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
1.3. Modeling local clusters of wavelet coefficients using Gaussian Scale Mixtures
Every feature of the image (like an edge, a corner, a dot, etc.) affects simultaneously the response of several wavelet coefficients, in a way that if we multiply the amplitude of that feature with a given factor, those wavelet responses most sensitive to it will be amplified or damped in a similar way. Ideally, and for isolated features, their responses will be affected by the same factor. This suggests the following model for clusters of neighbor coefficients [Wainwright 2000, Wainwright 2001]
where x represents a local cluster of N wavelet neighbor coefficients arranged in a vector, u is a zero-mean Gaussian vector of given covariance, and z is a hidden independent scalar random variable (sometimes termed hidden multiplier, or hidden factor or mixing scale factor) controlling the magnitude of the local response x. The random vector x is termed a Gaussian Scale Mixture (GSM) [Andrews 1974]. It can be interpreted as a continuous infinite mixture of zero-mean Gaussians with the same normalized covariance matrix but with different scale factors ( given by sqrt(z) ). In our model we choose for u the same covariance of x, which implies that z has an expected value of 1. Besides the covariance matrix of u, the other feature of the GSM is the mixing density p(z), that tells us the probability of z occurring for every given interval of scale values. The use of "sqrt(z)" in the definition instead of just "z" is chosen because it simplifies the expressions of p(x|z). The GSM's vectors of a given density form hyper-ellipses, and thus GSMs are a particular case of elliptically symmetric distributions.
Note how such a simple model can account for the statistical features of the image wavelet coefficients described in 1.2.. First, it is easy to see how a high kurtosis marginal 1-D density (sharp peak at zero, heavy tails) can be obtained by adding up (mixing) a few zero-mean Gaussian functions each with a different scale factor and different probability mass:
A high kurtosis marginal density can be obtained by integrating
zero-mean pdf's with different scale factors each (scale mixture)
If we use a continuous mixture we obtain a similar effect. If we look at the joint density of x (we have used for the illustration below an isotropic Gaussian u and a squared Gaussian for z) we see how the "bow-tie" shape [Simoncelli 1997] appears when we normalize column wise:
Left: An example of joint pdf of a GSM Right: Result of normalizing columns in (left)
and changing to gray level
We can see, thus, how the two basic features referred above for the statistics of local wavelet responses to images are reproduced using Gaussian Scale Mixtures.