Statistical models for wavelet coefficients with applications to denoising and deconvolution

Several signal processing tasks require the use of statistical models. The statistics of wavelet coefficients have been of particular interest due to the popularity of the wavelet transform and its many variations. Applications such as detection, estimation, and deconvolution often rely on accurate modeling of transform coefficients, and numerous models have been proposed in the literature, among them the Laplace, the generalized Gaussian, and the Bessel K form distributions. Modeling coefficients jointly has lead to the study of multivariate extensions of these and other distribution functions, with modest improvements in applications.;This thesis is developed in two parts. The first part deals with an example of statistical waveletbased denoising as applied to video. The utility of simple marginal and joint models for noise-free wavelet coefficients is illustrated in the context of Bayesian estimation of signal in additive white Gaussian noise (AWGN). The contribution is the development of one estimation/denoising strategy that extends estimators developed for images. Several efficient variations are examined. Performance of the proposed denoising algorithms is assessed through relative and comparative simulations, the latter of which draw into comparison three denoising strategies in recent literature. Relative comparisons assess performance of the algorithm's variations as they relate to each other and to the base denoising rule. Simulation conditions are described, and simulations are carried out for several noise levels and nine grayscale video sequences.;Building on the first part, the second part of the thesis looks at one extension of the multivariate Bessel K from (BKF) distribution, termed the multivariate generalized K form (GKF) distribution. The motivation for this is the fact that the expression for the BKF may not be tractable in applications due to the fact that the shape parameter appears as an argument to a special function, while the expression for the GKF distribution is shown to depend on a linear scaled sum of (simpler) Laplace distributions, with only the scaling factors depending on the shape parameter. Therefore while the GKF can be used to approximate the BKF, and retains its advantages such as the extra degree of freedom relative to the Laplace, it may also be simpler to apply. Two applications of the GKF to the problems of Bayesian estimation and deconvolution are presented.