a model for p(z). The BLS estimation for every central coefficient of every observed neighborhood is just a weighted average of the Wiener solutions according to the probability of each z value given the observed vector y. This strategy provides a smaller quadratic error than the classical (empirical Bayes) approach, which consists of first estimating the hidden variable (z, in this case), and then applying an estimator, as if the estimated value was exact.
In such a way we estimate all the wavelet coefficients of the image and then proceed to reconstruct the image estimate from those coefficients, by inverting the overcomplete wavelet.