Donovan, G. C., J. S. Geronimo, and D. P. Hardin, Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets, SIAM J. Math. Anal. 27 (1996), 1791-1815.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... at the cost of dropping orthogonality; in their construction two compactly supported dual refinable functions were needed, only one of which could be spline; in [CW] similar non orthogonal dual symmetric, spline wavelet bases were given, but only one of them could be compactly supported; in [DGH], symmetry, orthonormality and compact support were combined at the price of having multiwavelets, or vector MRA; in [DGH] it was shown that this could be done with spline vector MRA. In this paper, we are relaxing the non redundancy condition, which makes it possible to start from refinable # ....

.... needed, only one of which could be spline; in [CW] similar non orthogonal dual symmetric, spline wavelet bases were given, but only one of them could be compactly supported; in [DGH] symmetry, orthonormality and compact support were combined at the price of having multiwavelets, or vector MRA; in [DGH], it was shown that this could be done with spline vector MRA. In this paper, we are relaxing the non redundancy condition, which makes it possible to start from refinable # that satisfy no other conditions than those in Assumptions 1.3. At first sight, it is not clear how to use Proposition 1.7 ....

G. Donovan, J. S. Geronimo, and D. P. Hardin (1996), "Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets", SIAM J. Math. Anal. 27(6), 1791-1815.

....sequence of r r complex valued matrices on Z, called the (matrix) mask for . When r = 1, the re nable function vector in (1.2) is a scalar function. The re nement equation in (1. 2) as well as various properties of its re nable function vector has been well studied in the literature, see [3, 9, 14, 15, 17, 20, 22, 23, 28, 29, 31] and references therein. A wavelet system is usually generated by some wavelet function vectors ; 1; L which are derived from a d re nable function vector as follows: c ( b ( 1; L for some appropriate matrices b ( of 2 periodic trigonometric ....

G. C. Donovan, J. S. Geronimo, D. P. Hardin, Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets. SIAM J. Math. Anal., 27 (1996), 1791{ 1815.

....of the boundaries, and thus we suffer a deterioration in compression ratios. However, we still obtain superior compression ratio as observed in our results shown. We are in preparation of utilizing different families of wavelet functions, the so called multiwavelets introduced in Donovan, et al. [39] These wavelet functions have shorter support so that they achieve maximum compression ratio and provide better handling of boundary terms. The method is semi automated and achieved via fast filtering implementation of divergence free wavelets. The full automation of this algorithm requires ....

Donovan, G.C., Geronimo, J.S., and Hardin, D.P., "Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets," SIAM Journal of Mathematical Analysis, Vol. 27, No. 6, 1996, pp. 1791-1815. 31 List of Figures Fig. 1. Families of biorthogonal B-spline functions: (a) N x 2 ( ) , (b)

....in the construction of 9 53 22. Wavelets and for Example 3 59 23. Wavelets and for Example 3 60 24. Wavelets and for Example 3 61 25. Wavelets and for Example 3 62 26. Wavelets and for Example 3 64 27. Wavelets and for Example 3 65 28. Wavelets and for Example 3 65 29. An arbitrary partition of [0 2] 67 30. A dilation 3 partition of [0 2] 68 31. Scaling functions from Example 2 on the partition 70 32. An arbitrary triangulation 72 33. The dilation 3 triangulation 74 34. Scaling functions from Example 3 on the triangulation 83 35. Vertices of 84 36. An arbitrary cell in the triangulation 84 ....

....and for Example 3 59 23. Wavelets and for Example 3 60 24. Wavelets and for Example 3 61 25. Wavelets and for Example 3 62 26. Wavelets and for Example 3 64 27. Wavelets and for Example 3 65 28. Wavelets and for Example 3 65 29. An arbitrary partition of [0 2] 67 30. A dilation 3 partition of [0 2] 68 31. Scaling functions from Example 2 on the partition 70 32. An arbitrary triangulation 72 33. The dilation 3 triangulation 74 34. Scaling functions from Example 3 on the triangulation 83 35. Vertices of 84 36. An arbitrary cell in the triangulation 84 37. A bounded cell in the triangulation ....

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G. C. Donovan, J. S. Geronimo, and D. P. Hardin, Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets, , :6, pp. 1791-1815 (1996)

....These spaces generalize (1. 3) and are of the form V (# 1 , # r ) 8 : f = r X i=1 X j#ZZ d c i j # i (x j) 9 = Such spaces have been used in finite elements and approximation theory [83] and for the construction of multiresolution approximations and wavelet spaces [24, 25, 29, 41, 46, 55, 67, 69, 79, 83, 84, 85]. They have been extensively studied in recent years (for instance [5, 42, 53, 54] Sampling in (integer) shift invariant spaces that are not band limited is a suitable and realistic model for many applications, e.g. for taking into account real acquisition and reconstruction devices, for ....

G.C. Donovan, J.S. Geronimo, and D.P. Hardin. Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets. SIAM Journ. Math. Analysis, 26:1791--1815, 1996.

....results (for the scalar case) were obtained in [DDL] taking convolutions of indicator functions relative to different fundamental domains associated with M . Recently certain refinable functions for N 1 have been generated with the aid of techniques from the theory of iterated function systems [DGH]. In particular, they may be suitable for finite element applications [SS] Finally, we mention from a different point of view that the tuple h as generator of a vector field was used in [U] to construct compactly supported divergence free wavelets. 2.3 Stability and Linear Independence ....

G.C. Donovan, J.S. Geronimo, D.P. Hardin, Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets, Preprint 94-017, Department of Mathematics, Vanderbilt University, 1994.

....symmetric and antisymmetric compactly supported orthonormal multiscaling functions and multiwavelets. Geronimo, Hardin and Massopust in [19] used fractal interpolation to construct orthonormal multiscaling functions, and their corresponding multiwavelets were given in [29] and in [10] In [11], Donovan, Geronimo, and Hardin showed that there exist compactly supported orthogonal polynomial spline multiscaling functions with arbitrarily high regularity. For the bivariate setting, one easy way to get a multiscaling function is to use the tensor product of two univariate multiscaling ....

G.C. Donovan, J. S. Geronimo and D.P. Hardin, Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets, SIAM J. Math. Anal., 27(1996), pp. 1791--1815.

....scaling functions are the sum and difference of the two functions from the symmetric pair. In this article we will make use of several other nonsymmetric multiwavelets with desirable properties. More on the construction of multiscaling functions and multiwavelets can be found in [1, 9, 13, 18, 22, 24, 30, 31, 38, 39, 41]. 0.5 1 1.5 2 0.5 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 0.5 0.5 1 1.5 2 2.5 3 Figure 3: Symmetric pair of orthogonal scaling functions. 0.5 1 1.5 2 0.25 0.25 0.5 0.75 1 1.25 0.5 1 1.5 2 1.5 1 0.5 0.5 1 1.5 Figure 4: Chui Lian symmetric antisymmetric orthogonal scaling functions. 3 Multiwavelets ....

G. Donovan, J. Geronimo, and D. Hardin, "Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets," SIAM J. Math. Anal., vol. 27, pp. 1791-1815, 1996.

....lifting has many contacts with certain filter design algorithms used in signal processing. Those connections are pointed out in [105, 52] Over the last few years Donovan, Hardin, Geronimo, and Massopust have developed techniques to construct wavelets based on fractal interpolation functions [60, 61, 62, 70]. They also introduced the concept of several generating functions (multiwavelets) As this technique does not rely on the Fourier transform either, it too potentially can be used to construct second generation wavelets. Several spatial constructions of spline wavelets on irregular grids have been ....

G. C. Donovan, J. S. Geronimo, and D. P. Hardin, Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets, SIAM J. Math. Anal., 27 (1996), pp. 1791--1815.

....many contacts with certain filter design algorithms used in signal 4 WIM SWELDENS processing. Those connections are pointed out in [105, 52] Over the last few years Donovan, Hardin, Geronimo, and Massopust have developed techniques to construct wavelets based on fractal interpolation functions [60, 61, 62, 70]. They also introduced the concept of several generating functions (multi wavelets) As this technique does not rely on the Fourier transform either, it too potentially can be used to construct second generation wavelets. Several spatial constructions of spline wavelets on irregular grids have ....

G. C. Donovan, J. S. Geronimo, and D. P. Hardin. Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets. SIAM J. Math. Anal., 27:1791--1815, 1996.

....scaling functions are the sum and di#erence of the two functions from the symmetric pair. In this article we will make use of several other nonsymmetric multiwavelets with desirable properties. More on the construction of multiscaling functions and multiwavelets can be found in [1, 9, 13, 18, 22, 24, 30, 31, 38, 39, 41]. 0.5 1 1.5 2 0.5 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 0.5 0.5 1 1.5 2 2.5 3 Figure 3: Symmetric pair of orthogonal scaling functions. 0.5 1 1.5 2 0.25 0.25 0.5 0.75 1 1.25 0.5 1 1.5 2 1.5 1 0.5 0.5 1 1.5 Figure 4: Chui Lian symmetric antisymmetric orthogonal scaling functions. 3 Multiwavelets ....

G. Donovan, J. Geronimo, and D. Hardin, "Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets," SIAM J. Math. Anal., vol. 27, pp. 1791-1815, 1996.

....functions for this multiwavelet system are shown in Figure 3; observe that one is the reflection of the other about its center point. In this article we will make use of several other nonsymmetric multiwavelets with desirable properties. Additional constructions of multiwavelets are given in [1, 9, 28, 29, 21, 17]. 3 Multiwavelets and multirate filter banks Corresponding to each multiwavelet system is a matrix valued multirate filter bank [11] or multifilter. A multiwavelet filter bank [26] has taps that are N Theta N matrices (in this paper, we will be working with N = 2) Our principal example is ....

G. Donovan, J. Geronimo, and D. Hardin, "Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets," preprint, 1994.

....it leads to two new insights: a custom design construction of wavelets, and a faster, in place implementation of existing wavelet transforms. Over the last few years Donovan, Hardin, Geronimo, and Massopust have developed techniques to construct wavelets based on fractal interpolation functions [49, 50, 51, 58]. They also introduced the concept of several generating functions (multi wavelets) As this technique does not rely on the Fourier transform either, it too potentially can be used to construct second generation wavelets. The paper is organized as follows. In Sections 2, 3, 4, 5 we generalize ....

G. C. Donovan, J. S. Geronimo, and D. P. Hardin. Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets. SIAM J. Math. Anal., To appear.

....that a blockwise polynomial which is a solution of (2) with N = 1 and whose symbol matrix H( is a trigonometric polynomial must be a linear combination of box splines under the application of a homogeneous differential operator. The importance is that Donovan, Geronimo and Hardin constructed in [5] compactly supported orthonormal and symmetric wavelet which is piecewise polynomial in one dimension in which multiresolutions with large N are used. This result showed that the multiresolution for N 2 is interesting. In [9] Goodman and Lee constructed orthonormal wavelet bases with any ....

....another type of orthonormal scaling functions with compact support. We consider them seperately in two subsection. After finishing first manuscript, we learn from Professor D. P. Hardin that more general orthonornmal scaling functions via fractal interpolation functions have been constructed in [5] and [6] 4.1 Perturbation of Daubechies scaling functions Let fOE i g i=1;2 be two orthonormal functions with compact support, which satisfies (3) and fc ij (k)g in (3) has finite length, and let H( H 1 ( H 2 ( H 3 ( H 4 ( be its symbol matrix, where H i ( i = 1; Delta ....

Donovan G C, Geronimo J S, and Hardin D P, Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets, preprint, 1994.

....nonuniform grid, there is a unique squeeze map that preserves the accuracy of the space. In a sequel to this paper [7] we verify that spaces associated with uniform re nements of the initial grid then form a multiresolution and we describe how to nd the wavelets. In Example 6. 4, we use ideas from [5] to construct a multiresolution on an arbitrary nonuniform subdivision (the only requirement is that each interval is subdivided into two subintervals) Each space has a local orthogonal basis consisting of continuous piecewise quadratic functions. Finally, in Section 7 we construct a family of ....

....S( M ) equals S( M ) that is, the dilation by 1=M of the space S( The local linear independence conditions for minimal support must then be checked separately. Example 6.3 is constructed in this way. SQUEEZABLE ORTHOGONAL BASES: ACCURACY AND SMOOTHNESS 11 5.2. General construction. In [5] the authors developed a method for constructing orthogonal generators. For W L 2 (R) let PW denote the orthogonal projection onto W . Lemma 9. 5] Suppose is a minimally supported k generator. There exists an orthogonal minimally supported k generator such that S( S( if and only ....

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G. C. Donovan, J. S. Geronimo, and D. P. Hardin, \Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets" SIAM J. Math. Analysis 27:6, 1791-1815 (1996)

....a nonuniform grid, there is a unique squeeze map that preserves the accuracy of the space. In a sequel to this paper [7]weverify that spaces associated with uniform re nements of the initial grid then form a multiresolution and we describe how to nd the wavelets. In Example 6. 4, we use ideas from [5] to construct a multiresolution on an arbitrary nonuniform subdivision (the only requirement is that eachinterval is subdivided into two subintervals) Each space has a local orthogonal basis consisting of continuous piecewise quadratic functions. Finally, in Section 7 we construct a family of ....

....#( # ) equals #( # #) that is, the dilation by1##of the space #( The local linear independence conditions for minimal support must then be checked separately. Example 6.3 is constructed in this way. ########## ########## ###### ######## ### ########## ## 5.2. ####### ############# In [5] the authors developed a method for constructing orthogonal generators. For # # # # (#) let ## denote the orthogonal projection onto # . ##### ## ( 5] Suppose is a minimally supported # generator. There exists an orthogonal minimally supported # generator such that #( #( if and ....

[Article contains additional citation context not shown here]

G. C. Donovan, J. S. Geronimo, and D. P. Hardin, \Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets" SIAM J. Math. Analysis ##:6, 1791-1815 (1996)

....Orthonormal Scaling Vector 15 1 0.75 0.5 0.25 0.25 0.5 0.75 1 0.5 0.5 1 1.5 2 2.5 Figure 7: DGHM Scaling Vector with Approximation Order p = 2 The orthonormal scaling vector of Legendre polynomials has approximation order p = 3, but is not continuous. Following the intertwining techniques of [5], we construct a continuous compactly supported orthonormal scaling vector Phi with approximation order p = 3. We begin with a refinable space V 0 which has approximation order three, is not orthogonal, and consists of the two generators h(x) 1 Gamma jxj) Gamma1;1] x) q(x) 4x(1 Gamma ....

.... coefficients for scalar and vector data using periodic boundaries acting on a matrix of colums of column vectors function y=matconv1dcon(h,a) arows, acols] size(a) a= a; a(1:2, padding he= h( [5 6]) h( 1 2] Time Reversal and Transpose ho= h( 7 8] h( 3 4] hrows, hcols] size(he) y=zeros(2 arows,acols) for i=1:arows hrows y(4 i 3:4 i 2, he a(hrows i (hrows 1) hcols hrows (i 1) y(4 i 1:4 i, ho a(hrows i (hrows 1) hcols hrows (i 1) end 64 The following ....

G. C. Donovan, J. S. Geronimo, and D. P. Hardin, "Intertwining Multiresolution Analyses and the Construction of Piecewise-Polynomial Wavelets," SIAM Journ. Math. Analysis 27, 1791-1815 (1996)

....of reproduction in any form reserved. 2 G. Donovan, J. Geronimo and D. Hardin supported wavelets (called multiwavelets) 1 ; r that generate an orthogonal basis of W 0 = V 1 Psi V 0 . As in the single scaling function case, the work is in finding orthogonal scaling functions. In [4] the authors developed a general scheme for constructing univariate multiwavelets. This scheme is based on enlarging a given MRA (usually a classical spline space) by adding certain functions to obtain an orthogonal MRA. These constructions divide into two classes depending on the type of ....

....present a systematic approach to constructing orthogonal continuous MRA s and the associated multiwavelets in a forthcoming paper. The organization of the paper is as follows: In section 2 we review the construction of fractal interpolation functions. In section 3 we review some of the theory of [4] for constructing orthogonal MRA s, and in section 4 apply this to give a simplified construction of the orthogonal univariate scaling functions found in [5, 7] Proceeding by analogy with the univariate case, in section 5 we construct our bivariate example. x2. Fractal Interpolation Functions ....

[Article contains additional citation context not shown here]

Donovan, G. C., J. S. Geronimo, D. P. Hardin, Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets, preprint, 1994.

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Donovan, G. C., J. S. Geronimo, and D. P. Hardin, Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets, SIAM J. Math. Anal. 27 (1996), 1791-1815.

No context found.

Donovan, G. C., J. S. Geronimo, and D. P. Hardin, Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets, SIAM J. Math. Anal. 27 (1996), 1791-1815.

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Donovan, G., J. S. Geronimo, and D. P. Hardin, Intertwining multiresolution analyses and the construction of piecewise polynomial wavelets, SIAM J. Math. Anal. xxx (to appear), xx--xx.

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