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
Appendix
Origins and evolution of the BLS-GSM model
The method presented here comes from three different sources, besides my own contribution. The first source is the observations and modeling of Eero Simoncelli of the joint high-order statistical coupling between wavelet coefficients of natural images [Simoncelli 1997]. Second, Vasily Strela proposed [Strela 2000a] a denoising method that provided a local energy adaptation, which was conceptually close to some ideas of Simoncelli's research on the statistics of natural images. That was the precursor of the method presented here. A few months later, Strela got in touch with Simoncelli (I was a postdoc in Simoncelli's lab at that time), and the three of us started to work on an improved denoising method. After a little while Eero Simoncelli realized that we could apply the recent ideas and neat formalism of Martin Wainwright (the third source) and himself for using Gaussian Scale Mixtures (GSM) to model local image statistics in the wavelet domain [Wainwright 2000, Wainwright 2001]. A few years before some other people [Crouse 1998] had proposed using Gaussian mixtures for denoising, but a mixture of only two Gaussians, not a continuous mixture, and applied to a single coefficient, not to a neighborhood. We improved aspects of Strela's original work, such as the representation (overcomplete wavelets instead of blocks), extended the local Wiener solution from scalar to vectorial, used more advanced implementation techniques, and introduced some other conceptual improvements besides the GSM model, like including a prior for the hidden multiplier controlling the local energy (we were inspired by [Mihcak 1999], among others). In such a way, we arrived to a high-performance locally adaptive Wiener denoising method in the wavelet domain [Portilla 2001]. For that method, as it had been done before [e.g., Lee 1980, Michcak 1999, Simoncelli 1999] we applied a sequential estimation strategy in 2 separated steps (empirical Bayes): in our case, according to the GSM model, first we locally estimated the most likely (MAP) hidden scale factor for every wavelet neighborhood. Then, we applied a local Wiener solution using the estimated hidden scale factor (as if it was known exactly) for estimating the central coefficient of the neighborhood.
About a year later (2002), I proposed a new estimation strategy based on weighting all possible Wiener solutions according to their posterior probability given each observed cluster of coefficients (Bayes Least Squares - Gaussian Scale Mixture method, BLS-GSM). Besides being more correct conceptually, this also reduced the computational complexity of our previous GSM method and further improved the denoising performance [Portilla 2002, Portilla 2003b]. After that, we have also applied these ideas to deblur images in the presence of noise [Portilla 2003a] and I have found solutions for the estimation of the model parameters from a single degraded image, with application to full blind denoising of images corrupted by noise of any power spectral density [Portilla 2004a, Portilla 2004b].