Compactly Supported Tight Affine Frames with Integer Dilations and Maximum Vanishing Moments
نویسندگان
چکیده
When a Cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L = L(IR) with dilation integer factor M ≥ 2, the standard “matrix extension” approach for constructing compactly supported tight frames has the limitation that at least one of the tight frame generators does not annihilate any polynomial except the constant. The notion of vanishing moment recovery (VMR) was introduced in our earlier work (and independently by Daubechies, Han, Ron, and Shen) for dilation M = 2 to increase the order of vanishing moments. This present paper extends the tight frame results in the above mentioned papers from dilation M = 2 to arbitrary integer M ≥ 2 for any compactly supported M -dilation scaling functions. It is shown, in particular, that M compactly supported tight frame generators suffice, but not M − 1 in general. A complete characterization of the M dilation polynomial symbol is derived for the existence of M − 1 such frame generators. Linear spline examples are given forM = 3, 4 to demonstrate our constructive approach. 1) Research supported by NSF Grants CCR-9988289 and CCR-0098331, ARO Grant DAAD 19-00-1-0512, and a University of Missouri-St. Louis Research Award 2) This author is also with the Department of Statistics, Stanford University, Stanford, CA 94305 1
منابع مشابه
Compactly Supported Tight Affine Frames with Integer Dilations and Maximum Vanishing Moments1) by
When a Cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L2 = L2(IR) with dilation integer factor M ≥ 2, the standard “matrix extension” approach for constructing compactly supported tight frames has the limitation that at least one of the tight frame generators does not annihilate any polynomial except the constant. The not...
متن کاملCompactly supported tight and sibling frames with maximum vanishing moments
The notion of vanishing-moment recovery (VMR) functions is introduced in this paper for the construction of compactly supported tight frames with two generators having the maximum order of vanishing moments as determined by the given refinable function, such as the mth order cardinal B-spline Nm. Tight frames are also extended to “sibling frames” to allow additional properties, such as symmetry...
متن کاملTight Frames with Maximum Vanishing Moments and Minimum Support
The introduction of vanishing moment recovery (VMR) functions in our recent work (also called “fundamental functions” in an independent paper by Daubechies, Han, Ron, and Shen) modifies the so-called “unitary extension principle” to allow the construction of compactly supported affine frames with any desirable order of vanishing moments up to the order of polynomial reproduction of the given as...
متن کاملSome Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments
Let A ∈ Rd×d, d ≥ 1 be a dilation matrix with integer entries and | detA| = 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 o...
متن کاملA pr 2 00 7 Multivariate Wavelet Frames 1
We proved that for any matrix dilation and for any positive integer n, there exists a compactly supported tight wavelet frame with approximation order n. Explicit methods for construction of dual and tight wavelet frames with a given number of vanishing moments are suggested.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Adv. Comput. Math.
دوره 18 شماره
صفحات -
تاریخ انتشار 2003