Symmetric–Antisymmetric Orthonormal Multiwavelets and Related Scalar Wavelets
Applied and Computational Harmonic AnalysisPublished 1 May 2000
Lixin Shen, Hwee Hoon Tan, Jo Yew Tham
Citations93
SJR quartileQ1
SJR score2.05
SNIP2.06
Generate an AI Snapshot to get a quick, structured summary of this paper.
Study Snapshot
ObjectiveStudy objective
MethodsResearch methodology
PopulationPopulation studied
Sample sizeSample sizes
OutcomesStudy outcomes here
ResultsStudy results comes here
LimitationsResearch study limitations comes here
A concise AI-generated summary of the paper will appear here once you click Generate AI Snapshot.
TL;DR
It is shown that any length-2Nmultiwavelet system can be constructed from a length-(2N+1) multiwavelet system and vice versa, and two explicit formulations for the construction of multi wavelet functions directly from their associated multiscaling functions are presented.
Abstract
10.1006/acha.1999.0288
Keywords
Computer ScienceMathematicsEngineering
Computers in PhysicsTen Lectures on Wavelets
15,386 Citations1992Ingrid Daubechies, Christopher Heil
This paper presents a meta-analyses of the wavelet transforms of Coxeter’s inequality and its applications to multiresolutional analysis and orthonormal bases.
Communications on Pure and Applied MathematicsOrthonormal bases of compactly supported wavelets
8,187 Citations1988Ingrid Daubechies
This work construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity, by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction.
The Caltech Institute Archives (California Institute of Technology)Multirate Systems and Filter Banks
5,446 Citations1992P. P. Vaidyanathan
This chapter discusses Multirate Filter Bank Theory and Related Topics, which focuses on the role of the wavelet transform in the construction of filter banks.
Journal of Approximation TheoryFractal Functions and Wavelet Expansions Based on Several Scaling Functions
592 Citations1994Jeffrey S. Geronimo, Douglas P. Hardin +1 more
IEEE Transactions on Image ProcessingThe application of multiwavelet filterbanks to image processing
448 Citations1999Vasily Strela, P.N. Heller +3 more
An additional algorithm for multiwavelet processing of two-dimensional (2-D) signals, two rows at a time, is described, and a new family of multiwavelets (the constrained pairs) are developed that is well-suited to this approach.
Applied Numerical MathematicsA study of orthonormal multi-wavelets
370 Citations1996Charles K. Chui, Jian‐ao Lian
IEEE Transactions on Signal ProcessingDesign of prefilters for discrete multiwavelet transforms
295 Citations1996X.-G. Xia, Jeffrey S. Geronimo +2 more
The authors propose a general algorithm to compute multi wavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets, which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms.
SIAM Journal on Mathematical AnalysisConstruction of Orthogonal Wavelets Using Fractal Interpolation Functions
290 Citations1996George C. Donovan, Jeffrey S. Geronimo +2 more
IEEE Transactions on Signal ProcessingShort wavelets and matrix dilation equations
200 Citations1995Gilbert Strang, Vasily Strela
This construction is in parallel with recent progress by Hardin and Geronimo, to develop the underlying algebra from the matrix coefficients in the dilation equation-in another language, to build the 4/spl times/4 paraunitary polyphase matrix in the filter bank.
Transactions of the American Mathematical SocietyWavelets of multiplicity 𝑟
186 Citations1994T. N. T. Goodman, S. L. Lee
Transactions of the American Mathematical SocietyWavelets in wandering subspaces
153 Citations1993T. N. T. Goodman, S. L. Lee +1 more
SIAM Journal on Mathematical AnalysisRefinable Function Vectors
149 Citations1998Zuowei Shen
Regularity of function vectors gives smoothness orders in the time domain and decay rates at infinity in the frequency domain and regularity criteria are established in terms of the vanishing moment order of the matrix mask.
Numerische MathematikApproximation by translates of refinable functions
148 Citations1996Christopher Heil, Gilbert Strang +1 more
Conditions on the matrices $c_k$ that yield approximation of order $p$ from $V_0$ are found, equivalent to the Strang–Fix conditions on the Fourier transforms $\hat f_i(\omega)$ , but for refinable functions they can be explicitly verified from the c_k.
Mathematics of ComputationAn algorithm for matrix extension and wavelet construction
125 Citations1996Wayne Lawton, Sangyeol Lee +1 more
This paper gives a practical method of extending an n x r matrix P(z), r ≤ n, with Laurent polynomial entries in one complex variable z, to a square matrix also with LaurentPoison entries, applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter.
Constructive ApproximationApproximation order provided by refinable function vectors
123 Citations1997Gerlind Plonka
This result is a natural generalization of the result for a single functionϕ, where the refinement mask ofϕ contains the factor ((1 +e−iu)/2)m if approximation orderm is achieved.
Journal of Fourier Analysis and ApplicationsRegularity of refinable function vectors
121 Citations1997Albert Cohen, Ingrid Daubechies +1 more
The results on decay are used to prove uniqueness of solutions and convergence of the cascade algorithm and the existence and regularity of compactly supported solutions of vector refinement equations.
Constructive ApproximationBiorthogonal Wavelet Expansions
108 Citations1997Wolfgang Dahmen, Charles A. Micchelli
Advances in Computational MathematicsRegularity of multiwavelets
95 Citations1997Charles A. Micchelli, Thomas Sauer
This paper studies the close relationship between vector subdivision schemes and a generalized notion of scalar subdivision schemes based on bi-infinite matrices with certain periodicity properties and shows that differentiability of the limit functions can be characterized by factorization properties of the subdivision operator.
SIAM Journal on Mathematical AnalysisEnergy Moments in Time and Frequency for Two-Scale Difference Equation Solutions and Wavelets
84 Citations1992Lars Villemoes
IEEE Transactions on Signal ProcessingOrthogonal multiwavelets with optimum time-frequency resolution
72 Citations1998Qingtang Jiang
A procedure to design orthogonal multiwavelets with good time-frequency resolution is introduced and some optimal multifilter banks are provided.
SIAM Journal on Mathematical AnalysisOn the Regularity of Matrix Refinable Functions
61 Citations1998Qingtang Jiang
Constructive ApproximationOn Existence and Weak Stability of Matrix Refinable Functions
57 Citations1999Qingtang Jiang, Zuowei Shen
