I. Daubechies, B. Han, A. Ron, and Z. Shen, Framelets: MRA-Based Constructions of Wavelet Frames, 2001. Preprint.  Home/Search   Document Details and Download   Summary   Related Articles   Check

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I. Daubechies, B. Han, A. Ron, and Z. W. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal., to appear.

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I. Daubechies, B. Han, A. Ron and Z. W. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal., to appear.

....tight wavelet frame with n generators. More recently, Chui and He [1] also see Petukhov [12] showed that if the mask a for a symmetric refinable function satisfies # 1 T, 1.4) then one can derive a symmetric tight wavelet frame with three generators. Recently, Daubechies et al. [6] and Chui et al. 2] obtained the following interesting procedure that yields all possible MRA tight wavelet frames derived from a refinable function. Theorem 1.1. Let # be a refinable function in L 2 (R) such that #(2#) a(e i# ) #(#) for a Laurent polynomial a with a(1) 1. Suppose that ....

....frame in L 2 (R) According to Theorem 1.1, a framelet filter bank consists of a low pass filter a and r high pass filters a . In order to design a framelet filter bank, one has to split the matrix M# in (1.6) into the form of (1.5) Using Theorem 1.1, it was demonstrated in [2] also c.f. [6]) that for any refinable function # L 2 (R) whose integer shifts are stable, one can obtain an MRA tight wavelet frame with two generators. Unfortunately, when # is symmetric, the construction in [2, 6] cannot guarantee the symmetry of the two constructed generators which do not have symmetry in ....

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I. Daubechies, B. Han, A. Ron and Z. W. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comp. Harmon. Anal., 14 (2003), 1--46.

.... all the constructions of tight framelets are based, either implicitly or explicitly, on one of two construction principles : the first of which is called the Unitary Extension Principle (UEP) RS2] while the second, more general and more recent, one is the Oblique Extension Principle (OEP) [DHRS]. In all the specific novel constructions of compactly supported and other tight framelets (splines, box splines, pseudo splines) of [RS2] RS4] and [DHRS] X (#) is never orthogonal to X (#) in all these constructions, the set X j (#) lies in closed span of the set X j 1 (#) let al..one that ....

.... the Unitary Extension Principle (UEP) RS2] while the second, more general and more recent, one is the Oblique Extension Principle (OEP) DHRS] In all the specific novel constructions of compactly supported and other tight framelets (splines, box splines, pseudo splines) of [RS2] RS4] and [DHRS], X (#) is never orthogonal to X (#) in all these constructions, the set X j (#) lies in closed span of the set X j 1 (#) let al..one that those sets are not orthogonal one to the other. Correspondingly, for all these cases, the dimension trace function satisfies the inequalities 0 1, but ....

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I. Daubechies, B. Han, A. Ron, and Z. Shen (2001), "Framelets: MRA-based constructions of wavelet frames", preprint. Ftp site: ftp://ftp.cs.wisc.edu/Approx file dhrs.ps;

....5.3] It is easy to see that (2.3) can be rewritten as follows: a( 2 j=d)b( d ) j ( j = 0; d 1; 2.5) where denotes the Dirac sequence such that 0 = 1 and j = 0 for all j 2 Znf0g. Tight wavelet frames and dual wavelet frames have been investigated in [2, 6] for the case d = 2. In this paper, we shall give a systematic study of dual wavelet frames with a general dilation factor. We mention that Theorem 2.2 can also be veri ed using the characterization of dual wavelet frames in [7, 11] In order to prove the main results in this paper, the following ....

....1 1.5 (c) 2 0 2 4 0.2 0 0.2 0.4 0.6 (d) The functions have vanishing moments of orders 2; 4; 4; 2, respectively. f 2 wavelet frames. When the dilation factor d = 2 and = e = Bm , pairs of dual 2 wavelet frames derived from and e have been also constructed in [6]. It turns out that up to some integer shifts the construction in [6] for B spline functions coincides with the construction in Corollary 3.5 for B spline functions with the particular choice as given by ( Pm (sin =2) with 1 (2j 1) 2j) 2j 1) x 2m = Pm (x) O(jxj ....

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I. Daubechies, B. Han, A. Ron and Z. W. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal., to appear.

....continues to define the notion of MRA based wavelet constructions and to provide a complete characterization of all MRA based tight wavelet frames. The theory of [RS3] led several authors to developing various interesting constructions of compactly supported tight wavelet systems (see e.g. CHS] [DHRS], GR] RS3] RS4] RS5] Finally, band limited wavelet tight frames were constructed earlier through multiresolution analysis in [BL] Definition 3.20: wavelet systems. Let # ) be a finite set. Let s be a d expansive matrix, i.e. a matrix whose spectrum lies outside the closed unit ....

I. Daubechies, B. Han, A. Ron, and Z. Shen (2001), "Framelets: MRA-based constructions of wavelet frames", preprint. Ftp site: ftp://ftp.cs.wisc.edu/Approx file dhrs.ps;

....by half integers, such 2rd wavelet functions, which generate a pair of dual d wavelet frames, can be real valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d re nable function vector are also considered. This paper generalizes the work in [5, 12, 13] on constructing dual wavelet frames from scalar re nable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper. Key words: Dual wavelet frames, wavelet frames, re nable function vectors, multiwavelets, re nable Hermite interpolants, sum rules, ....

....the literature, for example, to mention only a few here, see [1] 32] and numerous references therein. Recently, there is a growing interest in both theory and application to study and construct redundant systems such as wavelet frames due to several attractive features of a redundant system (see [1, 2, 4, 5, 8, 10, 12, 13, 24, 25, 26, 27, 30]) Our work in this paper is following the lines developed in Chui, He and St ockler [5] Daubechies, Han, Ron and Shen [13] Daubechies and Han [12] and numerous work on multiwavelets in the literature. In particular, we shall generalize the results in [12] to the multiwavelet case. Before ....

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I. Daubechies, B. Han, A. Ron and Z. W. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal., to appear.

....artifacts are quite rare; thus, our need for good time localization together with our insistence on an artifact free wavelet system narrowed the search for the optimal system in a substantial way. The wavelet system we employ: We use a bi frame version of a system known as PS(4,1)Type II (cf. [25]) This is a framelet system, i.e. a redundant wavelet system (which essentially means that r, the number of high pass filters, is larger than 1; a simple count shows that, if r 1, the total number of wavelet coefficients exceeds the length of the original signal) In our work, the redundancy ....

I. Daubechies, B. Han, A. Ron, and Z. Shen, "Framelets: MRA-based constructions of wavelet frames," Preprint: ftp://ftp.cs.wisc.edu/Approx/dhrs.ps, 2001.

....In this paper the wavelet decomposition ideas are used only to seek out a set of potential knots. They are followed by statistical model selection which identifies the final set of the knots. The specific wavelets used in the paper are Haar wavelet and cubic spline wavelet frame (see [1] and [2]) The former leads to simple averaging as low pass and the first order difference as high pass. The latter leads to local smoothing as low pass and the fourth order difference as high pass. After the data pass through such filters, we take the local extrema as potential knots; see Section 3 for a ....

I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: MRAbased constructions of wavelet frames, preprint (2000).

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I. Daubechies, B. Han, A. Ron, and Z. Shen, Framelets: MRA-Based Constructions of Wavelet Frames, 2001. Preprint.

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I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (1) (January 2003) 1--46.

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I. Daubechies, B. Han, A. Ron, and Z. Shen. (2001) Framelets: MRA-based constructions of wavelet frames. [Online]. Available: ftp://ftp.cs.wisc.edu/Approx/dhrs.ps, preprint.

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I. Daubechies, B. Han, A. Ron, and Z. Shen. Framelets: MRA-based constructions of wavelet frames. Applied and Computational Harmonic Analysis, 14(1):1--46, 2003.

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I. Daubechies, B. Han, A. Ron, Z. W. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comp. Harmonic Anal., to appear, 2001.

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