A multitude of variational methods based on partial differential equations have been developed for a wide variety of images and applications [127,117], with some of these having applications to MRI [64,100,56]. However, such methods impose certain kinds of models on local image structure that are often too simple to capture the complexity of anatomical MR images. These methods, typically, do not take into account the bias introduced by Rician noise. Furthermore, such methods usually involve manual tuning of critical free parameters that control the conditions under which the models prefer one sort of structure over another; this has been an impediment to the widespread adoption of these techniques.
Another class of methods relies on statistical inference on multiscale representations of images. A prominent example includes methods based on wavelet transforms. Healy et al. [75] were among the first to apply soft-thresholding based wavelet techniques for denoising MR images. Hilton et al. [77] apply a threshold-based scheme for functional-MRI data. Nowak [115], operating on the square magnitude MR image, includes a Rician noise model in the threshold-based wavelet denoising scheme and thereby corrects for the bias introduced by the noise. Pizurica et al. [129] rely on the prior knowledge of the correlation of wavelet coefficients that represent significant features across scales. They first detect the wavelet coefficients that correspond to these significant features and then empirically estimate the PDFs of wavelet coefficients conditioned on the significant features. They employ these probabilities in a Bayesian denoising scheme.
In our previous work [9,6], we described UINTA which restores images by
generalizing the mean-shift to incorporate neighborhood information. UINTA, however, relies neither
on the knowledge of a noise model nor a prior model. Some MR-inhomogeneity correction methods are
based on the quantification of information content in MR images [157,103]. They follow
from the observation that inhomogeneities increase the entropy of the 
D gray scale
PDFs. However, entropy measures on first-order image statistics are insufficient for effective
denoising; thus this paper extends the information-theoretic strategy to higher-order Markov PDFs.
The proposed method takes the empirical-Bayes approach [141,140,24], pioneered by Robbins [141,140], for Bayesian denoising without making any ad hoc assumptions on the prior PDFs. The empirical-Bayes approach is applicable when we encounter multiple independent instances of a Bayesian decision problem (i.e., denoise each pixel) that all rely on exactly the same fixed, but unknown, prior PDF (i.e., uncorrupted-signal Markov PDF). In this special case, the empirical-Bayes approach allows accurate data-driven computation of the posterior PDF without the need to impose ad hoc or ill-fitting prior models. In this way, the decision procedure automatically adapts to the unknown prior PDFs. Robbins employed the empirical-Bayes approach to first obtain a maximum likelihood (ML) estimate of the prior distribution using the observations corrupted by a known noise model, and then employ the estimated prior model to compute the posterior [90]. The strategy in this paper closely follows Robbin's strategy.
Weismann et al. [175] address optimal image denoising using Markov statistics and empirical-Bayes approach [175]. Their discrete universal denoiser (DUDE) focuses on discrete signal intensities and, subsequently, relies on inverting the channel transition matrix (noise model) to give a closed-form estimate for source statistics from the observed statistics. The proposed method addresses continuous-valued signals, which is essential for medical-imaging applications, and thus entails estimating uncorrupted-signal statistics nonparametrically through the reduction of a Kullback-Leibler (KL) divergence. Snyder et al. [158] also use kernel density estimators for density deconvolution. The proposed approach also presents a method for practically dealing with the nonstationarity of real MRI data.
Cordy and Thomas [33] employ the expectation-maximization (EM) algorithm [43,104] for deconvolving PDFs corrupted with i.i.d. additive Gaussian noise. They model the uncorrupted-signal PDF as a Gaussian mixture model, but use the EM algorithm to estimate only the weights of Gaussians in the mixture--the means and variances of the Gaussians are tuned manually before EM is applied. They constrained the Gaussians to be spread uniformly over the entire domain of the PDF. Such a strategy, however, is not likely to be effective for density estimation in high-dimensional domains because of the enormous numbers of Gaussians needed to cover the space and sparsity of the data in the space--uniformly-distributed Gaussians will tend to oversmooth the PDF structure in high-curvature regions and will be inefficient in the tails of the PDF.