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.
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.
Short wavelets and matrix dilation equations Strang, G.; Strela, V.
IEEE transactions on signal processing,
1995-Jan., 1995, 1995-01-00, 19950101, Volume:
43, Issue:
1
Journal Article
Peer reviewed
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 /spl Phi//sub 1/ and /spl Phi//sub 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 /spl Phi/, apart from Haar's piecewise constants, those extra properties are known to be impossible. The novelty is to introduce 2/spl times/2 matrix coefficients while retaining orthogonality of the multiwavelets. This note derives the properties of /spl Phi//sub 1/ and /spl Phi//sub 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 by Geronimo, Hardin, and Massopust from the iterated interpolation that led to /spl Phi/1 and /spl Phi//sub 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 coefficients in the dilation equation-in another language, to build the 4/spl times/4 paraunitary polyphase matrix in the filter bank. The short support opens new possibilities for applications of filters and wavelets near boundaries.< >
This paper presents a new and efficient way to create multiscaling functions with given approximation order, regularity, symmetry, and short support. Previous techniques were operating in time domain ...and required the solution of large systems of nonlinear equations. By switching to the frequency domain and employing the latest results of the multiwavelet theory we are able to elaborate asimple and efficient method of construction of multiscaling functions. Our algorithm is based on a recently found factorization of the refinement mask through the two-scale similarity transform (TST). Theoretical results and new examples are presented.
The eigenvalue clustering of matrices S_n^{-1}A_n and C_n^{-1}A_n is experimentally studied, where A_n, S_n and C_n respectively are Toeplitz matrices, Strang, and optimal circulant preconditioners ...generated by the Fourier expansion of a function f(x). Some illustrations are given to show how the clustering depends on the smoothness of f(x) and which preconditioner is preferable. An original technique for experimental exploration of the clustering rate is presented. This technique is based on the bisection idea and on the Toeplitz decomposition of a three-matrix product CAC, where A is a Toeplitz matrix and C is a circulant. In particular, it is proved that the Toeplitz (displacement) rank of CAC is not greater than 4, provided that C and A are symmetric.
An important object in wavelet theory is the scaling functionφ(t), satisfying a dilation equationφ(t)=∑Ckφ(2t−k). Properties of a scaling function are closely related to the properties of the symbol ...or maskP(ω)=∑Cke−iωk. The approximation order provided by φ(t) is the number of zeros of P(ω) at ω=π, or in other words the number of factors (1+e−iω) in P(ω). In the case of multiwaveletsP(ω) becomes a matrix trigonometric polynomial. The factors (1+e−iω) are replaced by a matrix factorization of P(ω), which defines the approximation order of the multiscaling function. This matrix factorization is based on the two‐scale similarity transform (TST). In this article we study properties of the TST and show how it is connected with the theory of multiwavelets. This approach leads us to new results on regularity, symmetry, and orthogonality of multiscaling functions and opens an easy way to their construction.
We describe a statistical model for images decomposed in an overcomplete wavelet pyramid. Each coefficient of the pyramid is modeled as the product of two independent random variables: an element of ...a Gaussian random field, and a hidden multiplier with a marginal log-normal prior. The latter modulates the local variance of the coefficients. We assume subband coefficients are contaminated with additive Gaussian noise of known covariance, and compute a MAP estimate of each multiplier variable based on observation of a local neighborhood of coefficients. Conditioned on this multiplier, we then estimate the subband coefficients with a local Wiener estimator. Unlike previous approaches, we (a) empirically motivate our choice for the prior on the multiplier; (b) use the full covariance of signal and noise in the estimation; (c) include adjacent scales in the conditioning neighborhood. To our knowledge, the results are the best in the literature, both visually and in terms of squared error.
Finite element multiwavelets Strela, V.; Strang, G.
Proceedings of IEEE-SP International Symposium on Time- Frequency and Time-Scale Analysis,
1994
Conference Proceeding
Finite elements with support on two intervals span the space of piecewise polynomials with degree 2n-1 and n-1 continuous derivatives. Function values and n-1 derivatives at each mesh-point determine ...these "Hermite finite elements". The n basis functions satisfy a dilation equation with n by n matrix coefficients. Orthogonal to this scaling subspace is a wavelet subspace. It is spanned by the translates of n wavelets W/sub i/(t), each supported on three intervals. The wavelets are orthogonal to all rescalings W/sub i/(2/sup 0/t-k), but not to translates at the same level (j=0). These new multiwavelets achieve 2n vanishing moments and high regularity with symmetry and short support.< >
We present a simple denoising technique for geometric data represented as a semiregular mesh, based on locally adaptive Wiener filtering. The degree of denoising is controlled by a single parameter ...(an estimate of the relative noise level) and the time required for denoising is independent of the magnitude of the estimate. The performance of the algorithm is sufficiently fast to allow interactive local denoising.
Multiwavelet filter banks for data compression Heller, P.N.; Strela, V.; Strang, G. ...
1995 IEEE International Symposium on Circuits and Systems (ISCAS),
1995, Volume:
3
Conference Proceeding
This paper investigates the emerging notion of multiwavelets in the context of multirate filter banks, and applies a multiwavelet system to image coding. Multiwavelets are of interest because their ...constituent filters can be simultaneously symmetric and orthogonal (this combination is impossible for 2-band PR-QMFs), and because one can obtain higher orders of approximation (more vanishing wavelet moments) for a given filter length. We develop symmetric extension methods for finite-length signals under multiwavelet filtering. Techniques are then presented for pre- and post-processing one-dimensional signals in order to effectively exploit multiwavelet structures. Finally, we employ these new tools in a transform coding system and compare their performance with Daubechies' scalar wavelets.
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