Abstract-We describe a method for removing noise from digital images, based on a statistical model of the coefficients of an overcomplete multiscale oriented basis. Neighborhoods of coefficients 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 coefficients in the neighborhood, and is thus able to account for the empirically observed correlation between the coefficient amplitudes. Under this model, the Bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable. We demonstrate through simulations with images contaminated by additive white Gaussian noise that the performance of this method substantially surpasses that of previously published methods, both visually and in terms of mean squared error.Index Terms-Bayesian estimation, Gaussian scale mixtures, hidden Markov model, natural images, noise removal, overcomplete representations, statistical models, steerable pyramid.T HE artifacts arising from many imaging devices are quite different from the images that they contaminate, and this difference allows humans to "see past" the artifacts to the underlying image. The goal of image restoration is to relieve human observers from this task (and perhaps even to improve upon their abilities) by reconstructing a plausible estimate of the original image from the distorted or noisy observation. A prior probability model for both the noise and for uncorrupted images is of central importance for this application.Modeling the statistics of natural images is a challenging task, partly because of the high dimensionality of the signal. Two Manuscript received September 29, 2002; revised April 28, 2003. During the development of this work, V. Strela was on leave from Drexel University, and was supported by an AMS Centennial Fellowship. M. J. Wainwright was supported by a NSERC-1967 Fellowship. J. Portilla and E. P. Simoncelli were supported by an NSF CAREER grant and Alfred P. Sloan Fellowship to E. P. Simoncelli, and by the Howard Hughes Medical Institute. J. Portilla was also supported by an FPI fellowship, and subsequently by a "Ramón y Cajal" grant (both from the Spanish government). The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Mario A. T. Figueiredo basic assumptions are commonly made in order to reduce dimensionality. The first is that the probability structure may be defined locally. Typically, one makes a Markov assumption, that the probability density of a pixel, when conditioned on a set of neighbors, is independent of the pixels beyond the neighborhood. The second is an assumption of spatial homogeneity: the distribution of values in a neighborhood is the same for all such neighborhoods, regardless of absolute spatial position. The Markov random field model that results from these two assumptions i...

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