Denoising by wavelet thresholding using multivariate minimum distance partial density estimation
Scott, Alena I.
Scott, David W.
Doctor of Philosophy
In this thesis, we consider wavelet-based denoising of signals and images contaminated with white Gaussian noise. Existing wavelet-based denoising methods are limited because they make at least one of the following three unrealistic assumptions: (1) the wavelet coefficients are independent, (2) the signal component of the wavelet coefficient distribution follows a specified parametric model, and (3) the wavelet representations of all signals of interest have the same level of sparsity. We develop an adaptive wavelet thresholding algorithm that addresses each of these issues. We model the wavelet coefficients with a two-component mixture in which the noise component is Gaussian but the signal component need not be specified. We use a new technique in density estimation which minimizes an distance criterion (L2E) to estimate the parameters of the partial density that represents the noise component. The L2E estimate for the weight of the noise component, w&d4;L2E , determines the fraction of wavelet coefficients that the algorithm considers noise; we show that w&d4;L2E corresponds to the level of complexity of the signal. We also incorporate information on inter-scale dependencies by modeling across-scale (parent/child) groups of adjacent coefficients with multivariate densities estimated by L 2E. To assess the performance of our method, we compare it to several standard wavelet-based denoising algorithms on a number of benchmark signals and images. We find that our method incorporating inter-scale dependencies gives results that are an improvement over most of the standard methods and are comparable to the rest. The L2E thresholding algorithm performed very well for 1-D signals, especially those with a considerable amount of high frequency content. Our method worked reasonably well for images, with some apparent advantage in denoising smaller images. In addition to providing a standalone denoising method, L2E can be used to estimate the variance of the noise in the signal for use in other thresholding methods. We also find that the L2E estimate for the noise variance is always comparable and sometimes better than the conventional median absolute deviation estimator.
Statistics; Electronics; Electrical engineering