Faculty Bibliography

This paper describes a simple procedure, based on spectral factorization, for the design of a pair of orthonormal wavelet bases where the two wavelets form a Hilbert transform pair. The two scaling filters respectively have the numerator and denominator of the flat delay all-pass filter as factors. The design procedure allows for an arbitrary number of zero wavelet moments to be specified. A Matlab program for the procedure is given, and examples are also given to illustrate the results.

The FIR 2-band wavelets have found wide applications in practice. One of their disadvantages, however is that they cannot be made both symmetric and orthogonal. There have been some works on filters which are orthogonal and nearly symmetric. In this paper Grobner methods are used to design orthogonal filters with a subset of exactly symmetric coefficients of various lengths, as opposed to nearly symmetric coefficients.

This paper takes up the design of wavelet tight frames that are analogous to Daubechies orthonormal wavelets - that is, the design of minimal length wavelet filters satisfying certain polynomial properties, but now in the oversampled case. The oversampled dyadic DWT considered in this paper is based on a single scaling function and two distinct wavelets. Having more wavelets than necessary gives a closer spacing between adjacent wavelets within the same scale. As a result, the transform (like Kingsbury's dual-tree DWT) is nearly shift-invariant, and can be used to improve denoising. Because the associated time-frequency lattice preserves the dyadic structure of the critically sampled DWT (which the undecimated DWT does not) it can be used with tree-based denoising algorithms that exploit parent-child correlation.

This paper describes a basic difference between multiwavelets and scalar wavelets that explains, without using zero moment properties, why certain complications arise in the implementation of discrete multiwavelet transforms. Assuming we wish to avoid the use of prefilters in implementing the discrete multiwavelet transform, it is suggested that the behavior of the iterated filter bank associated with a multiwavelet basis of multiplicity r is more fully revealed by an expanded set of r(2) scaling functions phi(i,j). This paper also introduces new K-balanced orthogonal multiwavelet bases based on symmetric FIR filters. The nonlinear design equations arising in this work are solved using the Grobner basis. The minimal-length K-balanced multiwavelet bases based on even-length symmetric FIR filters are better behaved than those based on odd-length symmetric FIR filters, as illustrated by special relations they satisfy and by examples constructed.

This paper considers the design and application of wavelet tight frames based on iterated oversampled filter banks. The greater design freedom available makes possible the construction of wavelets with a high degree of smoothness, in comparison with orthonormal wavelet bases. Grobner bases are used to obtain the solutions to the nonlinear design equations. Following the dual-tree DWT of Kingsbury, one goal is to keep the redundancy-factor bounded by 2, instead of allowing it to grow as it does for the undecimated DWT (which is exactly shift-invariant). For the tight frame presented here, optimal-tree based denoising algorithms can be directly applied.

This paper considers the classical sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang, for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal (interpolating). They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which is not continuous, This paper addresses the same question, but in the multiwavelet context, where the situation is different. This paper presents the construction of compactly supported orthogonal multiscaling functions that are continuously differentiable and cardinal. The scaling functions thereby support a Shannon-like sampling theorem, Such wavelet bases are appealing because the initialization of the discrete wavelet transform (prefiltering) is the identity operator.

The discrete wavelet transform (DWT) is usually carried out by filterbank iteration; however, for a fixed number of zero moments, this does not yield a discrete-time basis that is optimal with respect to time localization. This paper discusses the implementation and properties of an orthogonal DWT, with two zero moments and with improved time localization. The basis is not based on filterbank iteration; instead, different filters are used for each scale. For coarse scales, the support of the discrete-time basis functions approaches two thirds that of the corresponding functions obtained by filterbank iteration, This basis, which is a special case of a class of bases described by Alpert, retains the octave-hand characteristic and is piecewise linear (but discontinuous). Closed-form expressions for the filters are given, an efficient implementation of the transform is described, and improvement in a denoising example is shown. This basis, being piecewise linear, is reminiscent of the slant transform, to which it is compared.

The Geronimo-Hardin-Massopust (GHM) multi-wavelet basis exhibits symmetry, orthogonality, short support, and approximation order K = 2, which is not possible for wavelet bases based on a single scaling-wavelet function pair. However, the filterbank associated with this basis does not inherit the zero moment properties of the basis, This work describes a version of the GHM multiscaling functions (constructed with Grobner bases) for which the zero moment properties do carry over to the associated filterbank, That is, the basis is balanced up to its approximation order K = 2.

This paper describes a new class of maximally Rat low-pass recursive digital filters. The filters are realizable as a parallel sum of two all-pass filters, a structure for which low-complexity low-noise implementations exist, Note that, with the classical Butterworth filter of degree N which is retrieved as a special case, it is not possible to adjust the delay (or phase linearity). However, with the more general class of filters described in this paper, the adjustment of the delay becomes possible, and the tradeoff between the delay and the phase linearity can be chosen. The construction of these low-pass filters depends upon a new maximally Bat delay allpole filter, for which the degrees of flatness at omega = 0 and omega = pi are not necessarily equal. For the coefficients of this flat delay filter, an explicit solution is introduced, which also specializes to a previously known result.