Design of prefilters for discrete multiwavelet transforms

Xia, X.-G.; Geronimo, Jeffrey S.; Hardin, Douglas P.; Suter, Bruce W.

doi:10.1109/78.482009

Cited by 279 publications

(125 citation statements)

References 24 publications

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“…Consequently, we introduced the concept of balanced multiwavelets, which is now also further investigated by several other authors [17], [27], [28]. One of the goals of this concept is to avoid the intricate steps of pre/post filtering [11], [37] that are required with systems based on multiwavelets that do not satisfy the interpolation/approximation properties of balancing. Inspired by some of the results from [4], [23], and [24], we will clarify the relations between balancing order (discrete-time property) and approximation power (continuous-time property) and prove that the notion of balancing order is truly central to the whole issue of regularity for multiwavelets.…”

Section: Introductionmentioning

confidence: 99%

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

Lebrun

Vetterli

2001

IEEE Trans. Signal Process.

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Abstract-This paper deals with multiwavelets and the different properties of approximation and smoothness associated with them. In particular, we focus on the important issue of the preservation of discrete-time polynomial signals by multifilterbanks. We introduce and detail the property of balancing for higher degree discrete-time polynomial signals and link it to a very natural factorization of the refinement mask of the lowpass synthesis multifilter. This factorization turns out to be the counterpart for multiwavelets of the well-known zeros at condition in the usual (scalar) wavelet framework. The property of balancing also proves to be central to the different issues of the preservation of smooth signals by multifilterbanks, the approximation power of finitely generated multiresolution analyses, and the smoothness of the multiscaling functions and multiwavelets. Using these new results, we describe the construction of a family of orthogonal multiwavelets with symmetries and compact support that is indexed by increasing order of balancing. In addition, we also detail, for any given balancing order, the orthogonal multiwavelets with minimum-length multifilters.

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Section: Introductionmentioning

confidence: 99%

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

Lebrun

Vetterli

2001

IEEE Trans. Signal Process.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Chui

Jiang

2004

Math. Comp.

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Abstract. The notion of K-balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi-wavelets to ensure the property of preservation/annihilation of scalarvalued discrete polynomial data of order K (or degree K − 1), when decomposed by the corresponding matrix-valued low-pass/high-pass filters. While this condition is indeed precise, to the best of our knowledge only the proof for K = 1 is known. In addition, the formulation of the K-balancing condition for K ≥ 2 is so prohibitively difficult to satisfy that only a very few examples for K = 2 and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the Kbalancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and bi-orthogonal settings for any K ≥ 1.

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“…Thus, for starting the algorithm, scalar data must be preprocessed to get the input vector. Repeated row and critically sampled approaches are two main preprocessing methods have been developed to do such preprocessing (23,26,27).…”

Section: Multiwavelet Transformmentioning

confidence: 99%

Functional magnetic resonance imaging activation detection: Fuzzy cluster analysis in wavelet and multiwavelet domains

Jahanian

Soltanian-Zadeh

Hossein-Zadeh

2005

Magnetic Resonance Imaging

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Purpose: To present novel feature spaces, based on multiscale decompositions obtained by scalar wavelet and multiwavelet transforms, to remedy problems associated with high dimension of functional magnetic resonance imaging (fMRI) time series (when they are used directly in clustering algorithms) and their poor signal-to-noise ratio (SNR) that limits accurate classification of fMRI time series according to their activation contents. Materials and Methods:Using randomization, the proposed method finds wavelet/multiwavelet coefficients that represent the activation content of fMRI time series and combines them to define new feature spaces. Using simulated and experimental fMRI data sets, the proposed feature spaces are compared to the cross-correlation (CC) feature space and their performances are evaluated. In these studies, the false positive detection rate is controlled using randomization. To compare different methods, several points of the receiver operating characteristics (ROC) curves, using simulated data, are estimated and compared. Results:The proposed features suppress the effects of confounding signals and improve activation detection sensitivity. Experimental results show improved sensitivity and robustness of the proposed method compared to the conventional CC analysis. Conclusion:More accurate and sensitive activation detection can be achieved using the proposed feature spaces compared to CC feature space. Multiwavelet features show superior detection sensitivity compared to the scalar wavelet features.

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Design of prefilters for discrete multiwavelet transforms

Cited by 279 publications

References 24 publications

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

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

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

Functional magnetic resonance imaging activation detection: Fuzzy cluster analysis in wavelet and multiwavelet domains

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