Short wavelets and matrix dilation equations

Scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter (in a two-band orthogonal filter bank). For “multifilters” those coefficients are matrices. This gives a new block structure for the filter bank, and leads to multiple scaling functions and wavelets. Geronimo, Hardin, and Massopust (see J. Approx. Theory, vol.78, p.373-401, 1994) constructed two scaling functions that have extra properties not previously achieved. The functions Φ₁ and Φ₂ are symmetric (linear phase) and they have short support (two intervals or less), while their translates form an orthogonal family. For any single function Φ, apart from Haar's piecewise constants, those extra properties are known to be impossible. The novelty is to introduce 2×2 matrix coefficients while retaining orthogonality of the multiwavelets. This note derives the properties of Φ₁ and Φ₂ 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 by Geronimo, Hardin, and Massopust from the iterated interpolation that led to Φ1 and Φ₂. 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 coefficients in the dilation equation-in another language, to build the 4×4 paraunitary polyphase matrix in the filter bank. The short support opens new possibilities for applications of filters and wavelets near boundaries     

Image denoising using scale mixtures of Gaussians in the wavelet domain

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.     

The application of multiwavelet filterbanks to image processing

Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrix-valued filterbanks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar two-channel wavelet systems. After reviewing this theory, we examine the use of multiwavelets in a filterbank setting for discrete-time signal and image processing. Multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filterbank. We describe two methods (repeated row and approximation/deapproximation) for obtaining such a vector input stream from a one-dimensional (1-D) 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 (2-D) 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 apply multiwavelet processing to images, frequently obtaining performance superior to the comparable scalar wavelet transform.