This dissertation was presented to the Faculty of the Graduate School of The University of Texas at Austin in partial fulfillment of the requirements for the degree of

Ph.D. in Electrical Engineering

During the last decade, the explosively developing wavelet theory has proven to be a powerful mathematical tool for signal analysis and synthesis and has found a wide range of successful applications in the area of digital signal processing (DSP). Compared to their counterparts in the Fourier realm, wavelet techniques permit significantly more flexibility in system design for many applications such as multirate filtering, sampling and interpolation, signal modeling and approximation, noise reduction, signal enhancement, feature extraction, and image data compression. Most classical wavelet systems have been constructed from a primarily mathematical point of view, and they are fundamentally suitable for representing continuous-domain functions rather than discrete-domain data. From a discrete-time or DSP perspective, we develop new wavelet systems.

This dissertation focuses on the theory, design, and applications of several novel classes of one-dimensional and multi-dimensional Coiflet-type wavelet systems. In particular, we propose a novel generalized Coifman criterion for designing high-performance wavelet systems, which emphasizes the vanishing moments of both wavelets and scaling functions. The resulting new wavelet systems are appropriate for representing discrete-domain data and enjoy a number of interesting and useful properties such as

• sparse representations for smooth signals,
• interpolating scaling functions,
• linear phase filterbanks, and
• dyadic fractional filter coefficients,

which are promising in solving a large variety of DSP problems. We show that some of the new wavelet systems achieve superior performance (e.g., better rate-distortion performance, better perceptual quality, and lower computational complexity) over the state-of-the-art ones in the field of image coding.