An algorithm for matrix extension and wavelet construction

Lawton, Wayne; Lee, S. L.; Shen, Zuowei

doi:10.1090/s0025-5718-96-00714-4

Cited by 113 publications

(59 citation statements)

References 17 publications

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“…In this article we will make use of several other nonsymmetric multiwavelets with desirable properties. More on the construction of multiscaling functions and multiwavelets can be found in [1,9,13,18,22,24,30,31,38,39,41]. …”

Section: Multiwavelets -Several Wavelets With Several Scaling Functionsmentioning

confidence: 99%

The application of multiwavelet filterbanks to image processing

Strela

Heller

Strang

et al. 1999

IEEE Trans. on Image Process.

376

213

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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.

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Section: Multiwavelets -Several Wavelets With Several Scaling Functionsmentioning

confidence: 99%

The application of multiwavelet filterbanks to image processing

Strela

Heller

Strang

et al. 1999

IEEE Trans. on Image Process.

376

213

View full text Add to dashboard Cite

show abstract

“…By the multiresolution analysis structure, , where is the orthogonal complement of in , and we can construct an orthonormal basis of generated by the multiwavelets and their integer translates by introducing by (2) where is a sequence of matrices of real coefficients obtained by completion of (a detailed exposition of the completion scheme is given in [18]). Introducing in the -domain the refinement masks and , (1) and (2) translate in the Fourier domain into and…”

Section: Multiwaveletsmentioning

confidence: 99%

“…For example, we get Sobolev smoothness by proving that for arbitrarily small, we have Now, in the special case the multifilterbanks has balancing order , we have the factorization for with . Assuming furthermore that and introducing (17) we get by Theorem 4.1 [4] that there exists a constant , such that (18) However, the computation of this supremum is highly impractical. Here, we introduce the heuristic of the invariant cycles that have been proved to be optimal in many cases [2].…”

Section: B Smoothness and Balancing Ordermentioning

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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“…This problem was solved by Lawton, Lee, and Shen in [22]. It will be interesting to know whether it is always possible to construct symmetric multiple wavelets if the multiple refinable functions φ 1 , .…”

Section: Multiple Waveletsmentioning

confidence: 99%

Vector subdivision schemes and multiple wavelets

Jia¹,

Riemenschneider

Zhou

1998

Math. Comp.

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Abstract. We consider solutions of a system of refinement equations written in the form φ = α∈Z a(α)φ(2 · −α), where the vector of functions φ = (φ 1 , . . . , φ r ) T is in (Lp(R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. Associated with the mask a is a linear operator Qa defined on (Lp(R)) r by Qaf := α∈Z a(α)f (2 · −α). This paper is concerned with the convergence of the subdivision scheme associated with a, i.e., the convergence of the sequence (Q n a f ) n=1,2,... in the Lp-norm. Our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the L 2 -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations.Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

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An algorithm for matrix extension and wavelet construction

Cited by 113 publications

References 17 publications

The application of multiwavelet filterbanks to image processing

The application of multiwavelet filterbanks to image processing

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

Vector subdivision schemes and multiple wavelets

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