Construction of multivariate compactly supported tight wavelet frames

Lai, Ming-Jun; Stöckler, Joachim

doi:10.1016/j.acha.2006.04.001

Cited by 68 publications

(117 citation statements)

References 24 publications

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“…We will use the following slight generalization of Theorem 3.4 of Lai and Stöckler [32]. The proof is similar, and is outlined in [41,Theorem 1], where the theorem is stated in a trivially equivalent form.…”

Section: Implicit In the Proofs Of [48 Lemma 31] [50 Proposition mentioning

confidence: 99%

“…In [41], San Antolín and Zalik developed a method to generate wavelet frames using the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler [32]. This method was used in [42] to construct, for a 2 × 2 expansive dilation matrix with integer entries and determinant ±2, families of compactly supported tight framelets with three generators and with any desired degree of smoothness.…”

Section: Introductionmentioning

confidence: 99%

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Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Antolín

Zalik

2015

J Fourier Anal Appl

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Abstract. Let A ∈ R d×d , d ≥ 1 be a dilation matrix with integer entries and | det A| = 2. We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is 2 d + d and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. For the particular case d = 2 and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values.

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Section: Implicit In the Proofs Of [48 Lemma 31] [50 Proposition mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Antolín

Zalik

2015

J Fourier Anal Appl

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“…In addition, such a decomposition can be generalized to the high dimensional case. Note that [17] showed that R(ω 1 , ω 2 ) corresponding to some type of 2-D box splines can always be decomposed as a finite sum of the squares of some 2-D trigonometric polynomials h j (ω 1 , ω 2 ). Hence R(ω) can be written as R(ω) = |h 0 (ω)| 2 + |h 1 (ω)| 2 with…”

Section: )mentioning

confidence: 99%

“…Now with 4 framelets added, {2 j/2 ψ ( ) (2 j · −k) : j ∈ Z, k ∈ Z 2 , 1 ≤ ≤ 7} is a tight frame of L 2 (R 2 ). Tight frames associated with B 222 with 7 framelets were constructed in [17] and [15]. All the highpass filters constructed here have the same size of support as the lowpass filter p. These highpass filters have a smaller size of support than those constructed in both [17] and [15].…”

Section: )mentioning

confidence: 99%

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Tight wavelet frames in low dimensions with canonical filters

Jiang

Shen

2015

Journal of Approximation Theory

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This paper is to construct tight wavelet frame systems containing a set of canonical filters by applying the unitary extension principle of [20]. A set of filters are canonical if the filters in this set are generated by flipping, adding a conjugation with a proper sign adjusting from one filter. The simplest way to construct wavelets of s-variables is to use the 2 s − 1 canonical filters generated by the refinement mask of a box spline. However, almost all wavelets (except Haar or the tensor product of Haar) defined by the canonical filters associated with box splines do not form a tight wavelet frame system. We consider how to build a filter bank by adding filters to a canonical filter set generated from the refinement mask of a box spline in low dimension, so that the wavelet system generated by this filter bank forms a tight frame system. We first prove that for a given low dimension box spline of s-variables, one needs at least additional 2 s filters to be added to the canonical filters from the refinement mask (that leads to the total number of highpass filters in the filter bank to be 2 s+1 −1) to have a tight wavelet frame system. We then provide several methods with many interesting examples of constructing tight wavelet systems with the minimal number of framelets that contain canonical filters generated by the refinement masks of box splines. The supports of the resulting framelets are not bigger than that of the corresponding box spline whose refinement mask is used to generate the first 2 s − 1 canonical filters in the filter bank. In many of our examples, the tight frame filter bank has the double-canonical property, meaning it is generated by adding another set of canonical filters generated from a highpass filter to the canonical filters generated by the refinement mask to make a tight frame system. Key words and phrases: Wavelet tight frame, symmetry, B-spline, box spline, canonical filters, semi-canonical tight frame filter bank, double-canonical tight frame filter bank.

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Multivariate Splines and Their Applications

Lai¹

2009

Encyclopedia of Complexity and Systems Science

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Construction of multivariate compactly supported tight wavelet frames

Cited by 68 publications

References 24 publications

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Tight wavelet frames in low dimensions with canonical filters

Multivariate Splines and Their Applications

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