| Signal recovery and noise reduction are closely related signal processing problems of both theoretical and practical interest. In this thesis, these classical problems are studied with a new tool--wavelets.;For a class of signal degradations where the degrading operator is linear shift-invariant and of lowpass filter type, the signal recovery problem is reformulated as finding missing data at the finest scale of a discrete wavelet transform. In this framework, the wavelet reproducing equation plays a fundamental role in determining a unique and stable recovery. It is shown that the solution to the stabilized wavelet reproducing equation is equivalent to a constrained least-squares solution. By discovering such a relationship, a new class of regularizing operators is obtained, and the wavelet-regularized solution to the signal recovery problem is derived in one and two dimensions. In experiments, the wavelet-regularized image restoration results in 75-85% reduction in mean square errors in noise-free cases, and 35-60% reduction in noisy cases.;For signal degradations caused solely by noise contamination, a new technique for noise reduction is developed based on the wavelet maxima representation. The concept of the wavelet maxima tree is formalized and algorithms for constructing such trees are developed for 1-D signals and 2-D images. Metrics are designed for the wavelet maxima tree to measure the scaling and spatial stabilities of wavelet maxima. These metrics reflect some perceptual criteria for discriminating objects from a noisy background. A complete computational framework for denoising digital signals and images is implemented. In experiments with both simulated and real signals and images, the new technique is able to reduce noise power by more than 90% and keep the edge gradients to 80-120% of their original values. With these results, the new technique outperforms widely used Wiener and median filters in solving the tradeoff between smoothing noise and preserving edges. |
