Multiwavelet prefilters. II. Optimal orthogonal prefilters

Abstract

Prefiltering a given discrete signal has been shown to be an essential and necessary step in applications using unbalanced multiwavelets. In this paper, we develop two methods to obtain optimal second-order approximation preserving prefilters for a given orthogonal multiwavelet basis. These procedures use the prefilter construction introduced in part I of this paper. The first prefilter optimization scheme exploits the Taylor series expansion of the prefilter combined with the multiwavelet. The second one is achieved by minimizing the energy compaction ratio (ECR) of the wavelet coefficients for an experimentally determined average input spectrum. We use both methods to find prefilters for the cases of the DGHM and Chui-Lian (CL) multiwavelets. We then compare experimental results using these filters in an image compression scheme. Additionally, using the DGHM multiwavelet with the optimal prefilters from the first scheme, we find that quadratic input signals are annihilated by the high-pass portion of the filter bank at the first level of decomposition.

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.…”

A robust audio watermarking based-on multiwavelet transform

“…The main motivation of using multiwavelet is that it is possible to construct multiwavelets that simultaneously possess desirable properties such as orthogonality, symmetry and compact support with a given approximation order [9]. These properties are not possible in any scalar wavelet.…”

Orthogonal 4-tap integer multiwavelet transforms using matrix factorization

“…While many researchers have investigated multiwavelet of multiplicity [3][4] [5][10] extensively, little work has been published on integer multiwavelet transforms. We will discuss integer transform on 4-tap multiwavelet of multiplicity .…”

“…Unlike scalar wavelet, little literature is available on integer multiwavelets transform due to the pre-and post-process of unbalanced multiwavelets [3][4] [5]. Cheung and Po [6] proposed an integer multiwavelet transform based on box-and-slope multiscaling system and used Haar transform as the associated integer prefilter, which is an approximation to non-trucated transform.…”