Faculty Bibliography

In this paper, the basic algebraic properties of the optimal PWM problem for single-phase inverters are revealed. Specifically, it is shown that the nonlinear design equations given by the standard mathematical formulation of the problem can be reformulated, and that the sought solution can be found by computing the roots of a single univariate polynomial P(x), for which algorithms are readily available. Moreover, it is shown that the polynomials P(x) associated with the optimal PWM problem are orthogonal and can therefore be obtained via simple recursions. The reformulation draws upon the Newton identities, Pade approximation theory, and properties of symmetric functions. As a result, fast O(n log(2) n) algorithms are derived that provide the exact solution to the optimal PWM problem. For the PWM harmonic elimination problem, explicit formulas are derived that further simplify the algorithm.

Most simple nonlinear thresholding rules for wavelet-based denoising assume the wavelet coefficients are independent. However, wavelet coefficients of natural images have significant dependency. In this paper, a new heavy-tailed bivariate pdf is proposed to model the statistics of wavelet coefficients, and a simple nonlinear threshold function (shrinkage function) is derived from the pdf using Bayesian estimation theory. The new shrinkage function does not assume the independence of wavelet coefficients.

Several authors have demonstrated that significant improvements can be obtained in wavelet-based signal processing by utilizing a pair of wavelet transforms where the wavelets form a Hilbert transform pair. This paper describes design procedures, based on spectral factorization, for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate Hilbert transform pair. Both orthogonal and biorthogonal FIR solutions are presented, as well as IIR solutions. In each case, the solution depends on an allpass filter having a flat delay response. The design procedure allows for an arbitrary number of vanishing wavelet moments to be specified. A Matlab program for the procedure is given, and examples are also given to illustrate the results.

The first nonlinear rules for wavelet based image denoising assume wavelet coefficients are independent. However it is well-known that there axe strong dependencies between coefficients such as interscale and intrascale dependencies. We have introduced a non-Gaussian bivariate pdf which exploits the interscale dependencies between a coefficient and its parent [7,8]. In this paper, how to extend this pdf in order to include the other dependencies will be discussed and in one example we will derive a multivaxiate shrinkage rule. The good performance of this new rule will be illustrated on an image denoising algorithm which capture also interscale dependencies.

This paper describes a simple formulation for the non-iterative design of narrow-band FIR linear-phase low-pass digital differentiators. The frequency response of the filters are flat around dc and have equally spaced nulls in the stopband. The design problem is formulated so as to avoid the complexity or ill-conditioning of standard methods for the design of similar filters when those methods are used to design narrow-band filters with long impulse responses.

The performance of image-denoising algorithms using wavelet transforms can be improved significantly by taking into account the statistical dependencies among wavelet coefficients as demonstrated by several algorithms presented in the literature. In two earlier papers by the authors, a simple bivariate shrinkage rule is described using a coefficient and its parent. The performance can also be improved using simple models by estimating model parameters in a local neighborhood. This letter presents a locally adaptive denoising algorithm using the bivariate shrinkage function. The algorithm is illustrated using both the orthogonal and dual tree complex wavelet transforms. Some comparisons with the best available results will be given in order to illustrate the effectiveness of the proposed algorithm.

Most simple nonlinear thresholding rules for wavelet-based denoising assume that the wavelet coefficients are independent. However, wavelet coefficients of natural images have significant dependencies. In this paper, we will only consider the dependencies between the coefficients and their parents in detail. For this purpose, new non-Gaussian bivariate distributions are proposed, and corresponding nonlinear threshold functions (shrinkage functions) are derived from the models using Bayesian estimation theory. The new shrinkage functions do not assume the independence of wavelet coefficients. We will show three image denoising examples in order to show the performance of these new bivariate shrinkage rules. In the second example, a simple subband-dependent data-driven image denoising system is described and compared with effective data-driven techniques in the literature, namely VisuShrink, SureShrink, BayesShrink, and hidden Markov models. In the third example, the same idea is applied to the dual-tree complex wavelet coefficients.

This paper considers the design 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. In particular, this paper takes up the design of systems 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. Grobner bases are used to obtain the solutions to the nonlinear design equations. Following the dual-tree DWT of Kingsbury, one goal is to achieve near shift invariance while keeping the redundancy factor bounded by 2, instead of allowing it to grow as it does for the undecimated DWT (which is exactly shift invariant). Like the dual tree, the overcomplete DWT described in this paper is less shift-sensitive than an orthonormal wavelet basis. Like the examples of Chui and He, and Ron and Shen, the wavelets are much smoother than what is possible in the orthonormal case. © 2001 Academic Press.

This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an alternative derivation and explanation for the result by Kingsbury, that the dual-tree DWT is (nearly) shift-invariant when the scaling filters satisfy the same offset.