Survey of image denoising techniques <s> ii. Joint Probabilistic Model <s> This paper describes a new method for the suppression of noise in images via the wavelet transform. The method relies on two measures. The first is a classic measure of smoothness of the image and is based on an approximation of the local Holder exponent via the wavelet coefficients. The second, novel measure takes into account geometrical constraints, which are generally valid for natural images. The smoothness measure and the constraints are combined in a Bayesian probabilistic formulation, and are implemented as a Markov random field (MRF) image model. The manipulation of the wavelet coefficients is consequently based on the obtained probabilities. A comparison of quantitative and qualitative results for test images demonstrates the improved noise suppression performance with respect to previous wavelet-based image denoising methods. <s> BIB001 </s> Survey of image denoising techniques <s> ii. Joint Probabilistic Model <s> ABSTRACT Wavelet-domain hidden Markov models have proven to be useful tools for statistical signal and image processing. The hiddenMarkov tree (HMT) model captures the key features of the joint density of the wavelet coefficients of real-world data. Onepotential drawback to the HMT framework is the need for cornputationallv expensive iterative training (using the Expectation-Iaximization algorithm, for example). In this paper, we propose two reduced-parameter HMT models that capture the generalstructure of a broad class of real-world images. In the image HMT (iHMT) model we use the fact that for a large class ofimages the structure of the HMT is self-similar across scale. This allows us to reduce the complexity of the iHMT to justnine easily trained parameters (independent of the size of the image and the number of wavelet scales). In the universal HMT(uHMT) we take a Bayesian approach and fix these nine parameters. The uHMT requires no training of any kind. Whilesimple, we show using a series of image estimation/denoising experiments that these two new models retain nearly all of thekey structure modeled by the full HvIT. Finally, we propose a fast shift-invariant HMT estimation algorithm that outperformsall other wavelet-based estimators in the current literature. both in mean-square error and visual metrics. <s> BIB002 </s> Survey of image denoising techniques <s> ii. Joint Probabilistic Model <s> Wavelet threshold algorithms replace small magnitude wavelet coefficients with zero and keep or shrink the other coefficients. This is basically a local procedure, because wavelet coefficients characterize the local regularity of a function. Although a wavelet transform has decorrelating properties, structures in images, like edges, are never decorrelated completely, and these structures appear in the wavelet coefficients: a classification based on a local criterion-like coefficient magnitude is not the perfect method to distinguish important, uncorrupted coefficients from coefficients dominated by noise. We therefore introduce a geometrical prior model for configurations of important wavelet coefficients and combine this with local characterization of a classical threshold procedure into a Bayesian framework. The local characterization is incorporated into the conditional model, whereas the prior model describes only configurations, not coefficient values. More precisely, local characterization favors co... <s> BIB003 </s> Survey of image denoising techniques <s> ii. Joint Probabilistic Model <s> We describe a method for removing noise from digital images, based on a statistical model of the coecien ts of an overcomplete multi-scale oriented basis. Neighborhoods of coecien ts at adjacent positions and scales are modeled as the product of two independent random variables: a Gaussian vector and a hidden positive scalar multiplier. The latter modulates the local variance of the coecien ts in the neighborhood, and is thus able to account for the empirically observed correlation between the amplitudes of pyramid coecien ts. Under this model, the Bayesian least squares estimate of each coecien t reduces to a weighted average of the local linear (Wiener) estimate over all possible values of the hidden multiplier variable. We demonstrate through simulations with images contaminated by additive Gaussian noise of known covariance that the performance of this method substantially surpasses that of previously published methods, both visually and in terms of mean squared error. In addition, we demonstrate the performance of the algorithm in removing sensor noise from high-ISO digital camera images. <s> BIB004
Hidden Markov Models (HMM)  models are efficient in capturing inter-scale dependencies, whereas Random Markov Field BIB001 models are more efficient to capture intrascale correlations. The complexity of local structures is not well described by Random Markov Gaussian densities whereas Hidden Markov Models can be used to capture higher order statistics. The correlation between coefficients at same scale but residing in a close neighborhood are modeled by Hidden Markov Chain Model where as the correlation between coefficients across the chain is modeled by Hidden Markov Trees. Once the correlation is captured by HMM, Expectation Maximization is used to estimate the required parameters and from those, denoised signal is estimated from noisy observation using wellknown MAP estimator. In BIB004 , a model is described in which each neighborhood of wavelet coefficients is described as a Gaussian scale mixture (GSM) which is a product of a Gaussian random vector, and an independent hidden random scalar multiplier. Strela et al.  described the joint densities of clusters of wavelet coefficients as a Gaussian scale mixture, and developed a maximum likelihood solution for estimating relevant wavelet coefficients from the noisy observations. Another approach that uses a Markov random field model for wavelet coefficients was proposed by Jansen and Bulthel BIB003 . A disadvantage of HMT is the computational burden of the training stage. In order to overcome this computational problem, a simplified HMT, named as uHMT BIB002 , was proposed.