Methods for digital image compression have been the subject of much study over the past decade. Advances in wavelet transforms and quantization methods have produced algorithms capable of surpassing the existing image compression standards like the Joint Photographic Experts Group (JPEG) algorithm. For best performance in image compression, wavelet transforms require filters that combine a number of desirable properties, such as orthogonality and symmetry. However, the design possibilities for wavelets are limited because they cannot simultaneously possess all of these desirable properties. The relatively new field of multiwavelets shows promise in removing some of the limitations of wavelets. Multiwavelets offer more design options and hence can combine all desirable transform features. The few previously published results of multiwavelet-based image compression have mostly fallen short of the performance enjoyed by the current wavelet algorithms. This thesis presents new multiwavelet transform methods and measurements that verify the potential benefits of multiwavelets. Using a zerotree quantization scheme modified to better match the unique decomposition properties of multiwavelets, it is shown that the latest multiwavelet filters can give performance equal to, or in many cases superior to, the current wavelet filters. The performance of multiwavelet packets is also explored for the first time and is shown to be competitive to that of wavelet packets in some cases. The wavelet and multiwavelet filter banks are tested on a much wider range of images than in the usual literature, providing a better analysis of the benefits and drawbacks of each. NOTE: (03/2007) An updated copy of this ETD was added after there were patron reports of problems with the file. / Master of Science

Compression of vector-valued datasets is increasingly needed for addressing the significant storage and transmission burdens associated with research activities in large-scale computational fluid dynamics and environmental science. However, vector-valued compression schemes have traditionally received few investigations within the data-compression community. Consequently, this dissertation conducts a systematic study of effective algorithms for the coding of vectorvalued datasets and builds practical embedded compression systems for both static and timevarying vector fields. In generalizing techniques from the relatively mature field of image and video coding to vector data, three critical issues must be addressed: the design of a vector wavelet transform (VWT) that is amenable to vector-valued compression applications, the implementation of vector-valued intraframe coding that enables embedded coding, and the investigation of interframe-compression techniques that are appropriate for the complex temporal evolutions of vector features. In this dissertation, we initially invoke multiwavelets to construct VWTs. However, a balancing problem arises when existing multiwavelets are applied directly to vector data. We analyze extensively this performance failure and develop the omnidirectional balancing (OB) design criterion to rectify it. Employing the OB principle, we derive with a family of biorthogonal multiwavelets possessing desired balancing and symmetry properties and yielding performance far superior to that of VWTs implemented via other multiwavelets. In the second part of the dissertation, quantization schemes for vector-valued data are studied, and a complete embedded coding system for static vector fields is designed by combining a VWT with suitable vector-valued successive-approximation quantization. Finally, we extend several interframecompression techniques from video-coding applications to vector sequences for the compression of time-varying vector fields. Since the complexity of temporal evolutions of vector features limits the efficiency of the simple motion models which have been successful for natural video sources, we develop a novel approach to motion compensation which involves applying temporal decorrelation to only low-resolution information. This reduced-resolution motion-compensation technique results in significant improvement in terms of rate-distortion performance.