Faculty Bibliography

This paper describes the design of type III and type IV linear-phase finite-impulse response (FIR) low-pass digital differentiators according to the maximally flat criterion. We introduce a two-term recursive formula that enables the simple stable computation of the impulse response coefficients. The same recursive formula is valid for both Type III and Type IV solutions.

Connexions is a new approach to authoring, teaching, and learning that aims to fully exploit modem information technology. Available free of charge to anyone under open-content and open-source licenses, Connexions offers custom-tailored, current course material, is adaptable to a wide range of learning styles, and encourages students to explore the links among courses and disciplines. In contrast to the traditional process of textbook writing and publishing, Connexions fosters world-wide, cross-institution communities of authors, instructors, and students, who collaboratively and dynamically fashion "modules" from which courses are constructed. We believe the ideas and philosophy embodied by Connexions have the potential to change the very nature of textbook writing and publishing, producing a dynamic, interconnected educational environment that is pedagogically sound, both time and cost efficient, and fun. This paper overviews the philosophy and technology behind Connexions and describes a nascent community developing material for DSP education.

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.

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.

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.

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.