Image processing via undecimated wavelet systems
Wells, Raymond O., Jr.
Doctor of Philosophy
We have studied undecimated wavelet transforms and their applications in image denoising. Because of the redundancy of the undecimated wavelet transform, the inversion scheme which implements the Moore-Penrose inverse of the forward transform makes undecimated wavelet systems have excellent performance in signal denoising. We propose an image denoising algorithm that prunes the complete undecimated discrete wavelet packet binary tree to select the best basis. Since we believe discarding the small coefficients permits to choose the best basis from the set of coefficients that will really contribute to the reconstructed image, we propose to select the best basis based on the thresholded wavelet coefficients rather than the original ones. We also propose an exponential decay model for autocorrelations of undecimated wavelet coefficients of real-world images. This is a model that captures the dependency of wavelet coefficients within a scale. With this model we present a parametric solution for FIR Wiener filtering in the undecimated wavelet domain. The persistence property of wavelet coefficients indicates strong dependency across scales. To capture the persistence of UDWT we propose an extension of the wavelet-domain hidden Markov tree model (HMT). By introducing the concept of composite coefficient, we simplify the general coefficient graph to be a tree-structure graph which is very suitable for training to obtain the HMT model parameters. This Bayesian framework allows us to formulate the image denoising problem as computing the posterior estimate.
Mathematics; Statistics; Electronics; Electrical engineering