Denoising via Empirical Bayesian Pursuit

219 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2002.Linear time-frequency and time-scale representations (e.g., the discrete Gabor representation or the discrete wavelet representation) provide useful tools for analyzing a variety of time-varying sampled signals including speech, medical and geophysical data, communications signals, and images. These representations often yield overdetermined signal expansions; for example, adaptive representations such as those arising from best window or best basis methods frequently compute highly overdetermined representations prior to selecting a subset of coefficients for the analysis representation. This dissertation addresses novel performance metrics and methods for blind signal recovery, or denoising, that employ all of the overdetermined representation coefficients. The introduction of L-unitary frames facilitates the analysis, for which many nondecimated, linear, time-frequency and time-scale representations qualify, as do mergers of multiple L-unitary frames. Worst-case bounds are derived on squared estimation error when denoising via hard-thresholding followed by efficient averaging-based synthesis in bounded noise environments; similar bounds guarantee minimum signal-to-interference ratios for spread-spectrum interference suppression. After this, the potential of denoising in a signal-adapted frame obtained via an eigendecomposition of the threshold-then-average denoising filter is considered, including proposing alternative eigendomain weightings as well as the derivation of lower bounds on signal concentration in the new eigenframe representation. Following the eigenanalysis of the threshold-then-average denoising filter, a hidden Gaussian mixture (GM) signal model is considered. Monte Carlo Markov-chain methods are developed for converging to optimal model parameter estimates, which are then used to generate a signal-dependent Wiener filter for denoising. Finally, the solution to the latent GM model problem is briefly related to traditional methods such as complexity-based signal reconstruction, projection onto (signal-adapted) convex subsets, and pursuit-based representation methods.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD