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...
Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrix-valued filter banks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar 2-channel wavelet systems. After reviewing this recently developed theory, we examine the use of multiwavelets in a filter bank setting for discrete-time signal and image processing. Multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filter bank. We describe two methods (repeated row and approximation/deapproximation) for obtaining such a vector input stream from a one-dimensional signal. Algorithms for symmetric extension of signals at boundaries are then developed, and naturally integrated with approximation-based preprocessing. We describe an additional algorithm for multiwavelet processing of two-dimensional signals, two rows at a time, and develop a new family of multiwavelets (the constrained pairs) that is well-suited to this approach. This suite of novel techniques is then applied to two basic signal processing problems, denoising via wavelet-shrinkage, and data compression. After developing the approach via model problems in one dimension, we applied multiwavelet processing to images, frequently obtaining performance superior to the comparable scalar wavelet transform.
Scaling functions and orthogonal wavelets are created from the coe cients of a lowpass and highpass lter (in a two-band orthogonal lter bank). For \multi lters" those coe cients are matrices. This gives a new block structure for the lter bank, and leads to multiple scaling functions and wavelets. Geronimo, Hardin, and Massopust constructed two scaling functions that have extra properties not previously achieved. The functions 1 and 2 are symmetric (linear phase) and they have short support (two intervals or less), while their translates form an orthogonal family. For any single function , apart from Haar's piecewise constants, those extra properties are known to be impossible. The novelty is to introduce 2 by 2 matrix coe cients while retaining orthogonality. This note derives the properties of 1 and 2 from the matrix dilation equation that they satisfy. Then our main step is to construct associated wavelets: two wavelets for two scaling functions. The properties were derived in 1] from the iterated interpolation that led to 1 and 2. One pair of wavelets was found earlier by direct solution of the orthogonality conditions (using Mathematica). Our construction is in parallel with recent progress by Hardin and Geronimo, to develop the underlying algebra from the matrix coe cients in the dilation equation | in another language, to build the 4 by 4 paraunitary polyphase matrix in the lter bank. The short support opens new possibilities for applications of lters and wavelets near boundaries.
The functions f 1 (x ), . . . , f r (x) are refinable if they are combinations of the rescaled and translated functions f i (2x − k ). This is very common in scientific computing on a regular mesh. The space V 0 of approximating functions with meshwidth h = 1 is a subspace of V 1 with meshwidth h = 1/2. These refinable spaces have refinable basis functions. The accuracy of the computations depends on p, the order of approximation, which is determined by the degree of polynomials 1, x , . . . , x p−1 that lie in V 0 .Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions f i (x ) are known only through the coefficients c k in the refinement equation-scalars in the traditional case, r ×r matrices for multiwavelets. The scalar "sum rules" that determine p are well known. We find the conditions on the matrices c k that yield approximation of order p from V 0 . These are equivalent to the Strang-Fix conditions on the Fourier transformŝ f i (ω), but for refinable functions they can be explicitly verified from the c k .
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