The application of multiwavelet filterbanks to image processing

TLDR

Multiwavelets extend wavelet theory by providing matrix‑valued filterbanks that achieve simultaneous orthogonality, symmetry, and short support—features unattainable with scalar two‑channel systems—and they require multiple input streams. The study investigates the application of multiwavelets within a filterbank framework for discrete‑time signal and image processing. The authors devise methods to generate vector input streams from one‑dimensional signals (repeated‑row and approximation/deapproximation), develop symmetric‑extension algorithms, and introduce a two‑row 2‑D processing scheme with a new constrained‑pair multiwavelet family. These techniques were tested on denoising via wavelet‑shrinkage and data compression, and on images they frequently achieved performance superior to comparable scalar wavelet transforms.

Abstract

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.