Faculty Bibliography

The denoising of video data should take into account both temporal and spatial dimensions, however, true 3D transforms are rarely used for video denoising. Separable 3-D transforms have artifacts that degrade their performance in applications. This paper describes the design and application of the non-separable oriented 3-D dual-tree wavelet transform for video denoising. This transform gives a motion-based multi-scale decomposition for video - it isolates in its subbands motion along different directions. In addition, we investigate the denoising of video using the 2-D and 3-D dual-tree oriented wavelet transforms, where the 2-D transform is applied to each frame individually.

The problem of image denoising has received more attention than the problem of image sharpening. In the paper, we propose that wavelet-based algorithms for image denoising can be used to perform image sharpening. Consequently, a variety of new image sharpening techniques becomes available. We examine the sharpening of natural images using an algorithm for image denoising with oriented complex 2D wavelets.

It is well documented that the statistical distribution of wavelet coefficients for natural images is non-Gaussian and that neighboring coefficients are highly dependent. In this paper, we propose a new multivariate non-Gaussian probability model to capture the dependencies among neighboring wavelet coefficients in the same scale. The model can be expressed as K exp(-parallel towparallel to) where w is an N-element vector of wavelet coefficients and parallel towparallel to is a convex combination of l(2) norms over subspaces of R-N. This model includes the commonly used independent Laplacian model as a special case but it has many more degrees of freedom. Based on this model, the corresponding non-linear threshold (shrinkage) function for denoising is derived using Bayesian estimation theory. Although this function does not have a closed-form solution, a successive substitution method can be used to numerically compute it.

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.

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.

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.