The application of multiwavelet filterbanks to image processing

Abstract

Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrix-valued filter banks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar 2-channel wavelet systems. After reviewing this recently developed theory, we examine the use of multiwavelets in a filter bank setting for discrete-time signal and image processing. Multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filter bank. We describe two methods (repeated row and approximation/deapproximation) for obtaining such a vector input stream from a one-dimensional 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 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 applied multiwavelet processing to images, frequently obtaining performance superior to the comparable scalar wavelet transform.

High-order balanced multiwavelets: theory, factorization, and design

“…Now, to prove the theorem, we will first prove that the minimal length condition with balancing and orthogonality implies that the refinement mask has a multiplexed filter structure. 2 The coefficients of BAT O1 already appeared in [3] and [34].…”

Balanced multi-wavelets in $\mathbb R^s$

“…The indirect approach is to apply certain appropriate prefiltering to the input data sequence {x k } as well as to the low-pass output of each wavelet decomposition level to be used as input for the next level of wavelet decomposition (see [1,7,19,20]). On the other hand, the direct approach is to design Φ and Ψ so that the decomposition algorithm (1.1) ensures polynomial output {y L k } of degree K −1 (or order K) and zero output {y H k }, when the polynomial data sequences {x k } = {v s,k,m }, k ∈ Z, for 0 ≤ s ≤ r − 1 and 0 ≤ m ≤ K − 1, are used as input sequences in (1.1), where {P k }/{Q k } are the refinement (or two-scale) sequences corresponding to the orthonormal Φ and Ψ.…”

Novel Video Coder Using Multiwavelets

“…Before applying the multiwavelet transform to the input images or residuals, the image is to be preprocessed. The prefilter (Strela, V., 1996; Strela, V., 1998) is chosen corresponding to the filters chosen for applying multiwavelet transforms (Strela, V & Walden A.T., 1998; Strela V et al, 1999). Similarly, the post processing is to be done at the receiver side.…”