Construction of Orthogonal Wavelets Using Fractal Interpolation Functions

Donovan, George C.; Geronimo, Jeffrey S.; Hardin, Douglas P.; Massopust, Peter

doi:10.1137/s0036141093256526

Cited by 265 publications

(134 citation statements)

References 16 publications

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“…Therefore, if an orthonormal multiwavelet system is balanced of order , then the associated multiscaling function has an approximation power of at least . We can notice that the converse is false: the DGHM [6] multiscaling function has an approximation power of 2 but is not even balanced [20]. However, we have the following theorem.…”

Section: A Approximation Power and Balancing Ordermentioning

confidence: 97%

“…5) The highpass filters are easily derived from the lowpass by imposing to be symmetric and to be antisymmetric. The orthonormality conditions (6) give unique solutions up to a change of sign. Using this approach, we have been able to construct all the shortest length (as defined below) orthonormal multiwavelets with flipped scaling functions and symmetric/antisymmetric wavelets for balancing order up to 4.…”

Section: A Bat Familymentioning

confidence: 99%

“…Similar relations are easily obtained for the biorthogonal scalar case. Without much loss of generality, we will, in this paper, look at the orthogonal case for multiwavelets (the interest of biorthogonality in the multiwavelet framework being not as obvious since there is no negative result [6] preventing us from constructing orthogonal, FIR, linear-phase multifilters).…”

Section: Introductionmentioning

confidence: 99%

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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: A Approximation Power and Balancing Ordermentioning

confidence: 97%

Section: A Bat Familymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Lebrun

Vetterli

2001

IEEE Trans. Signal Process.

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“…Typical examples of such scaling function vectors and multi-wavelets and their corresponding filters {P k } and {Q k } are the GHM multi-wavelets [6,5] and CL multi-wavelets [4], of dimension r = 2. We remark that the GHM and (one of the) CL scaling function vectors have polynomial reproduction order 2, and hence, their corresponding multi-wavelets have vanishing moments of order 2.…”

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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“…The next theorem gives the relationship between X and the fractal functions The interested reader may consult [4] or [6] for a proof. The fractal functions in Pd(R) may be used to construct finitely generated shift-invariant subspaces of L2(Q) (cf., for instance, [4, 61).…”

mentioning

confidence: 99%