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Construction of nearly orthogonal interpolating wavelets
Signal Processing 79 (1999) 289}300
Construction of nearly orthogonal interpolating wavelets Peng-Lang Shui*, Zheng Bao The Key Laboratory for Radar Signal Processing, Xidian University, Xi'an, 710071, People's Republic of China Received 29 October 1998; received in revised form 1 July 1999
Abstract This paper "rst studies the measure of redundancy in biorthogonal wavelet systems and gives the explicit parameter representations of the general interpolating "lters for lifting Donoho wavelets. Based on the measure and the parameter representations, a novel method is proposed to optimize co-redundancy degree which is then used to construct a family of nearly semi-orthogonal interpolating wavelets. Furthermore, by performing the local orthogonalization on the obtained wavelet functions we design a family of nearly orthogonal interpolating wavelets. Finally, some examples are given which show that our scheme is e$cient in designing practical wavelet systems with good performance indexes. ( 1999 Elsevier Science B.V. All rights reserved. Zusammenfassung In dm` esem Artikel wird zunaK chst das Redundanzma{ in bi-orthogonalen Wavelet-Systemen untersucht und eine explizite Parameterdarstellung des allgemeinen Interpolations"lters fuK r anhebende Donoho Wavelets angegeben. Ausgehend von den AusdruK cken fuK r das Ma{ und die Parameter wird ein neuartiges Verfahren vorgeschlagen, um den Grad der Co-Redundanz zu optimieren, der anschlie{end benutzt wird, um eine Familie von nahezu semi-orthogonalen Interpolationswavelets zu konstruieren. Durch lokale Orthogonalisierung der erhaltenen Wavelet-Funktionen entwerfen wir daruK ber hinaus eine Familie nahezu orthogonaler Interpolationswavelets. Schlie{lich werden einige Beispiele angegeben, die zeigen da{ sich unser Verfahren fuK r den Entwurf praktischer Wavelet Systeme mit gutem Leistungsgrad eignet. ( 1999 Elsevier Science B.V. All rights reserved. Re2 sume2 Nous eH tudions tout d'abord dans cet article la mesure de la redondance dans les syste`mes d'ondelettes bi-orthogonales et donnons les repreH sentations explicites des parame`tres des "ltres interpolants geH neH raux pour les ondelettes de Donoho. Sur la base de cette mesure et des repreH sentations des parame`tres, nous proposons une meH thode nouvelle pour optimiser le degreH de co-redondance, meH thode utiliseH e ensuite pour construire une famille d'ondelettes interpolantes presque semi-orthogonales. De plus, par orthogonalisation locale des ondelettes obtenues nous concevons une famille d'ondelettes interpolantes presqu'orthogonales. En"n, nous donnons des exemples qui montrent que notre meH thode est e$cace pour la conception de syste`mes d'ondelettes pratiques, preH sentant de bons indices de performances. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Interpolating "lter; Interpolating wavelets; Redundancy degree; Local orthogonalization; Nearly orthogonal
* Corresponding author. Tel.: #86-29-8221025; fax: #86-29-8236159. E-mail address: [email protected] (P.-L. Shui) 0165-1684/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 9 9 ) 0 0 1 0 2 - 4
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Nomenclature Z uD RY(u) RY(u,t) RY(t) h(m) D h, u a a015
integer set Deslauriers}Dubuc fundamental function redundancy degree of the scaling function u co-redundancy degree redundancy degree of the wavelet t Deslauriers}Dubuc "lter general interpolating "lters parameter vector optimal parameter vector
S f, gT fK q Q P Vj Wj cj( f ) dj( ( f )
inner product of f and g in ¸2(R) Fourier transform f optimal vector in local orthogonalization an all pole "lter inverse "lter of Q jth level approximation subspace jth level detail subspace jth approximation coe$cients of f jth detail coe$cients of f
Abbreviation Min minimum or minimizing
1. Introduction
holds:
Over the last 10 years, two-band interpolating wavelets have been widely studied and applied [4}7,10,14,16,17,19,20,23]. As an important property, the interpolation can remarkably simplify the initialization of the wavelet series transform (WST). Many applications often need to use the WSTs, which are performed by two steps. First, for a given multiresolution analysis (or scaling function u) and a single f (x)3L2(R), one determines a proper scaling factor J and the approximation coe$cients of f (x) in the multiresolution subspace V (u), which is j called the initialization. Second, one computes wavelet (or detail) coe$cients by the pyramid algorithm as one does in the case of discrete wavelet transform (DWT). Generally, the approximation coe$cients of a signal is not consistent with its uniform samples even if f (x)3V (u). But the initiaJ lization by the numerical integral is computationally costly, and the direct initialization by using of the uniform samples often has poor approximation power. Hence, the diverse pre-"ltering algorithms for the initialization [1,15,22] have been recently developed to improve the approximation power. But another problem exists with pre-"ltering, i.e., it is impossible to design perfect reconstruction FIR pre/post-"lters unless they are both all-pass. However, for cardinal interpolating scaling functions, the following standard sampling theorem [23]
n f (t)"+ f u(2J x!n), ∀f (x)3V (u), (1) J 2J n which is the generalization of Shannon sampling theorem. Consequently by using the interpolating wavelets the complicated pre-/post-"ltering process can be avoided, and thus much convenience results. Deslauriers, Dubuc, Donoho, Sweldens and Harten [5}7,19,20] have established the fundamental frame of interpolating wavelets. In this paper, we are only interested in the lifting Donoho wavelets, a family of biorthogonal interpolating wavelets. Using Deslauriers}Dubuc fundamental functions u (x) [5,7] as scaling functions and D t (x)"u (2x!1) as wavelets, Donoho construcD D ted a family of interpolating wavelets [6], later referred to as Donoho wavelets, which is closely connected to the Lazy wavelets by the dual lifting scheme [20]. However, such wavelets have the following disadvantages: wavelets do not have any vanishing moments and thus do not form a Riesz basis for L2(R); the duals do not even belong to L2(R), and the dual scaling "lters are full-pass without frequency localization. Later, by lifting scheme, Sweldens improved Donoho wavelets, thus the above disadvantages are avoided. Under the basic framework of lifting scheme [19,20], two-band compactly supported interpolating wavelets with compactly supported duals can be generated by
AB
P.-L. Shui, Z. Bao / Signal Processing 79 (1999) 289}300
applying the lifting scheme to Donoho wavelets, hereafter referred to as lifting Donoho wavelets. As is well known, there does not exist any twoband compactly supported orthogonal interpolating wavelets except for the Haar wavelet [23]. The lifting Donoho wavelets can have many advantages simultaneously, such as compact support, linear phase, high regularity, and the "nite digits of "lters [6,14,19]. However, one serious defect is comparatively much redundancy. Designing the compactly supported interpolating wavelets with less redundancy, namely, nearly orthogonal interpolating wavelets, is the main problem that will be studied in this paper. Section 2 reviews the concept and properties of the lifting Donoho wavelets and the relation with the lifting scheme, introduces the redundancy degrees to measure the redundancy of a biorthogonal wavelet system, studies the estimation of Riesz bounds and gives the parameter representations of general interpolating "lters. On the basis of these, enlightened the partial orthogonalization proposed by Lounsbery et al. [12], Section 3 constructs a family of nearly semi-orthogonal interpolating wavelets by optimizing of co-redundancy degree. In Section 4, a novel method called local orthogonalization is proposed to decorrelate the wavelet basis at the same level and by which we construct the nearly orthogonal interpolating wavelets. Finally, several examples are presented, which show that our method is highly e$cient. 2. Lifting Donoho wavelets and redundancy degrees
291
ing the Dirac impulse. Such wavelets have several disadvantages as mentioned in the above section. In 1995, Sweldens presented the lifting scheme [20,23], a novel way to construct biorthogonal wavelets. Under the basic framework of lifting scheme, Donoho wavelets can be obtained by using the dual lifting scheme for the Lazy wavelets. In order to overcome these disadvantages of Donoho wavelets, Sweldens [20] uses the lifting scheme for Donoho wavelets, and obtains a novel family of interpolating wavelets, which we term `lifting Donoho waveletsa. A lifting Donoho wavelet system can be described as follows: u(x)"u(2x)#+ h(k)u(2x!2k#1), k
(3)
t(x)"2u(2x!1)!+ u(k)u(x!k), k with the duals u8 (x)"2u8 (2x)#+ u(!k)tI (x!k), (4) k tI (x)"u8 (2x!1)!+ h(!k)u8 (2x!2k!2), k where h and u are a pair of real FIR general interpolating "lters. Obviously, when h"u"0, Eqs. (3) and (4) describe a Lazy wavelet; when u"0 these describe a Donoho wavelet. On the vanishing moments of lifting Donoho wavelets, there is the following conclusion. Proposition 1. u, u8 , t and tI in Eqs. (3) and (4) have D#1 vanishing moments if and only if the general interpolating xlters h and u satisfy [19, Theorem 12]
2.1. Lifting Donoho wavelet
+ kdh(k)"+ kdu(k)"(1/2)d, d"0,1,2, D. k k
In 1992, Donoho [6] suggested the idea of wavelets built from interpolating scaling functions, and constructed a family of interpolating wavelets, later referred to as Donoho wavelets, which can be described as follows:
A scaling function is called the D#1 vanishing moments (or D#1 order regular) if its satis"es :xdu(x) dx"d(d), d"0,1,2, D, which is closely connected with the approximation order of a scaling function and is usually referred to as the Coi#et property.
u(x)"u(2x)#+ h(k)u(2x!2k!1), k t(x)"2u(2x!1),
(2)
with the duals u8 (x)"d(x) and tI (x)" u8 (2x!1)!+ h(!k)u8 (2x!2k!2), d(x) denotk
(5)
2.2. Redundancy degrees The frame bounds and the redundancy ratio [3] are often used to measure the redundancy of
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a frame. Although they may be applied to biorthogonal wavelet system, their complicated estimations prevent them from being applied to optimal design of wavelets. As a result, we introduce the following three indexes to measure the redundancy of a biorthogonal wavelet system. Let u and t be the synthesis scaling function and wavelet; then the redundancy degree of the scaling function is de"ned as 1 RY(u), + Su(x),u(x!k)T2. (6) DDuDD4 2 kE0 The co-redundancy degree between the approximation subspace and the detail subspace at the same level is de"ned as 1 RY(u,t), + St(x),u(x!k)T2. (7) DDuDD2 DDtDD2 2 k 2 Lounsberg et al. [12] use the objective function analogous to (7) to construct near semi-orthogonal wavelets. The redundancy degree of the wavelet function is de"ned as 1 RY(t), + St(x),t(x!k)T2. (8) DDtDD4 2 kE0 Obviously, RY(u,t)"0 corresponds to semi-orthogonal wavelets, RY(u)"RY(t)"0 to shift-orthogonal wavelets [21] and RY(u,t)"RY(t)"0 to orthogonal wavelets. All existing orthogonal wavelets satisfy RY(u)"0 also, because their construction starts from orthogonal scaling functions. Whether or not there exist orthogonal wavelets with RY(u)O0 is an interesting problem, from which a novel method may be developed to construct orthogonal wavelets. Below we discuss the Riesz bounds of wavelets. Let u(x), t(x) be the compactly supported synthesis scaling and wavelet functions. We de"ne C (u)"+ Su(x),u(x!k)Te~+ku, r k C (u)"+ St(x),t(x!k)Te~+ku, t k C (u)"+ Su(x),t(x!k)Te~+ku, r,t k
(9)
Due to the compact support, C (u), C (u), C (u) r t r,t are trigonometric polynomials. Consider A "min MC (u)N, B "max MC (u)N, r r r r u u A "min MC (u)N, B "max MC (u)N, t t t t u u
(10)
A "min MDC (u)D2N, B "max MDC (u)D2N. r,t r,t r,t r,t u u Then, for the Riesz bounds of the wavelet the following theorem holds. Theorem 1. Let u(x), t(x) be the compactly supported synthesis scaling function and wavelet function in a biorthogonal wavelet system; then ∀Md N3l2(Z2), j,k 2 ADDtDD2 DDMd NDD22 ) + d t (x) 2 j,k l j,k j,k 2 j,k|Z )BDDtDD2 DDMd NDD22 , (11) 2 j,k l 1 B 1 A A" A ! r,t , B" B ! r,t , t DDtDD2 t A DDtDD2 B 2 2 r r where t (x)"2j@2t(2jx!k). When A A 'B , j,k r t r,t Mt , j, k3ZN constitutes a Riesz basis in ¸2(R). j,k
K
A
K
B
A
B
This theorem gives an e$cient estimator of Riesz bounds, and moreover, the ratio between A and B can be used to measure the closeness of a wavelet system to orthogonal one. The proof is given in Appendix A. 2.3. Parameter representation of general interpolating xlter The subdivision scheme has been successfully applied to the construction of wavelets on real line, two-dimensional nontenor-product wavelets [16], as well as wavelets on manifolds [12]. In one-dimensional subdivision scheme, Donoho wavelets, and lifting Donoho wavelets, Deslauriers}Dubuc "lters play an important role and their length are the shortest among D#1 order interpolating "lters. An interpolating "lter is of D#1 order if the interpolator is accurate for all polynomials whose degree is less than D#1 [10]. A D#1 order
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Deslauriers}Dubuc "lter can be formulated as m`D 2l!1 , h(m)(k)" < D 2l!2k l/m,lEk k"m, m#1,2, m#D.
(12)
Di!erent values of m correspond to di!erent "lters; such #exibility enables us to e$ciently perform the boundary processing [10]. More importantly, using them as bases, the general interpolating "lters can be parameterized. Proposition 2. Let h be a D#1 order interpolating xlter with support set MM, M#1,2, M#NN (N*D); then there exists a unique parameter vector a satisfying + a(m)"1 such that [13] m M`N~D (13) h" + a(m)h(m). D m~M The complexity of optimizing a wavelet system usually depends on the structure of the "lters. In the lifting scheme, the choice of the lifting "lter does not in#uence biorthogonolity; thus, the designing is easier and more #exible. The parameterization in (13) enables us to freely choose the parameter vector a in the designing of lifting Donoho wavelets but does not in#uence the biorthogonality and the number of vanishing moments. Given an interpolating function, the optimized wavelet is then derived as follows: "rst the co-redundancy degree RY(u,t) is minimized to derive the "lter u and thence a nearly semi-orthogonal interpolating wavelet; then design a nearly orthogonal interpolating wavelet by the local orthogonalization of the obtained wavelet function.
3. Optimal design of co-redundancy degree For the two-band orthogonal wavelets on real line, the compact support is incompatible with linear phase of "lters; therefore, one must either abandon the compact support or relax orthogonality. A successful case is the semi-orthogonal spline wavelets by Chui [3], whose duals are exponent decay rather than compact support. Another case is the biorthogonal wavelets by Cohen et al. [2], and
293
wavelets as well as their duals are compact support and linear phase. Moreover, in biorthogonal wavelets systems, to reduce the inter-level redundancy becomes a problem concerned by researchers. In multiresolution analysis for surfaces [12], Lounsbery proposed the idea of `partial orthogonalizationa to design near semi-orthogonal wavelets, the aim of which is that the approximation subspace and detail subspace on the same level are as orthogonal as possible, and this method can be seen as a particular instance of the lifting scheme. This method can be applied to a Donoho wavelet to design a nearly semi-orthogonal lifting Donoho wavelet. Below we give a brief description of this method. Let u(x) be the Donoho scaling function given in (2), X denotes any subset of integer set Z; typically X consists of the integer k such that u(x!k) is supported in some neighbourhood of u(2x!1). Take the wavelet t(x) to be [12, Eq. (17)] t(x)"2u(2x!1)! + u(k)u(x!k). (14) k|X In order to minimize the ¸2 norm of the projection of t(x) into V , the lifting "lter u are determined by 0 solving the linear system [12, Eq. (18)] + Su(x!k), u(x!l)Tu(k) l|X "S2u(2x!1), u(x!k)T, k3X.
(15)
When X"Z, a semi-orthogonal wavelet is obtained but is not compactly supported; when ) is "nite, the solution of the linear system (15) only guarantees that for arbitrary k3X, t(x) is orthogonal to u(x!k) but does not guarantee that the ¸2 norm of it projection into V is minimal. 0 Similar to the method in [12], in this section, we design nearly semi-orthogonal lifting Donoho wavelets by minimizing co-redundancy degree. In the designing, the interpolating scaling function is given and the length of the "lter u is "xed, and thus the regularity of wavelet function is invariant, but the regularity of the duals is changeable according to Eq. (3). Instead of the co-redundancy degree, the following objective function is employed: f (u)"+ St(x), u(x!k)T2. 0"+%#5 k
(16)
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This has slight di!erence with expression (15), where the square sum of the inner product between t(x) and u(x!k), k3X, rather than k3Z is taken as the objective function. For the sake of simplicity, we introduce the following notations: a(k),Su(x), u(x!k)T, b(k),2S(2x!1), u(x!k)T, A(k, l)"+ a(k!m)a(l!m), m B(k),+ a(k!m)b(m), m C"+ b2(k). k Therefore the objective function is rewritten as f (u)"uTAu!2uTB#C. (17) 0"+%#5 When the scaling function in (3) had D#1 regular moments, generally, the wavelet is required to have D#1 vanishing moments, i.e., the "lter u is D#1 order. This constraint condition and Eq. (17) lead to the following quadratic programming: MuTAu!2uTB#CN (18) 1 d + kdu(k)" , d"0.1,2,2, D. k 2 The parameter representation of Eq. (13) can greatly simplify the optimal algorithm. Let the support set of the "lter u be MM, M#1,2, M#NN; from Proposition 2, the programming (15) can be simpli"ed as the one with a linear constraint on the parameter vector a: min g s.t.
min a s.t.
AB
MaTHTAHa!2aTHTB#CN (19)
+a(k)"1, k where H is an (N#1)](N#1!D) matrix formed by Deslauriers}Dubuc "lters: H(m:m#D, m)"h(m), D m"M, M#1,2, M#N!D and the remaining entries are zeros. By solving the above quadratic programming, the optimal parameter vector a015 and the optimal "lter u015"Ha015 are obtained.
Sometimes, in order to retain other advantages, some proper constraints may be imposed on the parameter vector a. For example, when N and D are odd numbers, the symmetric a assures the linear phase; the "nite digits of a assures that the optimal "lter is of "nite length. These are advantageous for applications, especially for digital "ltering. Finally, we give a brief description on the di!erence between the optimal method in (15) and our method. First, the optimal method in (15) cannot be directly applied to the case with vanishing moments constraints. Imposing such constraints on the "lter u results in a super-determining linear system, which needs to search for the least-squares solution. Second, due to the slight di!erence of the objective functions, the optimal wavelet system obtained by our method is closer to semi-orthogonal one. For example, the scaling function u(x) is the hat function satisfying two scaling relation u(x)"u(2x)#1/2(u(2x!1)#u(2x#1)); consider the X"M0,1N, i.e., the wavelet function has the form of t(x)"2u(2x!1)!u(0)u(x)! u(1)u(x!1). Following the method in (15), the optimal "lter u "[3/5,3/5], Su(x#1), t (x)T"Su(x!2), 1 1 t (x)T"1/10, the correlation coe$cients else are 1 zero, and the co-redundancy degree is 0.0273. Using our method, the optimal "lter u " 2 [15/26,15/26] and Su(x),t (x)T"Su(x!1), 2 t (x)T"1/52, Su(x#1),t (x)T"Su(x!2),t (x)T 2 2 2 "!5/52, the correlation coe$cients else are zero, and the co-redundancy degree is 0.0262.
4. Decorrelation of wavelet bases } local orthogonalization The minimization of co-redundancy degree reduces the redundancy between the approximate subspace and the detail subspace at the same level. Thereby, there is hardly redundancy between the wavelet coe$cients at di!erent levels. However, because Mt(x!k), k3ZN does not constitute an orthogonal basis, the redundancy between the wavelet coe$cients of the same level becomes the main ingredient to degrade the performance of the system.
P.-L. Shui, Z. Bao / Signal Processing 79 (1999) 289}300
Gram}Schmidt orthogonalization and the normalized orthogonalization [3] are two commonly used methods to decorrelate. Unfortunately, the former does not perserve the good structure of the original basis, that is the new basis cannot be generated from integer shifts of a single function, which often complicates the processing in wavelet domain. The latter is often used to construct an orthogonal wavelet from semi-orthogonal one. Assume t as a compactly supported semi-orthogonal wavelet function with Riesz lower bound A'0, set tK (u) tK M(u)" ; J+DtK (u#2kn)D2 then MtM(x!k), k3ZN is an orthogonal basis in W but not compactly supported. Moreover, for 0 many applications, the nearly orthogonal basis is slightly di!erent from the orthonormal basis; thus the orthogonality can be traded for other good properties of the system. A new method, local orthogonalization, is proposed to minimize rather than to completely remove the redundancy of the wavelet basis Mt(x!k), k3ZN. It can be described as follows. Let t(x) be a wavelet function obtained from optimization (19); generally, the comparatively large coe$cients in r (k)"St(x),t(x!k)T cont centrate near k"0. Choose a proper length parameter ¸ and construct a new wavelet function L /(x),t(x)! + q(l)t(x!l), (20) l/0 such that the redundancy degree of the new wavelet / is minimal, i.e., min q
G
+L S/(x),/(x!l)T2 l/1
H
(21)
+L Dq(l)D)g(1. l/1 In this case, the decomposition and synthesis algorithms are as follows. Assume f (x)3< , the initial j approximation coe$cients are s.t.
cJ( f )(k)"2~J@2 f (k/2J), k3Z.
(22)
For j(J, the decomposition process is illustrated in Fig. 1 and the process "ts into the forward transform of the lifting scheme except for an ad-
295
Fig. 1. The block diagram for decomposition.
ditional post-position "lter Q on the high-pass channel to decorrelate the detail coe$cients. In the diagram, hH(m)"h(!m), uH(m)" u015(!m), Q is an all-pole "lter with the transfer function (1!+L q(l)z~l)~1, cj~1( f ) and dj~1( f ) ( l/1 denote the approximation coe$cients and the detail coe$cients of at j!1 level, respectively. Note that Q is a stable and causal IIR "lter when DDqDD (1. Therefore, the "lter Q performs the trans1 form from dj~1( f ) to dj~1( f ), a causal and stable ( t recursive process, L dj~1( f )(k)"dj~1( f )(k)# + q(l)dj~1( f )(k!l). ( t ( l/1 (23) For applications where the signals are of "nite length or boundary processing is taken, a simpli"ed implementation of (23) is as follows. Assume d ( f )(k)"0 when k(0; then j~1 dj~1( f )(0)"dj~1( f )(0), t ( k dj~1( f )(k)"dj~1( f )(k)# + q(l)dj~1( f )(k!1), ( t ( l/1 k"1,2,2, ¸!1, (24) L dj~1( f )(k)"dj~1( f )(k)# + q(l)dj~1( f )(k!1), ( t ( l/1 k"¸, ¸#1,2, N. Due to the causality of Q, the hypothesis dj~1( f )(k)"0, k(0, in (24) is reasonable. Even ( though the boundary processing results in the error of dj~1( f ) near the left boundary, the error at ( dj~1( f )(k) quickly decays as k increases. The small( er DDqDD (1, the faster does the error decay as 1 k increases.
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Though many numerical examples demonstrate that the optimal wavelet functions satisfy S/(x),/(x!l)T+0,
0(DlD)¸,
(26)
the conditions under which the solution of (21) is identical to that of Fig. 2. The block diagram for synthesis.
Therefore, a nearly orthogonal representation of f is obtained, which can be written as J~1 (25) f (x)" + + dj ( f )(k)2j@2/(2jx!k). ( j/~= k The reconstruction can be realized as described in Fig. 2 and the process "ts into the inverse transform of the lifting scheme except for an additional preposition "lter P on the high-pass channel which is the inverse "lter of the "lter Q with the transfer function 1!+L q(l)z~1. l/1 While the above descriptions are devoted to the implementation of analysis and synthesis of the designed wavelet system, the next paragraphs discuss the solving of the optimization problem (21) using the gradient method:
l"1,2,2, ¸,
(27)
remain undetermined and is left for future research. In non-orthogonal wavelet transform domain, the correlation between the detail coe$cients is simultaneously determined by the correlation of the signal and wavelet bases, which complicates the signal analysis. In many signal models, the signal is of short-range dependence [11]; in other words, the correlation sequence r of the signal decays quickly k as DkD increases. In this case, due to (27), the correlation between the wavelet coe$cients may e$ciently re#ect the property of the signal, which is an advantage of the local orthogonalization. By now a family of nearly orthogonal interpolating wavelet systems can be constructed using our optimal designing scheme.
5. Examples
(i) Initialization: q (l)"0, l"1,2,2, ¸, i.e., / (x)"t(x). 0 0 (ii) Calculate the gradient vector : Assume that q and / (x)"t(x)! k k +L q (l)t(x!1) can be obtained; then the gradil/1 k ent vector can be calculated by = grad(l)"!2 + c(k)(g(k#l)#g(k!l)), k/1 1,2,2, ¸, where c(m)"S/ (x),/ (x!m)T, k k t(x!m)T.
S/(x),/(x!l)T"0,
g(m)"S/ (x), k
(iii) Modify q : k q (l)"q (l)!jgrad(l), l"1,2,2, ¸, k`1 k j is a step. Sometimes, a changeable step may also be used. (iv) If DDq !q DD (e, then q015"q ; if not, go k`1 k 1 k`1 to (ii), where e is a threshold to end the iterative process.
Given an interpolating scaling function, "rstly, solve the quadric programming problem (19) and obtain the optimal parameter vector a015 and the optimal "lter u015; secondly, locally orthogonalize the obtained wavelet function and obtain the new wavelet function. Below Deslauriers}Dubuc fundamental functions are taken as examples. 5.1. The nearly orthogonal interpolating wavelets with two vanishing moments The Deslauriers}Dubuc fundamental function with two vanishing moments is the Hat function satisfying the two-scale relation u(x)"u(2x)#1/2(u(2x#1)#u(2x!1)). When the "lter u"[1/2,1/2], this is the famous Franklin wavelet, the corresponding wavelet function is denoted by t , and the correlation coe$0 cients between the wavelet function and the integer shifts of the scaling function are illustrated in
P.-L. Shui, Z. Bao / Signal Processing 79 (1999) 289}300
Fig. 3(a). When the support set of u is M!1, 0, 1, 2N, the optimal parameter vector and "lter are a015"[7,12,7]/26 and u015"[!7,33,33,!7]/52, respectively. The correlation coe$cients between the optimal wavelet function t and the integer shifts of the scaling function are illustrated in Fig. 3(b), and the co-redundancy degree is only about 1/20th of the original co-redundancy degree. The correlation coe$cients of the optimal wavelet function t are shown in Fig. 4(a). It is seen that the larger coe$cients concentrate on the set M!2,!1,0,1,2N. Furthermore, choose ¸"2 and locally orthogonalize t. The optimal vector q"[!0.34069109,!0.02501170]. The correlation coe$cients are shown in Fig. 4(b) and S/(x),/(x!1)T and S/(x),/(x!2)T are smaller than 10~16. This observation naturally leads to the hypothesis that the optimal q is the solution of S/(x),/(x!1)T"S/(x),/(x!2)T"0.
297
The comparison on redundancy degrees and Riesz bounds of wavelets are given in Table 1, where CRD, WRD, B and A denote the co-redundancy degree, redundancy degree of wavelet function, upper bound and lower bound in (11), respectively. 5.2. The nearly orthogonal interpolating wavelets with four vanishing moments Of all Deslauriers}Dubuc fundamental functions, the function with four vanishing moments has the best time}frequency localization (the smallest the product of time width and frequency width) and therefore is widely used. It satis"es the two-scale relation u(x)"u(2x)#9/16(u(2x#1)#u(2x!1)) !1/16(u(2x!3)#u(2x#3)).
Fig. 3. The comparison of correlation coe$cients.
Fig. 4. The comparison of the correlation structure with the wavelet functions.
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Table 1 The comparison about the redundancy indexes
Table 2 The comparison of the redundancy indexes
Wavelet type
CRD
WRD
B/A
Wavelet type
CRD
WRD
B/A
Mu, t N 0 Mu, tN Mu, /N
0.0556 0.0034 0.0035
0.2284 0.2380 1.4208]10~5
1.7777/0.1510 1.8979/0.4532 1.0067/0.9765
Mu, t N 0 Mu, tN Mu, /N
0.0242 0.0020 0.0018
0.2578 0.2545 2.586]10~6
1.9718/0.3934 2.0280/0.4994 1.0035/0.9875
When the interpolating "lter u"h(~1)" 3 [!1,9,9,!1]/16, formula (2) is a lifting Donoho wavelet and the corresponding wavelet function is denoted by t . When the designed support set is 0 M!2,!1,0,1,2,3N, the optimal results are a015"[0.54372520012555, !0.08745040025111,
Table 3 The comparison of the redundancy indexes Wavelet type
CRD
WRD
B/A
Mu, tN Mu, /N
2.1255]10~4 1.9665]10~4
0.1258 2.8766]10~8
1.7626/0.4392 1.0003/0.9987
0.54372520012555], u015"a015(1)h(~2)#a015(2)h(~1)#a015(3)h(0). 3 3 3 According to the correlation structure of the obtained wavelet function t, choose ¸"4, the obtained optimal result is q"[!0.329486, 0.007220, !0.002447, !0.000206]. The redundancy indexes of three wavelet systems are shown in Table 2. When the designed support set is M!3,!2, !1,0,1,2,3,4N, the optimal parameter vector is a015"[0.04389920, 0.11429242, 0.68361676, 0.11429242, 0.04389920], u015"a015(1)h(~3)#a015(2)h(~2)#a015(3)h(~1) 3 3 3 (28) #a015(4)h(0)#a015(5)h(1). 3 3 The corresponding wavelet function is t. Choose the parameter ¸"4, locally orthogonalize t, and the optimal result q"[!0.32872240, 0.00771976, !0.00219163, !0.00024376].
(29)
The comparison of redundancy indexes is given in Table 3. The results show that our designing scheme is e$cient. The nearly orthogonal interpolating wavelets designed by our scheme preserves the
regularity and the number of vanishing moments of the original system. And the fact that the Q is a stable, causal and all-pole "lter and the P is a short FIR promises the fast algorithm of the designed system. Except for Haar wavelet, the "lters in Daubechies wavelets are of in"nite digits. In applications, the truncation of the "lters will destroy the orthogonality. However, coe$cients of all D#1 order Deslauriers}Dubuc "lters are k/2D`1 type dyadic rational numbers and these enable us to perform the digital "ltering without the truncation error of "lters, which is one of the advantages of Donoho wavelets and lifting Donoho wavelets. In order to preserve this advantage, one may use the suboptimal parameter vectors a46"015 and q46"015 with "nite digits by, for example, quantifying the optimal vectors in (28) and (29) using the following formulas: a46"015"2~N roundM2Na015N, q46"015"2~(N`4) roundM2N`4qN, round(x) denotes the integer nearest x and N is a given positive integer. Sometime, the proper modi"cation of a46"015 is needed to promise sum(a46"015)"1. The adjustment of N realizes the trade-o! between the redundancy and the digits of "lters. The result is illustrated in Table 4.
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299
Table 4 The redundancy degrees versus the digits of "lters N
4
5
6
7
RY(u, /) RY(/) B/A
0.0030 9.14]10~4 1.0081/0.9713
9.113]10~4 8.93]10~4 1.0049/0.9910
2.630]10~4 1.73]10~4 1.0012/0.9962
2.997]10~4 1.806]10~4 1.0006/0.9967
Acknowledgements We would like to thank the anonymous reviewers for their suggestions on improving this paper. The authors would like to thank Dr. Xuejun Liao for his participation in the revision of the paper in English, too. Fig. 5. The comparison of sual scaling functions.
Appendix A. Proof of Theorem 1 6. Conclusions The con#ict between interpolation and orthogonality in two-band wavelets gives rise to the appearance of M-band wavelets and multiwavelets [8,9,18], and the relaxation of orthogonality produces semi-orthogonal wavelet and biorthogonal wavelets [3,2]. However, in practical applications, one pays more attention to the global performance rather than some single index of performance. In this paper, starting from the lifting Donoho interpolating wavelets which are #exible in designing, we design a novel family of nearly orthogonal interpolating wavelets. The designed wavelets preserve the main advantages of Donoho wavelets and yet markedly degrade the inter-level redundancy and the intra-level redundancy. Additionally, the low regularity of the duals is one of the drawbacks of the existing lifting Donoho wavelets. As the byproduct of our scheme, the optimal design of coredundancy degree improves the regularity of the duals. For example, the dual scaling functions of lifting Donoho wavelet with h"u"[!1/16, 9/16,9/16,!1/16] and those in (28) are illustrated in Fig. 5(a) and (b), respectively.
= ∀Md N3l2(Z2) and f (x)" + + d t (x) j,n j,n j,n j/~= n Let J~1 f (x)" + + d t (x), g (x)"+ d t (x). J j,n j,n J J,n J,n j/1 n n Then f (x)"f (x)#g (x) where, g (x)3= (t), J`1 J J J J spanMt (x), n3ZN, J,n f (x)3< (u) J J ,closeMu (x)"2J@2u(2Jx!n), n3ZN. J,n By Theorem 3.24 in [3], we have A + Dd D2) t n J,n DDg DD2 )B + Dd D2. t n J,n J 2 Let h (x) denote the part of f (x) in < W= , J J`1 J J then DD f DD2 "DD f DD2 #DDg DD2 !DDh DD2 . J`1 2 J 2 J 2 J 2
(A.1)
Obviously c ,Sh (x),u (x)T"Sg (x),u (x)T" J,n J J,n J J,n + c(n!l)d , l J,l c(l)"St(x),u(x!l)T. Using D (u), K(u) denote the Fourier series of J the sequences Md N and Mc(l)N, respectively. By J,n
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Parseval equation and the de"nitions in (11), A + Dd D2)DDc DD22 r,t J,n J,n l n 1 p DD (u)D2DK(u)D2 du)B + Dd D2. " J r,t J,n 2p ~p n
P
From Theorems 3.20 and 3.24 in [3], it is easy to obtain A DDh (x)DD2 )+ Dc D2)B DDh (x)DD2 . 2 2 J,n r J r J n Therefore, A B r,t + Dd D2)+ Dc D2) r,t + Dd D2. J,n J,n J,n B A r n r n n From (A.1) and (A.2), we have
A
B
(A.2)
B A ! r,t + Dd D2)DD f (x)DD2 !DD f (x)DD2 2 2 J t J,n J`1 A r n A ) B ! r,t + Dd D2, t J,n B r n = B + + Dd D2)DD f DD2 A ! r,t 2 j,n t A r j/~= n = " + (DD f DD2 !DD f DD2 ) j 2 J`1 2 j/~= = A ) B ! r,t + + Dd D2. j,n t B r j/~= n This completes the proof. h
A
A A
B
B B
References [1] P. Abry, P. Flandrin, On the initialization of the discrete wavelet transform algorithms, IEEE Signal Process. Lett. 1 (2) (1994) 32}34. [2] A. Cohen, I. Daubechies, J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Commun. Pure Appl. Math. XLV (1992) 485}560. [3] C.K. Chui, An Introduction to Wavelets, Academic Press, New York, 1992, Chapter 3, pp. 85}99. [4] I. Daubechies, Orthonormal bases of compactly supported wavelets II: Variations on a theme, SIAM J. Math. Anal. 24 (2) (1993) 499}519.
[5] G. Deslauriers, S. Dubuc, Symmetric iterative interpolation processes, Constr. Approx. 5 (1) (1989) 49}68. [6] D.L. Donoho, Interpolating wavelet transform, Technical Report, Dept. of Statistics Stanford Univ, 1992. [7] S. Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl. 144 (1986) 185}204. [8] T.N.T. Goodman, L.S. Lee, Wavelets of multiplicity r, Trans. Amer. Math. Soc. 342 (1994) 307}324. [9] T.N.T. Goodman, L.S. Lee, S.W. Tang, Wavelets in wandering subspace, Trans. Amer. Math. Soc. 338 (1993) 639}654. [10] A. Harten, Multiresolution representation of data: a general framework, SIAM J. Numer. Anal. 33 (3) (1996) 1205}1256. [11] I.M. Johnstone, B.W. Silverman, Wavelet threshold estimators for data with correlated noise, Technical Report, Dept. of Statistics, Stanford Univ., 1996. [12] M. Lounsbery, T. Derose, J. Warren, Multiresolution analysis for surfaces of arbitrary topological type, ACM Trans. Graphics 16 (1) (1997) 34}73. [13] P.-L. Shui, B. Zheng, Signal-adapted biorthogonal interpolating recursive wavelets, Electron. Lett. 34 (20) (1998) 1920}1921. [14] N. Saito, G. Beylkin, Multiscale representation using the auto-correlation functions of compactly supported wavelets, IEEE Trans. Signal Process. 41 (12) (1993) 3584}3590. [15] M.J. Shensa, The discrete wavelet transform: wedding a trous and Mallat algorithms, IEEE Trans. Signal Process. 40 (10) (1992) 2464}2483. [16] D.R. Sherman, Z.-W. Shen, Multidimensional interpolatory subdivision schemes, SIAM J. Numer. Anal. 34 (6) (1997) 2357}2381. [17] P.-L. Shui, Z. Bao, Biorthogonal interpolating wavelet with compactly supported duals, Electon. Lett. 34 (12) (1998) 1185}1186. [18] P. Ste!en, P.N. Heller, R.A. Gopinath, C.S. Burrus, Theory of regular M-band wavelet bases, IEEE Trans. Signal Process. 41 (12) (1993) 3497}3511. [19] W. Sweldens, The lifting scheme: a custom-design construction of biorthogonal wavelets, Appl. Comput. Harmon. Anal. 3 (2) (1996) 196}200. [20] W. Sweldens, The lifting scheme: a construction of second generation wavelets, SIAM J. Math. Anal. 29 (2) (1997) 511}546. [21] M. Unser, P. Thevenaz, A. Aldroubi, Shift-orthogonal wavelet bases, IEEE Trans. Signal Process. 46 (7) (1998) 1827}1836. [22] X.-G. Xia, J. Kuo, Z. Zhang, Wavelet coe$cients computation with optimal pre"ltering, IEEE Trans. Signal Process. 42 (8) (1994) 2191}2197. [23] X.G. Xia, Z. Zhen Zhang, On sampling theorem, wavelets, and wavelet transform, IEEE Trans. Signal Process. 41 (12) (1993) 3524}3535.
Construction of nearly orthogonal interpolating wavelets Peng-Lang Shui*, Zheng Bao The Key Laboratory for Radar Signal Processing, Xidian University, Xi'an, 710071, People's Republic of China Received 29 October 1998; received in revised form 1 July 1999
Abstract This paper "rst studies the measure of redundancy in biorthogonal wavelet systems and gives the explicit parameter representations of the general interpolating "lters for lifting Donoho wavelets. Based on the measure and the parameter representations, a novel method is proposed to optimize co-redundancy degree which is then used to construct a family of nearly semi-orthogonal interpolating wavelets. Furthermore, by performing the local orthogonalization on the obtained wavelet functions we design a family of nearly orthogonal interpolating wavelets. Finally, some examples are given which show that our scheme is e$cient in designing practical wavelet systems with good performance indexes. ( 1999 Elsevier Science B.V. All rights reserved. Zusammenfassung In dm` esem Artikel wird zunaK chst das Redundanzma{ in bi-orthogonalen Wavelet-Systemen untersucht und eine explizite Parameterdarstellung des allgemeinen Interpolations"lters fuK r anhebende Donoho Wavelets angegeben. Ausgehend von den AusdruK cken fuK r das Ma{ und die Parameter wird ein neuartiges Verfahren vorgeschlagen, um den Grad der Co-Redundanz zu optimieren, der anschlie{end benutzt wird, um eine Familie von nahezu semi-orthogonalen Interpolationswavelets zu konstruieren. Durch lokale Orthogonalisierung der erhaltenen Wavelet-Funktionen entwerfen wir daruK ber hinaus eine Familie nahezu orthogonaler Interpolationswavelets. Schlie{lich werden einige Beispiele angegeben, die zeigen da{ sich unser Verfahren fuK r den Entwurf praktischer Wavelet Systeme mit gutem Leistungsgrad eignet. ( 1999 Elsevier Science B.V. All rights reserved. Re2 sume2 Nous eH tudions tout d'abord dans cet article la mesure de la redondance dans les syste`mes d'ondelettes bi-orthogonales et donnons les repreH sentations explicites des parame`tres des "ltres interpolants geH neH raux pour les ondelettes de Donoho. Sur la base de cette mesure et des repreH sentations des parame`tres, nous proposons une meH thode nouvelle pour optimiser le degreH de co-redondance, meH thode utiliseH e ensuite pour construire une famille d'ondelettes interpolantes presque semi-orthogonales. De plus, par orthogonalisation locale des ondelettes obtenues nous concevons une famille d'ondelettes interpolantes presqu'orthogonales. En"n, nous donnons des exemples qui montrent que notre meH thode est e$cace pour la conception de syste`mes d'ondelettes pratiques, preH sentant de bons indices de performances. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Interpolating "lter; Interpolating wavelets; Redundancy degree; Local orthogonalization; Nearly orthogonal
* Corresponding author. Tel.: #86-29-8221025; fax: #86-29-8236159. E-mail address: [email protected] (P.-L. Shui) 0165-1684/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 9 9 ) 0 0 1 0 2 - 4
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Nomenclature Z uD RY(u) RY(u,t) RY(t) h(m) D h, u a a015
integer set Deslauriers}Dubuc fundamental function redundancy degree of the scaling function u co-redundancy degree redundancy degree of the wavelet t Deslauriers}Dubuc "lter general interpolating "lters parameter vector optimal parameter vector
S f, gT fK q Q P Vj Wj cj( f ) dj( ( f )
inner product of f and g in ¸2(R) Fourier transform f optimal vector in local orthogonalization an all pole "lter inverse "lter of Q jth level approximation subspace jth level detail subspace jth approximation coe$cients of f jth detail coe$cients of f
Abbreviation Min minimum or minimizing
1. Introduction
holds:
Over the last 10 years, two-band interpolating wavelets have been widely studied and applied [4}7,10,14,16,17,19,20,23]. As an important property, the interpolation can remarkably simplify the initialization of the wavelet series transform (WST). Many applications often need to use the WSTs, which are performed by two steps. First, for a given multiresolution analysis (or scaling function u) and a single f (x)3L2(R), one determines a proper scaling factor J and the approximation coe$cients of f (x) in the multiresolution subspace V (u), which is j called the initialization. Second, one computes wavelet (or detail) coe$cients by the pyramid algorithm as one does in the case of discrete wavelet transform (DWT). Generally, the approximation coe$cients of a signal is not consistent with its uniform samples even if f (x)3V (u). But the initiaJ lization by the numerical integral is computationally costly, and the direct initialization by using of the uniform samples often has poor approximation power. Hence, the diverse pre-"ltering algorithms for the initialization [1,15,22] have been recently developed to improve the approximation power. But another problem exists with pre-"ltering, i.e., it is impossible to design perfect reconstruction FIR pre/post-"lters unless they are both all-pass. However, for cardinal interpolating scaling functions, the following standard sampling theorem [23]
n f (t)"+ f u(2J x!n), ∀f (x)3V (u), (1) J 2J n which is the generalization of Shannon sampling theorem. Consequently by using the interpolating wavelets the complicated pre-/post-"ltering process can be avoided, and thus much convenience results. Deslauriers, Dubuc, Donoho, Sweldens and Harten [5}7,19,20] have established the fundamental frame of interpolating wavelets. In this paper, we are only interested in the lifting Donoho wavelets, a family of biorthogonal interpolating wavelets. Using Deslauriers}Dubuc fundamental functions u (x) [5,7] as scaling functions and D t (x)"u (2x!1) as wavelets, Donoho construcD D ted a family of interpolating wavelets [6], later referred to as Donoho wavelets, which is closely connected to the Lazy wavelets by the dual lifting scheme [20]. However, such wavelets have the following disadvantages: wavelets do not have any vanishing moments and thus do not form a Riesz basis for L2(R); the duals do not even belong to L2(R), and the dual scaling "lters are full-pass without frequency localization. Later, by lifting scheme, Sweldens improved Donoho wavelets, thus the above disadvantages are avoided. Under the basic framework of lifting scheme [19,20], two-band compactly supported interpolating wavelets with compactly supported duals can be generated by
AB
P.-L. Shui, Z. Bao / Signal Processing 79 (1999) 289}300
applying the lifting scheme to Donoho wavelets, hereafter referred to as lifting Donoho wavelets. As is well known, there does not exist any twoband compactly supported orthogonal interpolating wavelets except for the Haar wavelet [23]. The lifting Donoho wavelets can have many advantages simultaneously, such as compact support, linear phase, high regularity, and the "nite digits of "lters [6,14,19]. However, one serious defect is comparatively much redundancy. Designing the compactly supported interpolating wavelets with less redundancy, namely, nearly orthogonal interpolating wavelets, is the main problem that will be studied in this paper. Section 2 reviews the concept and properties of the lifting Donoho wavelets and the relation with the lifting scheme, introduces the redundancy degrees to measure the redundancy of a biorthogonal wavelet system, studies the estimation of Riesz bounds and gives the parameter representations of general interpolating "lters. On the basis of these, enlightened the partial orthogonalization proposed by Lounsbery et al. [12], Section 3 constructs a family of nearly semi-orthogonal interpolating wavelets by optimizing of co-redundancy degree. In Section 4, a novel method called local orthogonalization is proposed to decorrelate the wavelet basis at the same level and by which we construct the nearly orthogonal interpolating wavelets. Finally, several examples are presented, which show that our method is highly e$cient. 2. Lifting Donoho wavelets and redundancy degrees
291
ing the Dirac impulse. Such wavelets have several disadvantages as mentioned in the above section. In 1995, Sweldens presented the lifting scheme [20,23], a novel way to construct biorthogonal wavelets. Under the basic framework of lifting scheme, Donoho wavelets can be obtained by using the dual lifting scheme for the Lazy wavelets. In order to overcome these disadvantages of Donoho wavelets, Sweldens [20] uses the lifting scheme for Donoho wavelets, and obtains a novel family of interpolating wavelets, which we term `lifting Donoho waveletsa. A lifting Donoho wavelet system can be described as follows: u(x)"u(2x)#+ h(k)u(2x!2k#1), k
(3)
t(x)"2u(2x!1)!+ u(k)u(x!k), k with the duals u8 (x)"2u8 (2x)#+ u(!k)tI (x!k), (4) k tI (x)"u8 (2x!1)!+ h(!k)u8 (2x!2k!2), k where h and u are a pair of real FIR general interpolating "lters. Obviously, when h"u"0, Eqs. (3) and (4) describe a Lazy wavelet; when u"0 these describe a Donoho wavelet. On the vanishing moments of lifting Donoho wavelets, there is the following conclusion. Proposition 1. u, u8 , t and tI in Eqs. (3) and (4) have D#1 vanishing moments if and only if the general interpolating xlters h and u satisfy [19, Theorem 12]
2.1. Lifting Donoho wavelet
+ kdh(k)"+ kdu(k)"(1/2)d, d"0,1,2, D. k k
In 1992, Donoho [6] suggested the idea of wavelets built from interpolating scaling functions, and constructed a family of interpolating wavelets, later referred to as Donoho wavelets, which can be described as follows:
A scaling function is called the D#1 vanishing moments (or D#1 order regular) if its satis"es :xdu(x) dx"d(d), d"0,1,2, D, which is closely connected with the approximation order of a scaling function and is usually referred to as the Coi#et property.
u(x)"u(2x)#+ h(k)u(2x!2k!1), k t(x)"2u(2x!1),
(2)
with the duals u8 (x)"d(x) and tI (x)" u8 (2x!1)!+ h(!k)u8 (2x!2k!2), d(x) denotk
(5)
2.2. Redundancy degrees The frame bounds and the redundancy ratio [3] are often used to measure the redundancy of
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a frame. Although they may be applied to biorthogonal wavelet system, their complicated estimations prevent them from being applied to optimal design of wavelets. As a result, we introduce the following three indexes to measure the redundancy of a biorthogonal wavelet system. Let u and t be the synthesis scaling function and wavelet; then the redundancy degree of the scaling function is de"ned as 1 RY(u), + Su(x),u(x!k)T2. (6) DDuDD4 2 kE0 The co-redundancy degree between the approximation subspace and the detail subspace at the same level is de"ned as 1 RY(u,t), + St(x),u(x!k)T2. (7) DDuDD2 DDtDD2 2 k 2 Lounsberg et al. [12] use the objective function analogous to (7) to construct near semi-orthogonal wavelets. The redundancy degree of the wavelet function is de"ned as 1 RY(t), + St(x),t(x!k)T2. (8) DDtDD4 2 kE0 Obviously, RY(u,t)"0 corresponds to semi-orthogonal wavelets, RY(u)"RY(t)"0 to shift-orthogonal wavelets [21] and RY(u,t)"RY(t)"0 to orthogonal wavelets. All existing orthogonal wavelets satisfy RY(u)"0 also, because their construction starts from orthogonal scaling functions. Whether or not there exist orthogonal wavelets with RY(u)O0 is an interesting problem, from which a novel method may be developed to construct orthogonal wavelets. Below we discuss the Riesz bounds of wavelets. Let u(x), t(x) be the compactly supported synthesis scaling and wavelet functions. We de"ne C (u)"+ Su(x),u(x!k)Te~+ku, r k C (u)"+ St(x),t(x!k)Te~+ku, t k C (u)"+ Su(x),t(x!k)Te~+ku, r,t k
(9)
Due to the compact support, C (u), C (u), C (u) r t r,t are trigonometric polynomials. Consider A "min MC (u)N, B "max MC (u)N, r r r r u u A "min MC (u)N, B "max MC (u)N, t t t t u u
(10)
A "min MDC (u)D2N, B "max MDC (u)D2N. r,t r,t r,t r,t u u Then, for the Riesz bounds of the wavelet the following theorem holds. Theorem 1. Let u(x), t(x) be the compactly supported synthesis scaling function and wavelet function in a biorthogonal wavelet system; then ∀Md N3l2(Z2), j,k 2 ADDtDD2 DDMd NDD22 ) + d t (x) 2 j,k l j,k j,k 2 j,k|Z )BDDtDD2 DDMd NDD22 , (11) 2 j,k l 1 B 1 A A" A ! r,t , B" B ! r,t , t DDtDD2 t A DDtDD2 B 2 2 r r where t (x)"2j@2t(2jx!k). When A A 'B , j,k r t r,t Mt , j, k3ZN constitutes a Riesz basis in ¸2(R). j,k
K
A
K
B
A
B
This theorem gives an e$cient estimator of Riesz bounds, and moreover, the ratio between A and B can be used to measure the closeness of a wavelet system to orthogonal one. The proof is given in Appendix A. 2.3. Parameter representation of general interpolating xlter The subdivision scheme has been successfully applied to the construction of wavelets on real line, two-dimensional nontenor-product wavelets [16], as well as wavelets on manifolds [12]. In one-dimensional subdivision scheme, Donoho wavelets, and lifting Donoho wavelets, Deslauriers}Dubuc "lters play an important role and their length are the shortest among D#1 order interpolating "lters. An interpolating "lter is of D#1 order if the interpolator is accurate for all polynomials whose degree is less than D#1 [10]. A D#1 order
P.-L. Shui, Z. Bao / Signal Processing 79 (1999) 289}300
Deslauriers}Dubuc "lter can be formulated as m`D 2l!1 , h(m)(k)" < D 2l!2k l/m,lEk k"m, m#1,2, m#D.
(12)
Di!erent values of m correspond to di!erent "lters; such #exibility enables us to e$ciently perform the boundary processing [10]. More importantly, using them as bases, the general interpolating "lters can be parameterized. Proposition 2. Let h be a D#1 order interpolating xlter with support set MM, M#1,2, M#NN (N*D); then there exists a unique parameter vector a satisfying + a(m)"1 such that [13] m M`N~D (13) h" + a(m)h(m). D m~M The complexity of optimizing a wavelet system usually depends on the structure of the "lters. In the lifting scheme, the choice of the lifting "lter does not in#uence biorthogonolity; thus, the designing is easier and more #exible. The parameterization in (13) enables us to freely choose the parameter vector a in the designing of lifting Donoho wavelets but does not in#uence the biorthogonality and the number of vanishing moments. Given an interpolating function, the optimized wavelet is then derived as follows: "rst the co-redundancy degree RY(u,t) is minimized to derive the "lter u and thence a nearly semi-orthogonal interpolating wavelet; then design a nearly orthogonal interpolating wavelet by the local orthogonalization of the obtained wavelet function.
3. Optimal design of co-redundancy degree For the two-band orthogonal wavelets on real line, the compact support is incompatible with linear phase of "lters; therefore, one must either abandon the compact support or relax orthogonality. A successful case is the semi-orthogonal spline wavelets by Chui [3], whose duals are exponent decay rather than compact support. Another case is the biorthogonal wavelets by Cohen et al. [2], and
293
wavelets as well as their duals are compact support and linear phase. Moreover, in biorthogonal wavelets systems, to reduce the inter-level redundancy becomes a problem concerned by researchers. In multiresolution analysis for surfaces [12], Lounsbery proposed the idea of `partial orthogonalizationa to design near semi-orthogonal wavelets, the aim of which is that the approximation subspace and detail subspace on the same level are as orthogonal as possible, and this method can be seen as a particular instance of the lifting scheme. This method can be applied to a Donoho wavelet to design a nearly semi-orthogonal lifting Donoho wavelet. Below we give a brief description of this method. Let u(x) be the Donoho scaling function given in (2), X denotes any subset of integer set Z; typically X consists of the integer k such that u(x!k) is supported in some neighbourhood of u(2x!1). Take the wavelet t(x) to be [12, Eq. (17)] t(x)"2u(2x!1)! + u(k)u(x!k). (14) k|X In order to minimize the ¸2 norm of the projection of t(x) into V , the lifting "lter u are determined by 0 solving the linear system [12, Eq. (18)] + Su(x!k), u(x!l)Tu(k) l|X "S2u(2x!1), u(x!k)T, k3X.
(15)
When X"Z, a semi-orthogonal wavelet is obtained but is not compactly supported; when ) is "nite, the solution of the linear system (15) only guarantees that for arbitrary k3X, t(x) is orthogonal to u(x!k) but does not guarantee that the ¸2 norm of it projection into V is minimal. 0 Similar to the method in [12], in this section, we design nearly semi-orthogonal lifting Donoho wavelets by minimizing co-redundancy degree. In the designing, the interpolating scaling function is given and the length of the "lter u is "xed, and thus the regularity of wavelet function is invariant, but the regularity of the duals is changeable according to Eq. (3). Instead of the co-redundancy degree, the following objective function is employed: f (u)"+ St(x), u(x!k)T2. 0"+%#5 k
(16)
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This has slight di!erence with expression (15), where the square sum of the inner product between t(x) and u(x!k), k3X, rather than k3Z is taken as the objective function. For the sake of simplicity, we introduce the following notations: a(k),Su(x), u(x!k)T, b(k),2S(2x!1), u(x!k)T, A(k, l)"+ a(k!m)a(l!m), m B(k),+ a(k!m)b(m), m C"+ b2(k). k Therefore the objective function is rewritten as f (u)"uTAu!2uTB#C. (17) 0"+%#5 When the scaling function in (3) had D#1 regular moments, generally, the wavelet is required to have D#1 vanishing moments, i.e., the "lter u is D#1 order. This constraint condition and Eq. (17) lead to the following quadratic programming: MuTAu!2uTB#CN (18) 1 d + kdu(k)" , d"0.1,2,2, D. k 2 The parameter representation of Eq. (13) can greatly simplify the optimal algorithm. Let the support set of the "lter u be MM, M#1,2, M#NN; from Proposition 2, the programming (15) can be simpli"ed as the one with a linear constraint on the parameter vector a: min g s.t.
min a s.t.
AB
MaTHTAHa!2aTHTB#CN (19)
+a(k)"1, k where H is an (N#1)](N#1!D) matrix formed by Deslauriers}Dubuc "lters: H(m:m#D, m)"h(m), D m"M, M#1,2, M#N!D and the remaining entries are zeros. By solving the above quadratic programming, the optimal parameter vector a015 and the optimal "lter u015"Ha015 are obtained.
Sometimes, in order to retain other advantages, some proper constraints may be imposed on the parameter vector a. For example, when N and D are odd numbers, the symmetric a assures the linear phase; the "nite digits of a assures that the optimal "lter is of "nite length. These are advantageous for applications, especially for digital "ltering. Finally, we give a brief description on the di!erence between the optimal method in (15) and our method. First, the optimal method in (15) cannot be directly applied to the case with vanishing moments constraints. Imposing such constraints on the "lter u results in a super-determining linear system, which needs to search for the least-squares solution. Second, due to the slight di!erence of the objective functions, the optimal wavelet system obtained by our method is closer to semi-orthogonal one. For example, the scaling function u(x) is the hat function satisfying two scaling relation u(x)"u(2x)#1/2(u(2x!1)#u(2x#1)); consider the X"M0,1N, i.e., the wavelet function has the form of t(x)"2u(2x!1)!u(0)u(x)! u(1)u(x!1). Following the method in (15), the optimal "lter u "[3/5,3/5], Su(x#1), t (x)T"Su(x!2), 1 1 t (x)T"1/10, the correlation coe$cients else are 1 zero, and the co-redundancy degree is 0.0273. Using our method, the optimal "lter u " 2 [15/26,15/26] and Su(x),t (x)T"Su(x!1), 2 t (x)T"1/52, Su(x#1),t (x)T"Su(x!2),t (x)T 2 2 2 "!5/52, the correlation coe$cients else are zero, and the co-redundancy degree is 0.0262.
4. Decorrelation of wavelet bases } local orthogonalization The minimization of co-redundancy degree reduces the redundancy between the approximate subspace and the detail subspace at the same level. Thereby, there is hardly redundancy between the wavelet coe$cients at di!erent levels. However, because Mt(x!k), k3ZN does not constitute an orthogonal basis, the redundancy between the wavelet coe$cients of the same level becomes the main ingredient to degrade the performance of the system.
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Gram}Schmidt orthogonalization and the normalized orthogonalization [3] are two commonly used methods to decorrelate. Unfortunately, the former does not perserve the good structure of the original basis, that is the new basis cannot be generated from integer shifts of a single function, which often complicates the processing in wavelet domain. The latter is often used to construct an orthogonal wavelet from semi-orthogonal one. Assume t as a compactly supported semi-orthogonal wavelet function with Riesz lower bound A'0, set tK (u) tK M(u)" ; J+DtK (u#2kn)D2 then MtM(x!k), k3ZN is an orthogonal basis in W but not compactly supported. Moreover, for 0 many applications, the nearly orthogonal basis is slightly di!erent from the orthonormal basis; thus the orthogonality can be traded for other good properties of the system. A new method, local orthogonalization, is proposed to minimize rather than to completely remove the redundancy of the wavelet basis Mt(x!k), k3ZN. It can be described as follows. Let t(x) be a wavelet function obtained from optimization (19); generally, the comparatively large coe$cients in r (k)"St(x),t(x!k)T cont centrate near k"0. Choose a proper length parameter ¸ and construct a new wavelet function L /(x),t(x)! + q(l)t(x!l), (20) l/0 such that the redundancy degree of the new wavelet / is minimal, i.e., min q
G
+L S/(x),/(x!l)T2 l/1
H
(21)
+L Dq(l)D)g(1. l/1 In this case, the decomposition and synthesis algorithms are as follows. Assume f (x)3< , the initial j approximation coe$cients are s.t.
cJ( f )(k)"2~J@2 f (k/2J), k3Z.
(22)
For j(J, the decomposition process is illustrated in Fig. 1 and the process "ts into the forward transform of the lifting scheme except for an ad-
295
Fig. 1. The block diagram for decomposition.
ditional post-position "lter Q on the high-pass channel to decorrelate the detail coe$cients. In the diagram, hH(m)"h(!m), uH(m)" u015(!m), Q is an all-pole "lter with the transfer function (1!+L q(l)z~l)~1, cj~1( f ) and dj~1( f ) ( l/1 denote the approximation coe$cients and the detail coe$cients of at j!1 level, respectively. Note that Q is a stable and causal IIR "lter when DDqDD (1. Therefore, the "lter Q performs the trans1 form from dj~1( f ) to dj~1( f ), a causal and stable ( t recursive process, L dj~1( f )(k)"dj~1( f )(k)# + q(l)dj~1( f )(k!l). ( t ( l/1 (23) For applications where the signals are of "nite length or boundary processing is taken, a simpli"ed implementation of (23) is as follows. Assume d ( f )(k)"0 when k(0; then j~1 dj~1( f )(0)"dj~1( f )(0), t ( k dj~1( f )(k)"dj~1( f )(k)# + q(l)dj~1( f )(k!1), ( t ( l/1 k"1,2,2, ¸!1, (24) L dj~1( f )(k)"dj~1( f )(k)# + q(l)dj~1( f )(k!1), ( t ( l/1 k"¸, ¸#1,2, N. Due to the causality of Q, the hypothesis dj~1( f )(k)"0, k(0, in (24) is reasonable. Even ( though the boundary processing results in the error of dj~1( f ) near the left boundary, the error at ( dj~1( f )(k) quickly decays as k increases. The small( er DDqDD (1, the faster does the error decay as 1 k increases.
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Though many numerical examples demonstrate that the optimal wavelet functions satisfy S/(x),/(x!l)T+0,
0(DlD)¸,
(26)
the conditions under which the solution of (21) is identical to that of Fig. 2. The block diagram for synthesis.
Therefore, a nearly orthogonal representation of f is obtained, which can be written as J~1 (25) f (x)" + + dj ( f )(k)2j@2/(2jx!k). ( j/~= k The reconstruction can be realized as described in Fig. 2 and the process "ts into the inverse transform of the lifting scheme except for an additional preposition "lter P on the high-pass channel which is the inverse "lter of the "lter Q with the transfer function 1!+L q(l)z~1. l/1 While the above descriptions are devoted to the implementation of analysis and synthesis of the designed wavelet system, the next paragraphs discuss the solving of the optimization problem (21) using the gradient method:
l"1,2,2, ¸,
(27)
remain undetermined and is left for future research. In non-orthogonal wavelet transform domain, the correlation between the detail coe$cients is simultaneously determined by the correlation of the signal and wavelet bases, which complicates the signal analysis. In many signal models, the signal is of short-range dependence [11]; in other words, the correlation sequence r of the signal decays quickly k as DkD increases. In this case, due to (27), the correlation between the wavelet coe$cients may e$ciently re#ect the property of the signal, which is an advantage of the local orthogonalization. By now a family of nearly orthogonal interpolating wavelet systems can be constructed using our optimal designing scheme.
5. Examples
(i) Initialization: q (l)"0, l"1,2,2, ¸, i.e., / (x)"t(x). 0 0 (ii) Calculate the gradient vector : Assume that q and / (x)"t(x)! k k +L q (l)t(x!1) can be obtained; then the gradil/1 k ent vector can be calculated by = grad(l)"!2 + c(k)(g(k#l)#g(k!l)), k/1 1,2,2, ¸, where c(m)"S/ (x),/ (x!m)T, k k t(x!m)T.
S/(x),/(x!l)T"0,
g(m)"S/ (x), k
(iii) Modify q : k q (l)"q (l)!jgrad(l), l"1,2,2, ¸, k`1 k j is a step. Sometimes, a changeable step may also be used. (iv) If DDq !q DD (e, then q015"q ; if not, go k`1 k 1 k`1 to (ii), where e is a threshold to end the iterative process.
Given an interpolating scaling function, "rstly, solve the quadric programming problem (19) and obtain the optimal parameter vector a015 and the optimal "lter u015; secondly, locally orthogonalize the obtained wavelet function and obtain the new wavelet function. Below Deslauriers}Dubuc fundamental functions are taken as examples. 5.1. The nearly orthogonal interpolating wavelets with two vanishing moments The Deslauriers}Dubuc fundamental function with two vanishing moments is the Hat function satisfying the two-scale relation u(x)"u(2x)#1/2(u(2x#1)#u(2x!1)). When the "lter u"[1/2,1/2], this is the famous Franklin wavelet, the corresponding wavelet function is denoted by t , and the correlation coe$0 cients between the wavelet function and the integer shifts of the scaling function are illustrated in
P.-L. Shui, Z. Bao / Signal Processing 79 (1999) 289}300
Fig. 3(a). When the support set of u is M!1, 0, 1, 2N, the optimal parameter vector and "lter are a015"[7,12,7]/26 and u015"[!7,33,33,!7]/52, respectively. The correlation coe$cients between the optimal wavelet function t and the integer shifts of the scaling function are illustrated in Fig. 3(b), and the co-redundancy degree is only about 1/20th of the original co-redundancy degree. The correlation coe$cients of the optimal wavelet function t are shown in Fig. 4(a). It is seen that the larger coe$cients concentrate on the set M!2,!1,0,1,2N. Furthermore, choose ¸"2 and locally orthogonalize t. The optimal vector q"[!0.34069109,!0.02501170]. The correlation coe$cients are shown in Fig. 4(b) and S/(x),/(x!1)T and S/(x),/(x!2)T are smaller than 10~16. This observation naturally leads to the hypothesis that the optimal q is the solution of S/(x),/(x!1)T"S/(x),/(x!2)T"0.
297
The comparison on redundancy degrees and Riesz bounds of wavelets are given in Table 1, where CRD, WRD, B and A denote the co-redundancy degree, redundancy degree of wavelet function, upper bound and lower bound in (11), respectively. 5.2. The nearly orthogonal interpolating wavelets with four vanishing moments Of all Deslauriers}Dubuc fundamental functions, the function with four vanishing moments has the best time}frequency localization (the smallest the product of time width and frequency width) and therefore is widely used. It satis"es the two-scale relation u(x)"u(2x)#9/16(u(2x#1)#u(2x!1)) !1/16(u(2x!3)#u(2x#3)).
Fig. 3. The comparison of correlation coe$cients.
Fig. 4. The comparison of the correlation structure with the wavelet functions.
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Table 1 The comparison about the redundancy indexes
Table 2 The comparison of the redundancy indexes
Wavelet type
CRD
WRD
B/A
Wavelet type
CRD
WRD
B/A
Mu, t N 0 Mu, tN Mu, /N
0.0556 0.0034 0.0035
0.2284 0.2380 1.4208]10~5
1.7777/0.1510 1.8979/0.4532 1.0067/0.9765
Mu, t N 0 Mu, tN Mu, /N
0.0242 0.0020 0.0018
0.2578 0.2545 2.586]10~6
1.9718/0.3934 2.0280/0.4994 1.0035/0.9875
When the interpolating "lter u"h(~1)" 3 [!1,9,9,!1]/16, formula (2) is a lifting Donoho wavelet and the corresponding wavelet function is denoted by t . When the designed support set is 0 M!2,!1,0,1,2,3N, the optimal results are a015"[0.54372520012555, !0.08745040025111,
Table 3 The comparison of the redundancy indexes Wavelet type
CRD
WRD
B/A
Mu, tN Mu, /N
2.1255]10~4 1.9665]10~4
0.1258 2.8766]10~8
1.7626/0.4392 1.0003/0.9987
0.54372520012555], u015"a015(1)h(~2)#a015(2)h(~1)#a015(3)h(0). 3 3 3 According to the correlation structure of the obtained wavelet function t, choose ¸"4, the obtained optimal result is q"[!0.329486, 0.007220, !0.002447, !0.000206]. The redundancy indexes of three wavelet systems are shown in Table 2. When the designed support set is M!3,!2, !1,0,1,2,3,4N, the optimal parameter vector is a015"[0.04389920, 0.11429242, 0.68361676, 0.11429242, 0.04389920], u015"a015(1)h(~3)#a015(2)h(~2)#a015(3)h(~1) 3 3 3 (28) #a015(4)h(0)#a015(5)h(1). 3 3 The corresponding wavelet function is t. Choose the parameter ¸"4, locally orthogonalize t, and the optimal result q"[!0.32872240, 0.00771976, !0.00219163, !0.00024376].
(29)
The comparison of redundancy indexes is given in Table 3. The results show that our designing scheme is e$cient. The nearly orthogonal interpolating wavelets designed by our scheme preserves the
regularity and the number of vanishing moments of the original system. And the fact that the Q is a stable, causal and all-pole "lter and the P is a short FIR promises the fast algorithm of the designed system. Except for Haar wavelet, the "lters in Daubechies wavelets are of in"nite digits. In applications, the truncation of the "lters will destroy the orthogonality. However, coe$cients of all D#1 order Deslauriers}Dubuc "lters are k/2D`1 type dyadic rational numbers and these enable us to perform the digital "ltering without the truncation error of "lters, which is one of the advantages of Donoho wavelets and lifting Donoho wavelets. In order to preserve this advantage, one may use the suboptimal parameter vectors a46"015 and q46"015 with "nite digits by, for example, quantifying the optimal vectors in (28) and (29) using the following formulas: a46"015"2~N roundM2Na015N, q46"015"2~(N`4) roundM2N`4qN, round(x) denotes the integer nearest x and N is a given positive integer. Sometime, the proper modi"cation of a46"015 is needed to promise sum(a46"015)"1. The adjustment of N realizes the trade-o! between the redundancy and the digits of "lters. The result is illustrated in Table 4.
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299
Table 4 The redundancy degrees versus the digits of "lters N
4
5
6
7
RY(u, /) RY(/) B/A
0.0030 9.14]10~4 1.0081/0.9713
9.113]10~4 8.93]10~4 1.0049/0.9910
2.630]10~4 1.73]10~4 1.0012/0.9962
2.997]10~4 1.806]10~4 1.0006/0.9967
Acknowledgements We would like to thank the anonymous reviewers for their suggestions on improving this paper. The authors would like to thank Dr. Xuejun Liao for his participation in the revision of the paper in English, too. Fig. 5. The comparison of sual scaling functions.
Appendix A. Proof of Theorem 1 6. Conclusions The con#ict between interpolation and orthogonality in two-band wavelets gives rise to the appearance of M-band wavelets and multiwavelets [8,9,18], and the relaxation of orthogonality produces semi-orthogonal wavelet and biorthogonal wavelets [3,2]. However, in practical applications, one pays more attention to the global performance rather than some single index of performance. In this paper, starting from the lifting Donoho interpolating wavelets which are #exible in designing, we design a novel family of nearly orthogonal interpolating wavelets. The designed wavelets preserve the main advantages of Donoho wavelets and yet markedly degrade the inter-level redundancy and the intra-level redundancy. Additionally, the low regularity of the duals is one of the drawbacks of the existing lifting Donoho wavelets. As the byproduct of our scheme, the optimal design of coredundancy degree improves the regularity of the duals. For example, the dual scaling functions of lifting Donoho wavelet with h"u"[!1/16, 9/16,9/16,!1/16] and those in (28) are illustrated in Fig. 5(a) and (b), respectively.
= ∀Md N3l2(Z2) and f (x)" + + d t (x) j,n j,n j,n j/~= n Let J~1 f (x)" + + d t (x), g (x)"+ d t (x). J j,n j,n J J,n J,n j/1 n n Then f (x)"f (x)#g (x) where, g (x)3= (t), J`1 J J J J spanMt (x), n3ZN, J,n f (x)3< (u) J J ,closeMu (x)"2J@2u(2Jx!n), n3ZN. J,n By Theorem 3.24 in [3], we have A + Dd D2) t n J,n DDg DD2 )B + Dd D2. t n J,n J 2 Let h (x) denote the part of f (x) in < W= , J J`1 J J then DD f DD2 "DD f DD2 #DDg DD2 !DDh DD2 . J`1 2 J 2 J 2 J 2
(A.1)
Obviously c ,Sh (x),u (x)T"Sg (x),u (x)T" J,n J J,n J J,n + c(n!l)d , l J,l c(l)"St(x),u(x!l)T. Using D (u), K(u) denote the Fourier series of J the sequences Md N and Mc(l)N, respectively. By J,n
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Parseval equation and the de"nitions in (11), A + Dd D2)DDc DD22 r,t J,n J,n l n 1 p DD (u)D2DK(u)D2 du)B + Dd D2. " J r,t J,n 2p ~p n
P
From Theorems 3.20 and 3.24 in [3], it is easy to obtain A DDh (x)DD2 )+ Dc D2)B DDh (x)DD2 . 2 2 J,n r J r J n Therefore, A B r,t + Dd D2)+ Dc D2) r,t + Dd D2. J,n J,n J,n B A r n r n n From (A.1) and (A.2), we have
A
B
(A.2)
B A ! r,t + Dd D2)DD f (x)DD2 !DD f (x)DD2 2 2 J t J,n J`1 A r n A ) B ! r,t + Dd D2, t J,n B r n = B + + Dd D2)DD f DD2 A ! r,t 2 j,n t A r j/~= n = " + (DD f DD2 !DD f DD2 ) j 2 J`1 2 j/~= = A ) B ! r,t + + Dd D2. j,n t B r j/~= n This completes the proof. h
A
A A
B
B B
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