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There are many noise sources for images. Images are, in many cases, degraded even before they are encoded. Previously, we focused on Poisson noise (Huang, X. et al., IEEE Int. Conf. on Multimedia and Expo, vol.1, p.593, 2003). Unlike additive Gaussian noise, Poisson noise is signal-dependent and separating signal from noise is a difficult task. A wavelet-based maximum likelihood method for a Bayesian estimator that recovers the signal component of the wavelet coefficients in the original images by using an alpha-stable signal prior distribution is demonstrated for Poisson noise removal. The paper extends, via Levy process analysis, our previous results to more complex cases of noise comprised of compound Poisson and Gaussian. As an example, an improved Bayesian estimator that is a natural extension of other wavelet denoising (soft and hard threshold methods) via a colour image is presented to illustrate our discussion; even though computers did not know the noise, this method works well.