Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrix-valued filterbanks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar two-channel wavelet systems. After reviewing this theory, we examine the use of multiwavelets in a filterbank setting for discrete-time signal and image processing. Multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filterbank. We describe two methods (repeated row and approximation/deapproximation) for obtaining such a vector input stream from a one-dimensional (1-D) signal. Algorithms for symmetric extension of signals at boundaries are then developed, and naturally integrated with approximation-based preprocessing. We describe an additional algorithm for multiwavelet processing of two-dimensional (2-D) signals, two rows at a time, and develop a new family of multiwavelets (the constrained pairs) that is well-suited to this approach. This suite of novel techniques is then applied to two basic signal processing problems, denoising via wavelet-shrinkage, and data compression. After developing the approach via model problems in one dimension, we apply multiwavelet processing to images, frequently obtaining performance superior to the comparable scalar wavelet transform.

This paper investigates the emerging notion of multiwavelets in the context of multirate filter banks, and applies a multiwavelet system to image coding. Multiwavelets are of interest because their constituent filters can be simultaneously symmetric and orthogonal (this combination is impossible for 2-band PR-QMFs), and because one can obtain higher orders of approximation (more vanishing wavelet moments) for a given filter length. We develop symmetric extension methods for finite-length signals under multiwavelet filtering. Techniques are then presented for pre- and post-processing one-dimensional signals in order to effectively exploit multiwavelet structures. Finally, we employ these new tools in a transform coding system and compare their performance with Daubechies' scalar wavelets.

Four-coefficient dilation equations are examined and results converse to a theorem of Daubechies-Lagarias are given. These results complete the characterization of those four-coefficient dilation equations having a continuous solution.>