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
1.4. Including Gaussian noise of known autocovariance in the model
When we add Gaussian noise to an image,
then this also results in additive Gaussian noise in every wavelet coefficient:
y
= x + w
If we assume knowledge of the noise autocovariance (if we do not, then we need to apply some estimation techniques for doing blind denoising - see [Portilla 2004a, Portilla 2004b]), then it is easy to estimate the covariance of clusters of neighbor coefficients, for a given neighborhood structure, for each subband. Now, for the neighborhoods around coefficients of a given subband, we can write: Cy = Cx + Cw, where Cx, Cw and Cy are the covariance matrices of the original x (clean coeffs.), noise w and noisy observations y, respectively. For a given hidden factor z the observed vectors are zero-mean Gaussian with covariance
Cy|z = z Cx + Cw = z (Cy - Cw) + Cw = z Cy + (1 - z ) Cw.
Note that we can estimate Cy directly from the observations, and that we have assumed knowledge of Cw. As p(x | z), p(y | z) is also Gaussian (if we freeze the hidden factor, then the observed vector is the sum of the two Gaussians sqrt(z)u and w). This property will allow us to estimate the solution using a kind of "local energy adaptive Wiener filter", as explained next.