CHUI,C .AND QUAK, E. 1992. Wavelets on a bounded interval. In Numerical Methods of Approximation Theory, D. Braess and L. L. Schumaker, Eds. Birkhauser-Verlag, Basel, 1--24.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is a coarsest level j 0 : 1. The in nite set of all possible indices will be denoted by II. We will employ (tensor products of) the piecewise linear boundary adapted B spline (pre)wavelets f g 2II constructed in [9] which are a special case of the (pre)wavelets on the interval constructed in [3]. They have the following properties: R) they constitute a Riesz basis for L 2( O) they are semi orthogonal, i.e. for j j 6= j j one always has Z (x) x) dx = 0; 3) L) they are compactly supported and satisfy for each 2 II diam (supp ) 2 j j : 4) The particular ....

Chui, C. K., and E. G. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, Vol. 9, D. Braess and L. L. Schumaker (eds.), Int. Ser. Numer. Math. 105, Birkhauser, Basel, 1992, 53-75.

....to di erential equations with boundary conditions and image processing. One excellent construction of orthogonal wavelet bases on the interval [0; 1] was given by Cohen, Daubechies and Vial [4] by adapting the famous Daubechies orthogonal wavelets on the real line to the interval [0; 1] see also [1, 3]) The motivation to construct wavelets on the interval and the fast wavelet transforms associated with wavelets on the interval were explained in detail in [4] In the literature, several other approaches were also reported in [6, 9, 12, 20] to obtain wavelets on the interval by adapting the ....

C. Chui, and E. Quak, Wavelets on a bounded interval, in \Numer. Methods of Approx. Theory" (D. Braess, and L. L. Schumaker, Eds.), pp. 1-24, Birkhauser, Basel, 1992.

....and animation [9] because of its lack of continuity. There are a variety of ways to construct wavelets with k continuous derivatives. One such class of wavelets can be constructed from piecewise polynomial splines. These spline wavelets have been developed to a large extent by Chui and colleagues [3, 4]. The Haar basis is in fact the simplest instance of spline wavelets, resulting when the polynomial degree is set to zero. In the following, we briefly sketch the ideas behind the construction of endpoint interpolating B spline wavelets. Finkelstein and Salesin [8] developed a collection of ....

....instance of spline wavelets, resulting when the polynomial degree is set to zero. In the following, we briefly sketch the ideas behind the construction of endpoint interpolating B spline wavelets. Finkelstein and Salesin [8] developed a collection of wavelets for the cubic case, and Chui and Quak [4] presented constructions for arbitrary degree. Although the derivations for arbitrary degree are too involved to present here, we give the synthesis filters for the piecewise constant (Haar) linear, quadratic, and cubic cases in Appendix A. The next three sections parallel the three steps ....

[Article contains additional citation context not shown here]

Charles K. Chui and Ewald Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical Methods in Approximation Theory, volume 9, pages 53--75. Birkhauser Verlag, Basel, 1992.

....properties. Thus, the common ground for all the above approaches is to have a sufficiently versatile collection of wavelet bases on . Such wavelets are conveniently constructed via tensor products of wavelets on the interval [0; 1] These, in turn, have been intensely studied in the literature [3, 9, 13, 18]. The above comments indicate that versatility in the present context means good localization of primal and dual bases as well as a possibly flexible choice of the order d of exactness (controlling the accuracy of the discretization scheme) and the order d of vanishing moments (controlling the ....

....S j to [0; 1] The fact that only very small portions of some functions contribute to the interval would seriously hurt the stability of the corresponding bases. Also since the supports of and generally differ the count would not match. The common strategy employed in all the quoted papers [3, 9, 13, 18] is to retain only those functions [j;k] j;k] whose support is fully contained in [0; 1] while forming in addition certain modified basis functions near the end points of the interval by taking fixed linear combinations of functions [j;k] near 0 and 1. These linear combinations have to ....

C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, D. Braess and L.L. Schumaker (eds.), Birkhauser, 1992, 1--24.

....ffl Intervals: When working with finite data it is desirable to have basis functions adapted to life on an interval. This way no awkward solutions such as zero padding, periodization, or reflection are needed. We point out that many wavelet constructions on the interval already exist, see [1, 6, 3], but we would like to use the subdivision schemes adapted to boundaries since they lead to more straightforward constructions and implementations. ffl Irregular samples: In many practical applications, the samples do not necessarily live on a regular grid. Resampling is fraught with pitfalls and ....

....of cubic B splines we need to worry about the endpoints of a finite sized interval. Because of their support the scaling functions close to the endpoints would overlap the outside of 69 the interval. This issue can be addressed in a number of different ways. One treatment, used by Chui and Quak [3], uses multiple knots at the endpoints of the interval. The appropriate subdivision weights then follow from the evaluation of the de Boor algorithm for those control points. The total number of scaling functions at level j becomes 2 3 in this setting. Consequently it is not so easy anymore ....

C. Chui and E. Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical Methods of Approximation Theory, pages 1--24. Birkhauser-Verlag, Basel, 1992.

....traditional, biorthogonal wavelet transforms have difficulties dealing with boundaries. Traditionally, wavelets # are defined as translates and dilates of one mother wavelet # ; i.e. # #(2 k) Orthogonal and biorthogonal wavelet transforms have been extended to the interval [20 22], but a better solution is to abandon the translation dilation relationship. This leads to what are referred to as second generation wavelets in the literature [23] The main advantage of second generation wavelets is that wavelets are constructed in the spatial domain and can be custom designed ....

C. K. Chui and E. Quak, Wavelets on a bounded interval, in Numerical Methods of Approximation Theory, edited by D. Braess and L. L. Schumaker, International Series of Numerical Mathematics (Birkhauser Verlag, Basel, 1992), Vol. 105, p. 53.

....with respect to a B spline wavelet basis results. This representation is unique and results in a number of computational advantages, such as preconditioning. Finkelstein and Salesin [20] describe a system which uses semi orthogonal cubic B spline wavelets [11] adapted to the interval [10] in an interactive curve editing environment. A curve, g(t) x(t) y(t) is given as a sequence of B spline control knots at some finest resolution L. Performing a wavelet transform on these coefficients results in a wavelet representation of the underlying curve. While all internal computations ....

CHUI,C.,AND QUAK, E. Wavelets on a Bounded Interval. In Numerical Methods of Approximation Theory, D. Braess and L. L. Schumaker, Eds., 1--24, 1992.

....the local bases on i build global bases on 35 In order to retain the option of realizing high order cancellation properties suggested by Theorem 5.1 all presently known approaches along the above receipe concern biorthogonal wavelets. i) Wavelets on (0; 1) have been studied in several papers [27, 5, 32, 56, 94]. The starting point is a multi resolution setting for IR generated by a suitable scaling function . The common strategy is then to construct generator bases [0;1] j on [0; 1] consisting of three groups of basis functions. The rst two are formed by the left and right boundary functions ....

C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, D. Braess and L.L. Schumaker (eds.), Birkhauser, Basel, 1992, 1-24.

....to differential equations with boundary conditions and image processing. One excellent construction of orthogonal wavelet bases on the interval [0; 1] was given by Cohen, Daubechies and Vial [4] by adapting the famous Daubechies orthogonal wavelets on the real line to the interval [0; 1] see also [1, 3]) The motivation to construct wavelets on the interval and the fast wavelet transforms associated with wavelets on the interval were explained in detail in [4] In the literature, several other approaches were also reported in [6, 11, 18] to obtain wavelets on the interval by adapting the ....

C. Chui and E. Quak, Wavelets on a bounded interval, in "Numer. Methods of Approx. Theory", D. Braess and L. L. Schumaker (eds.), Birkhauser, Basel, 1992, 1--24.

....L W n . For n; k 2 ZZ, f nk g forms an orthonormal basis for L 2 [a; b] For L 2 (IR) Daubechies [8] was the first to construct an orthonormal basis f nk g generated from one compactly supported mother wavelet . For L 2 [a; b] Gamma1 a b 1, similar constructions can be found in [5, 6, 13] among others. It is desirable to use an orthonormal basis in the subsequent computations since each iterative step involves solving a linear finite dimensional system. In addition, the compact support properties of the wavelet basis greatly reduce the number of numerical integrations that must be ....

....by the classical Legendre polynomials. ii) S 2 1 ( 0; 1] the space of piecewise continuously differentiable quadratic polynomials with possible breakpoints at 0; 1 2 ; 1, spanned by orthonormalized B splines. These splines serve as a basis for a V 1 ( 0; 1] space given in Chui and Quak [5]. iii) S 1 0 ( 0; 1] the space of continuous linear polynomials with possible breakpoints at 0; 1 4 ; 1 2 ; 3 4 ; 1, spanned by orthonormalized B splines. These splines serve as a basis for a V 1 ( 0; 1] space given in [5] Remark. The spaces (ii) and (iii) can be viewed as generated ....

[Article contains additional citation context not shown here]

C.K. Chui, and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, Vol. 9, Dietrich Braess and Larry L. Schumaker (eds.), 53-75, International Series of Numerical Mathematics, Vol. 105, Birkhauser Verlag, Basel, 1992.

....of wavelets on any domain that can be represented as a disjoint union of parametric images of cubes. This includes closed surfaces arising in connection with boundary integral equations (see Section 2. 2) Wavelets on the interval have been discussed in several papers; see for instance [5, 46, 41, 68]. The basic idea common to all these approaches is to construct multiresolution sequences S on [0; 1] which up to local boundary effects, agree with the restriction of the stationary spaces defined on all of IR. Thus one retains possibly many translates 2 j=2 OE(2 j Delta Gammak) whose ....

C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, D. Braess and L.L. Schumaker (eds.), Birkhauser, Basel, 1992, 1--24.

....3.2 Wavelets on the interval As the simplest example of a bounded domain let us consider first Omega = 0; 1] This case deserves particular attention because it will also serve as a core ingredient of constructions for more complex situations. 11 The common starting point (see e.g. [1, 8, 12, 17]) is to construct collections Phi k = fOE k;m : m 2 Delta k g ae L 2 ( 0; 1] such that the spaces S k : span Phi k are nested and contain all polynomials up to a certain desired degree. Taking some dual pair ; as in (3.1.15) and fixing such that for k k 0 we have supp (2 k Delta ....

Chui, C. K. and E. Quak, Wavelets on a bounded interval, in Numerical Methods of Approximation Theory, D. Braess and L. L. Schumaker (eds.), Birkhauser, 1992, 1--24.

....For example, most of the popular examples of wavelet spaces (see [30, 16] are derived in a one dimensional, shift invariant setting on lR. Multivariate examples on lR d are mostly obtained by tensor product techniques. Adaptions to bounded intervals and domains have been studied in, e.g. [2, 17, 19, 18]. However, up to now there is no comprehensive study of the practical potential of discretizations using multilevel structures based on shift invariance and dyadic dilation (modulo boundary modifications) in the case of general, non rectangular geometries. It is not completely clear to the author ....

Chui, C. K. and E. Quak, Wavelets on a bounded interval, in Numerical Methods of Approximation Theory , vol. 9, D. Braess, L. L. Schumaker (eds.), Birkhauser, Basel, 1992, pp. 53--75.

....biorthogonal wavelet transforms have difficulties dealing with boundaries. Traditionally, wavelets j k are defined as translates and 1 dilates of one mother wavelet , i.e. j k (x) 2 j x Gamma k) Orthogonal and biorthogonal wavelet transforms have been extended to the interval [17, 18, 19], but a better solution is to abandon the translation dilation relationship. This leads to what are referred to as second generation wavelets in the literature [20, 21] The main advantage of second generation wavelets is that wavelets are constructed in the spatial domain and can be custom ....

C. K. Chui and E. Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical Methods of Approximation Theory, volume 105 of International Series of Numerical Mathematics, pages 53--75. Birkhauser Verlag, Basel, 1992.

....by Schumaker and Traas [22] where the sphere is parametrized by polar coordinates, a tensor product of trigonometric and polynomial splines is employed for scattered data approximation. Also in [9] tensor products are employed, there E splines ( 20] 11] are connected with interval wavelets ([2]) A characteristic feature of an approach which yields C 1 functions is the reproduction of trigonometric functions, this is assured with the help of appropriate E splines . Before summarizing the contents of the present work we describe shortly some other approaches which also treat the ....

C.K. Chui and E. Quak. Wavelets on a Bounded Interval. In D. Braess and L.L. Schumaker (Hrsg.), Numerical Methods of Approximation Theory, 1--24. Birkhauser Verlag, 1992.

....Computing, October 21, 1999 (24) The global cube algorithm that we present in this paper is based on a wavelet representation that uses the Haar scalar and wavelet basis functions. Further research may shed light on the advantages that higher order bases such as B splines on a bounded interval [5, 10] or orthonormalized basis functions on the bounded interval [25] can bring. Hierarchical Representation of Volume Data Sets In the following we describe the relationship between global illumination and wavelet theory by describing hierarchical representations of relevant parameters, and by ....

C. Chui and E. Quak. "Wavelets on a bounded interval." In D. Braess and L. L. Schumaker, editors, Numerical Methods of Approximation Theory, pages 1--24. Birkhuser Verlag, Basel, 1992.

....easy to fulfil when the underlying domain is the full Euclidean space IR d or the torus. However, the situation changes drastically when dealing with more realistic domain geometries in which case very little is known about appropriate bases. Thanks to numerous studies of wavelets on an interval [AHJP, CDV, CQ, DKU] wavelet bases on cubes are also available. In this paper we shall show that suitable biorthogonal multiresolution spaces on cubes which satisfy certain boundary conditions can be composed via parametric lifting to bases on unions of parametric images of cubes. The resulting composite ....

.... Delta Gammak) which are supported inside [0; 1] supplemented by certain additional linear combinations of the translates overlapping the end points of the interval. These linear combinations are formed in such a way that the resulting span still contains all polynomials of a desired order (see [AHJP, CQ, CDV, DKU]) To our knowledge only in [DKU] the dual multiresolution spaces S j induced by also exhibit the original order of polynomial exactness which is crucial in the present context. It is well known that the order of polynomial exactness determines the approximation order of the spaces. An ....

C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, D. Braess and L.L. Schumaker (eds.), Birkhauser, 1992, 1-24.

....properties. Thus, the common ground for all the above approaches is to have a sufficiently versatile collection of wavelet bases on . Such wavelets are conveniently constructed via tensor products of wavelets on the interval [0; 1] These, in turn, have been intensely studied in the literature [3, 9, 13, 18]. The above comments indicate that versatility in the present context means good localization of primal and dual bases as well as a possibly flexible choice of the order d of exactness (controlling the accuracy of the discretization scheme) and the order d of vanishing moments (controlling the ....

....S j to [0; 1] The fact that only very small portions of some functions contribute to the interval would seriously hurt the stability of the corresponding bases. Also since the supports of and generally differ the count would not match. The common strategy employed in all the quoted papers [3, 9, 13, 18] is to retain only those functions [j;k] j;k] whose support is fully contained in [0; 1] while forming in addition certain modified basis functions near the end points of the interval by taking fixed linear combinations of functions [j;k] near 0 and 1. These linear combinations have ....

C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, D. Braess and L.L. Schumaker (eds.), Birkhauser, 1992, 1--24.

....Sobolev spaces and their duals on [0; 1] within a range depending on the regularity and degree of exactness of the involved multiresolution analyses. 1. 1 Background and Motivation The issue of constructing wavelets on the interval has been recently addressed in several papers (see e.g. [AHJP, CQ, CDJV, CDV, Me]) However, as far as we know none of these approaches meets the above complete list of requirements. While [CDV, CQ] focus on orthogonal decompositions [AHJP] does address biorthogonal multiresolution but fails to build in any polynomial exactness of the dual spaces. Furthermore, neither is it ....

....multiresolution analyses. 1.1 Background and Motivation The issue of constructing wavelets on the interval has been recently addressed in several papers (see e.g. AHJP, CQ, CDJV, CDV, Me] However, as far as we know none of these approaches meets the above complete list of requirements. While [CDV, CQ] focus on orthogonal decompositions [AHJP] does address biorthogonal multiresolution but fails to build in any polynomial exactness of the dual spaces. Furthermore, neither is it proved there that the central biorthogonalization of properly adjusted spanning sets is actually possible nor are the ....

C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, D. Braess and L.L. Schumaker (eds.), Birkhauser, 1992, 1--24.

.... Gamma k)i IR (x Gamma k) 0 r d Gamma 1; 5.7) where d d 2. It is known that this is equivalent to the fact that the symbols a(z) X k2ZZ a k z k ; a(z) X k2ZZ a k z k ; contain (1 z) d , 1 z) d as a factor, respectively. The usual approach taken in [CDV, AHJP, CQ] is to introduce for k = 0; d Gamma 1 boundary near functions L j;L Gammad k : L Gamma1 X m= Gamma1 ff L j;m;k 2 j=2 (2 j Delta Gammam) fi fi [0;1] R j;2 j GammaR d Gammak : 1 X m=2 j GammaR 1 ff R j;m;k 2 j=2 (2 j Delta Gammam) fi fi [0;1] ....

C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, D. Braess, L.L. Schumaker (eds.), Birkhauser, 1992, 1-24.

.... are also other constructions of wavelets on [0; 1] In fact, for historical perspective it is interesting to notice that Franklin s original construction [56] was given for [0; 1] Another interesting one, in the case of semiorthogonal spline wavelets, has been given by Charles Chui and Ewald Quak [13]; we refer to the original paper for details. 12 Wavelet packets A simple, but most powerful extension of wavelets and multiresolution analysis are wavelet packets [29, 31] In this section it will be useful to switch to the following notation, m e ( H e ( G 1 Gammae ( for e = 0; 1: ....

C. Chui and E. Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical Methods of Approximation Theory, pages 1--24. Birkhauser Verlag, Basel, 1992.

....be constructed with the aid of tensor products of wavelets on the interval. The above mentioned boundary conditions immediately follow from corresponding complementary boundary conditions of wavelets defined on [0; 1] Although wavelets on the unit interval have recently received much attention [AHJP, CQ, DKU2] it seems that not much is known with regard to the type of boundary conditions needed in the above context. Therefore we have to revisit this issue in Section 4.1 where we begin with preparing the necessary univariate ingredients. This construction of suitable univariate wavelets which is of ....

.... Gammak) which are supported inside [0; 1] supplemented by certain additional linear combinations of the translates overlapping the end points of the interval. These linear combinations are formed in such a way that either the resulting span still contains all polynomials of a desired order (see [AHJP, CQ, CDV, DKU2]) or that certain homogeneous boundary conditions hold. We will consider here dual pairs where the primal generator is a B spline. Denoting by [x 0 ; x d ]f the dth order divided difference of f at the points x 0 ; x d 2 IR, the dth order centered cardinal B spline is defined by ....

C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, D. Braess and L.L. Schumaker (eds.), Birkhauser, Basel, 1992, 1--24.

....ffl Intervals: When working with finite data it is desirable to have basis functions adapted to life on an interval. This way no awkward solutions such as zero padding, periodization, or reflection are needed. We point out that many wavelet constructions on the interval already exist, see [1, 6, 3], but we would like to use the subdivision schemes adapted to boundaries since they lead to more straightforward constructions and implementations. ffl Irregular samples: In many practical applications, the samples do not necessarily live on a regular grid. Resampling is fraught with pitfalls and ....

....of cubic B splines we need to worry about the endpoints of a finite sized interval. Because of their support the scaling functions close to the endpoints would overlap the outside of the interval. This issue can be addressed in a number of different ways. One treatment, used by Chui and Quak [3], uses multiple knots at the endpoints of the interval. The appropriate subdivision weights then follow from the evaluation of the de Boor algorithm for those control points. The total number of scaling functions at level j becomes 2 j 3 in this setting. Consequently it is not so easy anymore ....

C. Chui and E. Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical Methods of Approximation Theory, pages 1--24. Birkhauser-Verlag, Basel, 1992.

....B spline techniques. As will be shown later, the orthogonal multiresolution pyramid originally proposed by S. Mallat [40] and the biorthogonal pyramid [28] 46] 47] in wavelet theory can all be derived from B splines [30] 33] 36] Other types of wavelets such as the wavelets on a interval [48], the periodic wavelets [37] and the cardinal spline wavelets [29] are all related to B splines. Motivated by these observations, the purpose of this paper is to build a more general framework of scale space representation in the context of B splines as an improvement of the traditional ....

....et al. 20] It was shown that the cubic B spline rather than the Gaussian kernel is optimal for edge detection. B splines are the shortest basis functions that provide a stable multiresolution analysis of a signal [33] This explains why many wavelet models of a vision [40] 28] 46] 47] [48], 37] are derived from B splines [33] 36] For the derivative operations, B spline approach is very intrinsic which elucidates the relationship between derivative and difference which are usually characterized by the two scale difference relations. B splines play an important role to bridge the ....

C. K. Chui and E. Quak, Wavelets on a bounded interval, in D. Brasess and L. L. Schumaker, eds., Numerical Methods in approximation Theory, vol. 9, pp. 53-65, Birhauser Verlag, Basel, 1992.

....analysis for domains Omega which preserves the important properties mentioned above. In order to motivate the multivariate constructions given later in this paper, it is useful to recall the techniques that have been proposed to handle multiresolution in the univariate case (see e.g. [CDV, CQ]) The approach in [CQ] makes use of Bspline techniques and therefore seems to be confined to piecewise polynomial scaling functions. Moreover, it doesn t seem to permit an extension to the multivariate case except for rectangular domains whose boundaries coincide with grid lines. The approach in ....

....which preserves the important properties mentioned above. In order to motivate the multivariate constructions given later in this paper, it is useful to recall the techniques that have been proposed to handle multiresolution in the univariate case (see e.g. CDV, CQ] The approach in [CQ] makes use of Bspline techniques and therefore seems to be confined to piecewise polynomial scaling functions. Moreover, it doesn t seem to permit an extension to the multivariate case except for rectangular domains whose boundaries coincide with grid lines. The approach in [CDV] is similar to ....

[Article contains additional citation context not shown here]

C.K. Chui, E. Quak, Wavelets on a bounded interval, in: Numerical methods of approximation theory, Vol.9, D.Braess and L.L.Schumaker eds, International series of numerical mathematics, Vol. 105, Birkha¨user, Basel, 1992.

....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 Micchelli and Xu. The aim of this paper is to construct a (semi orthogonal ) cubic spline wavelet basis for the Sobolev space on a bounded interval and to present its properties. Let I ....

Chui, C.K. and Quak, E., Wavelets on a bounded interval, in "Numerical Methods of Approximation Theory, Vol. 9." D.Braess and L. L. Schumaker, Eds. Birkh'auser Verlag, Basel, (1992), 53--75.

....and dilation, however, implies that the Fourier transform can no longer be used as a construction tool. A proper substitute is needed. Several results concerning the construction of wavelets adapted to some of the cases in G1 G3 already exist. For example, we have wavelets on an interval [8, 10, 18, 30, 31, 88], wavelets on bounded domains [27, 74] spline wavelets for irregular samples, 15, 7, 45] and weighted wavelets [11, 12, 104] These constructions are tailored toward one specific setting. Other instances of second generation wavelets have been reported in the literature, e.g. the construction ....

C. Chui and E. Quak, Wavelets on a bounded interval, in Numerical Methods of Approximation Theory, D. Braess and L. L. Schumaker, eds., Birkhauser-Verlag, Basel, 1992, pp. 1--24.

....the Fourier transform can no longer be used as a construction tool. LIFTING AND SECOND GENERATION WAVELETS 3 A proper substitute is needed. Several results concerning the construction of wavelets adapted to some of the cases in G1 G3 already exist. For example, we have wavelets on an interval [8, 10, 18, 30, 31, 88], wavelets on bounded domains [27, 74] spline wavelets for irregular samples, 15, 7, 45] and weighted wavelets [11, 12, 104] These constructions are tailored toward one specific setting. Other instances of second generation wavelets have been reported in the literature, e.g. the construction ....

C. Chui and E. Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical Methods of Approximation Theory, pages 1--24. Birkh¨auser-Verlag, Basel, 1992.

....wavelet analysis to the unit interval. In particular, it is no longer possible to have all wavelets be simple dilations and translations of a single mother wavelet as in (2) Wavelets on the boundary must be adapted to retain orthonormality. Some approaches to this problem include Meyer (1991) Chui and Quak (1992) and Cohen, Daubechies, Jawerth, and Vial (1993) Simpler approaches use reflection or symmetric handling of the boundary. This work will presuppose the choice of an appropriate orthonormal wavelet basis on [0; 1] denoted f j;k ; k = 0; 2 j Gamma 1; j = 0; 1; g. Given such a basis ....

Chui, C. K., & Quak, E. (1992). Wavelets on a bounded interval. In D. Braess, & L. L. Schumaker (Eds.), Numerical Methods of Approximation Theory, pp. 1--31. Birkhauser Verlag, Basel, Switzerland.

....boundary adaptation here. Throughout the rest of this paper let Omega = 0; 1] n . Employing as above tensor products of univariate functions (3.2. 6) then reduces the problem to constructing suitable multiscale bases on [0; 1] Several such constructions have been described in the literature [1, 7, 10, 12, 21]. In particular, 1] treats the case of biorthogonal wavelets which is needed here. Unfortunately, one cannot apply these results directly since, on one hand, the stability in the sense of (2:11) is not addressed there and, on the other hand, the stability properties needed here require estimates ....

C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, D. Braess and L.L. Schumaker (eds.), Birkh¨auser, 1992, 1-24.

....k)i IR (x Gamma k) 0 r d Gamma 1; 5.7) where d d 2. It is known that this is equivalent to the fact that the symbols a(z) X k2ZZ a k z k ; a(z) X k2ZZ a k z k ; contain (1 z) d , 1 z) d as a factor, respectively. The usual approach taken in [CDV, AHJP, CQ] is to introduce for k = 0; d Gamma 1 boundary near functions L j;L Gammad k : L Gamma1 X m= Gamma1 ff L j;m;k 2 j=2 (2 j Delta Gammam)j [0;1] R j;2 j GammaR d Gammak : 1 X m=2 j GammaR 1 ff R j;m;k 2 j=2 (2 j Delta Gammam)j [0;1] 5.8) where ....

C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods of Approximation Theory, D. Braess, L.L. Schumaker (eds.), Birkhauser, 1992, 1-24.

....(N = 3) average interpolation at j = 3 and k = 0; 1; 2; 3. On the bottom the scaling functions of cubic (N = 4) interpolation at j = 3 and k = 0; 1; 2; 3. Note how the boundary scaling functions are still interpolating. out that many wavelet constructions on the interval already exist, see [2, 9, 4], but we would like to use the subdivision schemes of the previous sections since they lead to easy implementations. 2. Weighted inner products: Often one needs a basis adapted to a weighted inner product instead of the regular L 2 inner product. A weighted inner product of two functions f and g ....

CHUI, C., AND QUAK, E. Wavelets on a Bounded Interval. In Numerical Methods of Approximation Theory, D. Braess and L. L. Schumaker, Eds., 1--24, 1992.

....wavelet transform on the sphere is discretized. In [26] one first maps the rectangle [0; Theta [0; 2 ] or box in higher dimensions) to the sphere via standard spherical coordinates, and then one constructs wavelets by taking tensor products of Euclidean wavelets designed for intervals [5, 6] with periodic wavelets [4, 14, 18, 20, 22] Of course, doing this results in singularities and distortions near the poles, and these must be dealt with. The mathematical motivation for the work presented here stems from work by several researchers. Chui, Jetter, Stockler, and Ward [3] employ ....

C. K. Chui and E. Quak, "Wavelets on a bounded interval," pgs. 53-75 in Numerical Methods of Approximation Theory, D. Braess and L. Schumaker (eds.), Birkhauer, Basel, 1992.

....method is needed and it is natural to turn to wavelet estimators. The spatially adaptive properties of wavelet shrinkage estimators are discussed by Donoho and Johnstone (1995b) First, a brief introduction to wavelets. For more detailed information see, for example, Daubechies (1992) and C. K. Chui (1992). Wavelets on the real line IR may be formed from a mother wavelet function via dyadic dilation and integer translation operations: j;k (x) 2 j=2 (2 j x Gamma k) 2) The entire set of wavelets f j;k ; j 2 ZZ; k 2 ZZg forms an orthonormal basis for L 2 (IR) provided the mother ....

....wavelet analysis to the unit interval. In particular, it is no longer possible for all wavelets to be simple dilations and translations of a single mother wavelet as in (2) Wavelets on the boundary must be adapted to retain orthonormality. Some approaches to this problem include Meyer (1991) Chui and Quak (1992) and Cohen, Daubechies, Jawerth, and Vial (1993) Simpler approaches use reflection or symmetric handling of the boundary. This work will presuppose the choice of an appropriate orthonormal wavelet basis on [0; 1] denoted f j;k ; k = 0; 2 j Gamma 1; j = 0; 1; g. 1.2 The discrete ....

Chui, C. K., & Quak, E. (1992). Wavelets on a bounded interval. In D. Braess, & L. L. Schumaker (Eds.), Numerical Methods of Approximation Theory, pp. 1--31. Birkhauser Verlag, Basel, Switzerland.

....on the line. Y. Meyer [45] introduced orthogonal wavelet transforms on the interval, CDJV] offered a numerically more stable orthogonal transform; bi orthogonal Boxcar Wavelet Decompositions 25 transforms have been proposed by Auscher[49] by Jouini and Lemari e [38] and by Chui and Quak [7]. In this volume, see the articles of Daubechies [13] and Andersson, Hall, Jawerth, and Peters [1] Our presentation stresses the resemblance of the B W Interval transform with these other interval transforms: it has a similar formal structure all the basis functions are dilations and ....

Chui, C. and E. Quak, Wavelets on a bounded interval, in Numerical Methods of Approximation Theory, Dietrich Braess and Larry L. Schumaker (eds.), Birkhauser Verlag, Basel, 1992, 53--75.

....wavelets j i . There are many alternatives to the Haar basis, each with advantages and disadvantages. Like the Haar basis, flatlets and multiwavelets are suited to the bounded domains over which we define radiance distributions [18] B spline wavelets can also be adapted to a bounded interval [11, 38, 39]. These higher order basis functions are appealing because of their improved convergence properties, but they also require more costly numerical integration rules than the Haar basis functions. Our algorithm uses the Haar basis because of its simplicity and convenience, but further research may ....

Charles K. Chui and Ewald Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical Methods in Approximation Theory, volume 9, pages 53--75. Birkhauser Verlag, Basel, 1992.

....and dilation, however, implies that the Fourier transform can no longer be used as a construction tool. A proper substitute is needed. Several results concerning the construction of wavelets adapted to some of the cases in G1 G3 already exist. For example, we have wavelets on an interval [6, 8, 13, 25, 26, 74], wavelets on bounded domains [22, 61] and weighted wavelets [9, 10, 88] These constructions are tailored toward one specific setting. In this paper, we present the lifting scheme, a simple, general construction of second generation wavelets. The basic idea, which inspired the name, is to start ....

C. Chui and E. Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical Methods of Approximation Theory, pages 1--24. Birkhauser-Verlag, Basel, 1992.

....and animation [9] because of its lack of continuity. There are a variety of ways to construct wavelets with k continuous derivatives. One such class of wavelets can be constructed from piecewise polynomial splines. These spline wavelets have been developed to a large extent by Chui and colleagues [3, 4]. The Haar basis is in fact the simplest instance of spline wavelets, resulting when the polynomial degree is set to zero. In the following, we briefly sketch the ideas behind the construction of endpoint interpolating B spline wavelets. Finkelstein and Salesin [8] developed a collection of ....

....instance of spline wavelets, resulting when the polynomial degree is set to zero. In the following, we briefly sketch the ideas behind the construction of endpoint interpolating B spline wavelets. Finkelstein and Salesin [8] developed a collection of wavelets for the cubic case, and Chui and Quak [4] presented constructions for arbitrary degree. Although the derivations for arbitrary degree are too involved to present here, we give the synthesis filters for the piecewise constant (Haar) linear, quadratic, and cubic cases in Appendix A. The next three sections parallel the three steps ....

[Article contains additional citation context not shown here]

Charles K. Chui and Ewald Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical Methods in Approximation Theory, volume 9, pages 53--75. Birkhauser Verlag, Basel, 1992.

....determine the analysis filters A j and B j by equation (9) We use the set of 2 j minimally supported functions that span W j . Appendix A contains more details on the specific wavelets we use and their derivation. A similar construction has also been independently proposed by Chui and Quak [9]. Note that multiresolution constructions can be built for other types of splines as well, such as uniform B splines [8] and nonuniform B splines with arbitrary knot sequences [21] A recent construction applicable to subdivision surfaces is discussed by Lounsbery et al. 19] Note that ....

C. K. Chui and E. Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical Methods in Approximation Theory, volume 9, pages 53--75. Birkhauser Verlag, Basel, 1992.

....a biorthogonal wavelet basis was constructed by A. Cohen, I. Daubechies, B. Jawerth and P. Vial [5] For our purpose it is sufficient to work with pre wavelets in the sense of (1.2) On an interval and for the case of B splines such a wavelet basis was constructed by C. K. Chui and E. Quak [3]. We will use their approach in the sequel, see Section 2.2 for details. For more general manifolds, one approach could be to use specific charts that enables us to reduce the problem to these well known cases. As we will see below, for S 2 and S 3 such specific charts exist which make it ....

....equivalent to G (2 ) 0, we obtain det a (z) b (z) a ( Gammaz) b ( Gammaz) z Gamma1 (ja (z)j 2 G =2 ( ja ( Gammaz)j 2 G =2 ( 2 ) 4z Gamma1 G (2 ) 0 : 2 2. 2 Wavelets on an Interval In this section we briefly recall the construction presented in [3]. There one starts with a knot sequence t (j) t (j) m : ft (j) ff g 2 j m Gamma1 ff= Gammam 1 given by t (j) Gammam 1 = t (j) Gammam 2 = t (j) 0 = 0 (knot of multiplicity m) t (j) ff = ff2 Gammaj ; ff = 1 ; 2 j Gamma 1 ; t (j) 2 j = t (j) 2 j 1 = ....

[Article contains additional citation context not shown here]

C. K. Chui and E. Quak, Wavelets on a bounded interval, in "Numerical Methods of Approximation Theory, Vol. 9" ( D. Braess and L. L. Schumaker, Eds.), Birkh¨auser, Basel, 1992.

....idea is similar to previous approaches in particular to the construction of spline wavelets [13] 10] Moreover, with standard techniques of multiresolution analysis in [5] orthonormal bases and in [3] biorthonormal bases of compactly supported wavelets in L 2 (R) were developed. Chui and Quak [2] studied B splines and mulitresolution analysis. They can give an explicit but complicated spline wavelet for their wavelet space. Furthermore, an adaptive approximation scheme was proposed by de Boor and Rice [9] and modified by Dahmen in [4] The basic new feature of our approach is a simple ....

C. K. Chui and E. Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, International Series of Numerical Mathematics, volume 105, pages 53--75, Basel, 1992. Birkhauser.

....A( and then look for the minimax approximation of e ( in the interval [ Gamma ; 5 Wavelets on an interval So far our discussion only involved the case of the real line which is invariant for integer shifts. Recently several constructions of wavelet on an interval became available [4, 8, 12, 13, 29]. These constructions all have in common that the functions that are supported, in some sense, away from the endpoints correspond to the ones from the real line, while new basis functions are constructed near the boundary. One of the problems is that the shift invariance is lost at the boundary. ....

C. Chui and E. Quak. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical Methods of Approximation Theory, pages 1--24. Birkhauser-Verlag, Basel, 1992.

....by introducing special boundary splines. These boundary functions, however, are typically not obtained by restricting B splines with uniform knots, but by introducing B splines with multiple knots at the ends of the interval. The construction of spline wavelets in this framework was carried out in [8]. In the more general setting, where direct sum decompositions are considered instead of orthogonal ones, biorthogonal spline wavelets on the interval were investigated in [12] In this paper, we will restrict our attention to mutually orthogonal spline wavelet spaces, but for general nonuniform ....

....used in the interior of the interval, with knots of multiplicity d 1 at the interval endpoints, i.e. for nonnegative integers d, n the knot sequence is u d,n = d 1 z 0, 0 # k 2 n : 1 # k # 2 n 1 # d 1 z 1, 1 . This setting was first considered in [8]. The corresponding decomposition and reconstruction algorithms were investigated in [28] their tensor product versions in [29] and a spline wavelet packet approach in [30] As a typical and very frequently used example, let us consider the cubic case d = 3, for which we want to provide all ....

Chui, C. K. and E. G. Quak, Wavelets on a bounded interval, in Numerical Methods in Approximation Theory, ISNM 105, D. Braess, L.L. Schumaker (ed), Birkhauser, Basel, 1992, 53--75.

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CHUI,C .AND QUAK, E. 1992. Wavelets on a bounded interval. In Numerical Methods of Approximation Theory, D. Braess and L. L. Schumaker, Eds. Birkhauser-Verlag, Basel, 1--24.

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Chui C., Quak E.: Wavelets on a bounded interval. In Braess D., Schumaker L.L. (eds.) Numerical Methods of Approximation Theory. Birkhauser Verlag, Basel, 1--24 (1992).

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C. K. Chui and E. Quak, Wavelets on a bounded interval, in Numerical Methods of Approximation Theory, D. Braess and L. L. Schumaker, eds., vol. 6, Birkhuser-Verlag, Basel, 1992, pp. 5375.

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Chui, C. K. and Quak, E., Wavelets on a bounded interval, in Numerical Methods of Approximation Theory, Braess, D. and Schumaker, L. L., Eds., vol. 105 of International Series of Numerical Mathematics, pp. 53--75. Birkhauser Verlag, Basel, (1992).

No context found.

CHUI,C .AND QUAK, E. 1992. Wavelets on a bounded interval. In Numerical Methods of Approximation Theory, D. Braess and L. L. Schumaker, Eds. Birkhauser-Verlag, Basel, 1--24.

No context found.

C. K. Chui and E. Quack. Wavelets on a bounded interval. In D. Braess and L. L. Schumaker, editors, Numerical methods in approximation theory, volume 9 of International Series of Numerical Mathematics, 105, pages 53--75. Birkhauser, Basel, 1992.

No context found.

Chui, C. and Quak, E. (1992) Wavelets on a bounded interval. Numerical Methods of Approximation Theory, Dietrich Braess and Larry L. Schumaker, Eds. Birkhauser Verlag, Basel, pp. 1-24.

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