Chui, C. K. and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. 113 (1991), 785-793.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we conclude the following Proposition 1. The Stromberg wavelet n is associated with a multiresolution approximation built from spline functions of order n with knot sequence Z. Next, let us look at the connection between with Stromberg s wavelets and ChuiWang s cardinal spline wavelets in [Chui and Wang 91] Let L 2n (x) X k b k N 2n (x Gamma k) be the cardinal spline interpolant satisfying L 2n (k) ae 1; if k = 0 0; if k 2 Znf0g. Since L 2n;t is of exponential decay, we can see that s(x) L 2n;t (x) Gamma X k1 L 2n;t (2k)L 2n (x Gamma 2k) is a spline function of order 2n with ....

.... (x) L 2n (x) X k1 L 2n;t (2k)L 2n (x Gamma 2k) By taking n times derivatives both sides of the above equation, we have L (n) 2n;t (2x Gamma 1) X k0 L 2n;t (2k)L (n) 2n (2x Gamma 1 Gamma 2k) Recall Chui Wang s cardinal spline wavelet n (x) L (n) 2n (2x Gamma 1) in [Chui and Wang 91] We have Proposition 2. The Stromberg wavelet n (x) and Chui Wang s cardinal spline wavelet n have the following relation: n (x) c n X k0 L 2n;t (2k) n (x Gamma k) 6 4. A Generalization of Stromberg s Wavelets. The construction of Stromberg s wavelet may be generalized to ....

Chui, C. K. and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc., 113(1991), pp. 785--793.

....about some point are preferred. It is well known, however, that continuous compactly supported (real valued) orthonormal wavelets cannot be symmetric. On the other hand, it is possible to construct symmetric continuous compactly supported semi 5 orthogonal wavelets. In fact, Chui and Wang [10], 9] have constructed such wavelets using piecewise polynomials. Early constructions of wavelets were, in general, orthonormal wavelets which were not compactly supported (cf. 30] 31] Some of the first semi orthogonal wavelets were constructed in [1] The first compactly supported continuous ....

Chui, C. K. and Jian-Zhong Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. 113(3) (1991), 785--793.

....been carried out which improve on them in some respects, while giving up on other properties. For instance, one can give up some of the orthogonality in the constructions above, and construct a Riesz basis rather than an orthonormal basis of wavelets (together with the dual Riesz basis) as in Chui Wang (1991), Chui Wang (1992) Auscher (1989) or Cohen et al. 1992) this relaxing of orthonormality buys more smoothness and or symmetry for the wavelets. Another useful construction restricts these wavelet bases to an interval while retaining their powerful mathematical properties (see e.g. Cohen et ....

Chui, C.K., & Wang, J.Z. 1991. A cardinal spline approach to wavelets. Proc. Amer. Math. Soc., 113, 785-793.

....supported wavelets in 1987 [18] Later many generalizations to the biorthogonal or semiorthogonal (pre wavelet) case were introduced. Biorthogonality allows the construction of symmetric wavelets and thus linear phase filters. Examples are: the construction of semiorthogonal spline wavelets [1, 10, 12, 13, 50], fully biorthogonal compactly supported wavelets [14, 57] and recursive filter banks [28] Recently a new angle to study these constructions was provided by the so called lifting scheme [46] The basic idea behind lifting is that it provides a simple relationship between all multiresolution ....

C. K. Chui and J. Z. Wang. A cardinal spline approach to wavelets. Proc. Amer. Math. Soc., 113:785--793, 1991.

....themselves can be computed with an accurate quadrature technique. Besides hierarchical splats [15] wavelet splatting [17] is a sophisticated extension. 2.2. B spline Wavelets Since we aim at a unified data representation for both volume data compression and rendering, we use Chui and Wang [3] B spline wavelet bases of order j. They have many usefull properties, such as smoothness and vanishing moments. In addition, analytic expressions for scaling and wavelet functions and their duals in the frequency and spatial domain are given. The mother scaling function of order 1 is defined ....

....Blackwell Publishers. 2] B. Cabral, N. Cam, and J. Foran. Accelerated volume rendering and tomographic reconstruction using texture mapping hardware. In A. Kaufman and W. Krueger, editors, 1994 Symposium on Volume Visualization, pages 91 98. ACM SIGGRAPH, Oct. 1994. ISBN 0 89791 741 3. [3] C. K. Chui and J. Z. Wang. A cardinal spline approach to wavelets. Proc. Amer. Math. Soc. 113:785 793, 1991. 4] R. Crawfis and N. Max. Direct volume visualization of three dimensional vector fields. 1992 Workshop on Volume Visualization, pages 55 60, 1992. Figure 16: Relation between ....

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C. K. Chui and J. Z. Wang. "A cardinal spline approach to wavelets." Proc. Amer. Math. Soc., 113:785--793, 1991.

....continuous (or smoother) However, these wavelets do not have closed form representations; they are defined via a limiting process. Also, it is known that these wavelets cannot have certain other desirable properties (e.g. symmetry) By giving up compact support (cf. 1] or orthogonality (cf. [4]) symmetric wavelet bases have been constructed using piecewise polynomial splines. For some applications the symmetry and simple representation are more important than having both compact support and orthogonality. Recently, wavelet constructions generated by a finite collection of scaling ....

C. K. Chui, and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc., 113, pp. 785-793 (1991)

....ones have compact support and linear phase. Moreover, all filter coefficients are dyadic rationals. A disadvantage is that for small filter lengths, the dual functions have very low regularity. ffl Examples of semiorthogonal wavelets are the ones constructed by Charles Chui and Jianzhong Wang in [17, 18, 19]. The scaling functions are cardinal B splines of order m and the wavelet functions are splines with compact support [0; 2m Gamma 1] All primary and dual functions still have generalized linear phase and all scaling and wavelet parameters are rationals. A powerful feature here is that analytic ....

C. K. Chui and J. Z. Wang. A cardinal spline approach to wavelets. Proc. Amer. Math. Soc., 113:785--793, 1991.

....3g is an orthonormal basis for L 2 (R 2 ) In recent years, multiscaling functions and multiwavelets have been studied extensively. Goodman, Lee and Tang [20] established a characterization of multiscaling functions and multiwavelets with semi orthogonality which was introduced by Chui and Wang[4], in particular, they gave the examples of spline wavelets with multiple knots. Examples of cubic and quintic finite elements and their corresponding multiwavelets were studied by Strang and Strela [28] In [3] Chui and Lian introduced a scheme for constructing symmetric and antisymmetric ....

Chui, C. K. and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc., 113(1991), pp. 785--793.

....the numerical solution of di#erential and integral equations, and noise reduction. Many first generation wavelet families have been constructed over the last ten years. We refer to the work of (in alphabetical order) Aldroubi and Unser [2, 3, 108, 107] Battle and Lemarie [13, 78] Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28] Cohen, Daubechies, and Feauveau [29] Daubechies [47, 49, 48] Donoho [57, 56] Frazier and Jawerth [65, 67, 66] Herley and Vetterli [73, 110] Kovacevic and Vetterli [77, 111] Mallat [85, 84, 86] Meyer [87] and many more. Except for Donoho, they all rely on the ....

C. K. Chui and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc., 113 (1991), pp. 785--793.

....multiresolution approach has lead Daubechies [16] to the discovery of a family of compactly supported wavelets. This family includes the Haar function [23] as a special case, which was not considered as a wavelet by earlier definitions because the lack of Utilizing the same approach, Chui and Wang [9, 10] found another framework of constructing wavelets and pre wavelets from B splines. Battle [4] proposed yet another approach based on renormalization groups in the study of physical systems with infinitely many degrees of freedom. regularities. Applications of wavelets are better established in the ....

C. K. Chui and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc., 113 (1991), pp. 785--793.

....spline wavelet bases by using different methods. Their wavelets are identical and are cardinal spline functions. Cohen, Daubechies, and Feauveau constructed compactly supported spline wavelets with compactly supported dual wavelets. Around 1990, several authors (Auscher [1] Battle [3] Chui Wang [9, 10, 11], Micchelli [24] and Unser Aldroubi [28] studied the so called cardinal spline pre wavelets m and or j m . The terminology pre wavelet was given by Battle [3] This means that the wavelet spaces remain mutually orthogonal and the wavelet systems are non orthogonal only in any given wavelet ....

.... cardinal interpolation spline [25] which satisfies the interpolation conditions L 2m (k) ffi k;0 ; k 2 Z: By using the spline multiresolution analysis one can prove that the above functions m and j m are m th order cardinal spline wavelets having m th order cardinal spline dual wavelets [1, 3, 9, 10, 11, 24, 28]. However, Chui and Wang studied the characterization of cardinal spline pre wavelets. They even proved that the function m is the minimally supported m th order cardinal spline pre wavelet. The present paper is mainly motivated by Chui and Wang s work [9, 10, 11] The advantage of the above ....

[Article contains additional citation context not shown here]

C. K. Chui and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. 113 (1991), 785-793.

....the numerical solution of differential and integral equations, and noise reduction. Many first generation wavelet families have been constructed over the last ten year. We refer to the work of (in alphabetical order) Aldroubi Unser [2, 3, 108, 107] Battle Lemari e [13, 78] Chui Wang [19, 25, 24, 23], Cohen Daubechies [28] CohenDaubechies Feauveau [29] Daubechies [47, 49, 48] Donoho [57, 56] Frazier Jawerth [65, 67, 66] Herley Vetterli [73, 110] Kovacevi c Vetterli [77, 111] Mallat [85, 84, 86] Meyer [87] and many more. Except for Donoho, they all rely on the Fourier transform as a ....

C. K. Chui and J. Z. Wang. A cardinal spline approach to wavelets. Proc. Amer. Math. Soc., 113:785--793, 1991.

....themselves can be computed with an accurate quadrature technique. Besides hierarchical splats [15] wavelet splatting [17] is a sophisticated extension. 2. 2 B spline Wavelets Since we aim at a unified data representation for both volume data compression and rendering, we use Chui and Wang [3] B spline wavelet bases of order j. They have many usefull properties, such as smoothness and vanishing moments. In addition, analytic expressions for scaling and wavelet functions and their duals in the frequency and spatial domain are given. The mother scaling function of order 1 is defined ....

....spatial domain F u v , frequency domain a b u v n w 3 Figure 3: Illustration of the multiview concept. f y y z x view 2 view 1 view 3 view 1 view 2 view 3 f 0000 3 1 [ f 0000 3 1 [ Table 1: Permutations and negations for multi view arrangements PERMUTATIONS NEGATIONS VIEW 1 [ 0 1 2 3 4 5 6 7 ] [ 1 1 1 1 1 1 1 1 ] VIEW 2 [ 0 1 4 5 2 3 6 7 ] 1 1 1 1 1 1 1 1] VIEW 3 [ 0 4 2 6 1 5 3 7 ] 1 1 1 1 1 1 1 1 ] ffy yff u 1 v 1 , u 1 a sin a cos 0 u x u y u z = v 1 a cos b sin a sin b sin b cos v x v y v z = u 2 v 2 , u 2 a cos a sin 0 u ....

[Article contains additional citation context not shown here]

C. K. Chui and J. Z. Wang. "A cardinal spline approach to wavelets." Proc. Amer. Math. Soc., 113:785--793, 1991.

....data compression, data transmission, the numerical solution of differential and integral equations, and noise reduction. Many first generation wavelet families have been constructed over the last ten year. We refer to the work of (in alphabetical order) Aldroubi Unser [1, 2, 92, 91] Chui Wang [14, 20, 19, 18], CohenDaubechies [23] Cohen Daubechies Feauveau [24] Daubechies [37, 39, 38] Donoho [46, 45] FrazierJawerth [53, 55, 54] Herley Vetterli [60, 94] Kovacevi c Vetterli [64, 95] Mallat [71, 70, 72] Meyer [73] and many more. Except for Donoho, they all rely on the Fourier transform as a ....

C. K. Chui and J. Z. Wang. A cardinal spline approach to wavelets. Proc. Amer. Math. Soc., 113:785--793, 1991.

....since A 2n = kB 2n (x)k 1 = jB 2n j ; where B 2n is the 2nth Bernoulli number. The leading term of the expansion is exactly the same for Battle Lemari e orthogonal spline wavelets [2, 22] Cohen Daubechies Feauveau biorthogonal spline wavelets [7] and Chui Wang semiorthogonal spline wavelets [4, 5, 6]. The dependency of the higher order terms is studied in the following lemmas. Lemma 5. Given e N , the first N tot = N e N moments of the dual scaling function are dependent. Proof. Since e (x) is a dual function, it holds that X k b e ( k2 ) b ( k2 ) 1 : Taking the pth ....

Chui, C.K., Wang, J.Z. (1991): A cardinal spline approach to wavelets. Proc. AMS 113, 785--793

....fa j g is a real sequence, finite or infinite. We ask: is the single function N (l) m l (2x Gamma j) a wavelet In this paper we prove that it is a wavelet when m and l satisfy some mild conditions. Some proofs in this paper are borrowed from Cohen Daubechies Feauveau [10,11,13] and Chui Wang [7,8,9]. Using the derivative of certain functions to construct wavelets is a typical method in the construction of wavelet frames and dyadic wavelet transforms [16,22] In this paper we show that this method is also a source of wavelet bases. First we define the term wavelet . For a given function f , ....

C. K. Chui and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. 113 (1991), 785-793.

....and Computational Mathematics, Princeton University, Princeton NJ 08544. Lucent Technologies, Bell Laboratories, Rm. 2C 175, 700 Mountain Avenue, Murray Hill NJ 07974. ingrid math.princeton.edu, wim bell labs.com. filters. Examples are: the construction of semiorthogonal spline wavelets [1, 8, 10, 11, 49], fully biorthogonal compactly supported wavelets [12, 56] and recursive filter banks [25] Various techniques to construct wavelet bases, or to factor existing wavelet filters into basic building blocks are known. One of these is lifting. The original motivation for developing lifting was to ....

C. K. Chui and J. Z. Wang. A cardinal spline approach to wavelets. Proc. Amer. Math. Soc., 113:785--793, 1991.

....since A 2n = kB 2n (x)k 1 = jB 2n j ; where B 2n is the 2nth Bernoulli number. The leading term of the expansion is exactly the same for Battle Lemari e orthogonal spline wavelets [2, 22] Cohen Daubechies Feauveau biorthogonal spline wavelets [7] and Chui Wang semiorthogonal spline wavelets [4, 5, 6]. The dependency of the higher order terms is studied in the following lemmas. Lemma 5. Given e N , the first N tot = N e N moments of the dual scaling function are dependent. Proof. Since e (x) is a dual function, it holds that X k e ( k2 ) k2 ) 1 : Taking the pth derivative ....

Chui, C.K., Wang, J.Z. (1991): A cardinal spline approach to wavelets. Proc. Amer. Math. Soc. 113, 785--793

....the dual ones have compact support and linear phase. Moreover, all filter coefficients are dyadic rationals. A disadvantage is that for small filter lengths, the dual functions have very low regularity. ii) Semiorthogonal spline wavelets were constructed by Charles Chui and Jianzhong Wang in [23, 24, 25]. The scaling functions are cardinal B splines of order m and the wavelet functions are splines with support [0; 2m Gamma 1] All primary and dual functions still have generalized linear phase and all coefficients used in the fast 22 BJORN JAWERTH AND WIM SWELDENS Table 1 A quick comparison ....

C. K. Chui and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc., 113 (1991), pp. 785--793.

....1 Introduction The use of polynomial splines is widespread in numerical analysis and many other fields because they have simple structure, good localization properties, and computing stability. See [7] Based on this fact, recently splines are often used to construct wavelet bases. See [1] [2], 3] 6] 11] 12] 14] 17] In applications, wavelets on a bounded interval are also very useful. Chui and Quak first construct spline wavelet bases of L 2 (I) See [4] A general discussion of the construction of wavelets on interval [0; 1] can be found in paper [13] written by ....

Chui, C.K. and Wang, J.Z., A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. 113(1991), 785--793.

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Chui, C. K. and J. Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. 113 (1991), 785-793.

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Chui, C. K., & Wang, J. Z. 1991. A cardinal spline approach to wavelets. Proc. Amer.

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Chui, C. K. and Wang, J. Z. A cardinal spline approach to wavelets. Proc. Amer. Math. Soc. to appear.

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