基于提升方法的多元小波构造研究
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摘要
小波的绝大多数应用在高维情形,因此,具有良好性能指标的多元小波基构造是它成功应用的重要前提与关键保障,多元非分离小波的构造研究成为小波分析的一个重要组成部分。基于提升格式的双正交小波构造方法,它不依赖于伸缩和平移的概念,也不需要频谱分析工具,它的提出对多元小波的构造极具指导意义。
借鉴提升格式在多元插值单小波构造中取得的成果,本论文重点对多元非分离小波、非分离多小波的一般构造理论进行系统深入的研究,所获得的结果系统深刻,具有十分重要的理论意义和实际应用价值。主要结果包括:
1)系统研究了多元单小波的多步提升构造算法,重点研究了多步提升算子与小波各重要性质之间的关系,提出了同时满足对称、短支撑、高阶消失矩的任意元任意任意取样矩阵双正交单小波的统一提升构造算法,为构造性质优美的多元小波基提供了最优提升设计方案;
2)系统研究了基于提升格式的多元Lagrange插值多小波构造算法,提出了一类新的预测多滤波器——Multi-Neville滤波器,系统研究其性质及与提升算子之间的关系,首次建立了任意维任意取样矩阵任意重具有任意阶消失矩性质优美的Lagrange插值平衡多小波的统一构造框架;得到了一系列具有紧支、对称、光滑的双正交多小波,丰富了多元非分离小波函数库;
3)系统研究了多元Hermite插值多小波的提升构造算法,提出了一类新的预测多滤波器——Hermite-Neville滤波器,首次提出了任意维任意取样矩阵任意阶消失矩任意重导数信息的Hermite插值(单)小波的提升构造格式,为多元Hermite插值算子的设计提供了统一的构造框架,几乎所有已有多元插值小波成为本方法的特例;首次提出了基于提升格式的任意带任意重Hermite插值多小波预滤波器设计。
In recent years, multivariate wavelet has generated considerable interest because of their widely applications. And the construction of multivariate non-separable wavelet systems with good synthetic performance becomes a very important aspect on the theory of wavelet analysis. On the other hand, the lifting scheme has recently emerged as a new way for wavelet construction. The main feature of lifting is that it provides an entirely spatial-domain interpretation of the transform, as opposed to the more traditional frequency-domain based constructions. It is worthwhile to simplify the higher dimensional non-separable wavelet construction to some extent.
Motivated by the pioneering work on the construction of multivariate Lagrange interpolating wavelet with lifting schemes, in this paper, we focus on the construction of multivariate nonseparable wavelet and nonseparable multiwavelet. The main results include the following:
1) We study the wavelet construction based on multi-step lifting scheme, and systematically investigate the relationship between the lifting operators and some important properties of wavelet system. And then, we propose a general framework based on lifting for constructing filter banks and wavelets in any dimension, for any lattice with desired properties such as linear phase, short support length and high order vanishing moment. And it also presents a guideline for the selection of optimal lifting scheme to satisfy the given wavelet properties.
2) We study the construction of multivariate Lagrange multiwavelet based on lifting scheme. A new multifilter named as Multi-Neville filter was defined for prediction, and the relations between the Multi-Neville filter and the properties of multiwavelet were systematically investigated. And then, we propose a general recipe for constructing biorthogonal balanced multiwavelets in any dimension, for any lattice with any multiplicity and any order of vanishing moment. A series of biorthogonal multiwavelet with some desired properties were obtained. To our knowledge, there is no formal publication on such systematic construction.
3) We firstly define a new predictor named as Hermite-Neville filter, and from which a general design framework is developed for building Hermite interpolation filter banks of any derivative order for any lattice in any dimension with any number of primal and dual vanishing moments. And almost all of the existed multivariate interpolation wavelets become a special case of our algorithm. Moreover, based on lifting scheme, a novel pre-filter design for arbitrary dimensional Hermite interpolation multiwavelet with arbitrary multiplicity is proposed.
借鉴提升格式在多元插值单小波构造中取得的成果,本论文重点对多元非分离小波、非分离多小波的一般构造理论进行系统深入的研究,所获得的结果系统深刻,具有十分重要的理论意义和实际应用价值。主要结果包括:
1)系统研究了多元单小波的多步提升构造算法,重点研究了多步提升算子与小波各重要性质之间的关系,提出了同时满足对称、短支撑、高阶消失矩的任意元任意任意取样矩阵双正交单小波的统一提升构造算法,为构造性质优美的多元小波基提供了最优提升设计方案;
2)系统研究了基于提升格式的多元Lagrange插值多小波构造算法,提出了一类新的预测多滤波器——Multi-Neville滤波器,系统研究其性质及与提升算子之间的关系,首次建立了任意维任意取样矩阵任意重具有任意阶消失矩性质优美的Lagrange插值平衡多小波的统一构造框架;得到了一系列具有紧支、对称、光滑的双正交多小波,丰富了多元非分离小波函数库;
3)系统研究了多元Hermite插值多小波的提升构造算法,提出了一类新的预测多滤波器——Hermite-Neville滤波器,首次提出了任意维任意取样矩阵任意阶消失矩任意重导数信息的Hermite插值(单)小波的提升构造格式,为多元Hermite插值算子的设计提供了统一的构造框架,几乎所有已有多元插值小波成为本方法的特例;首次提出了基于提升格式的任意带任意重Hermite插值多小波预滤波器设计。
In recent years, multivariate wavelet has generated considerable interest because of their widely applications. And the construction of multivariate non-separable wavelet systems with good synthetic performance becomes a very important aspect on the theory of wavelet analysis. On the other hand, the lifting scheme has recently emerged as a new way for wavelet construction. The main feature of lifting is that it provides an entirely spatial-domain interpretation of the transform, as opposed to the more traditional frequency-domain based constructions. It is worthwhile to simplify the higher dimensional non-separable wavelet construction to some extent.
Motivated by the pioneering work on the construction of multivariate Lagrange interpolating wavelet with lifting schemes, in this paper, we focus on the construction of multivariate nonseparable wavelet and nonseparable multiwavelet. The main results include the following:
1) We study the wavelet construction based on multi-step lifting scheme, and systematically investigate the relationship between the lifting operators and some important properties of wavelet system. And then, we propose a general framework based on lifting for constructing filter banks and wavelets in any dimension, for any lattice with desired properties such as linear phase, short support length and high order vanishing moment. And it also presents a guideline for the selection of optimal lifting scheme to satisfy the given wavelet properties.
2) We study the construction of multivariate Lagrange multiwavelet based on lifting scheme. A new multifilter named as Multi-Neville filter was defined for prediction, and the relations between the Multi-Neville filter and the properties of multiwavelet were systematically investigated. And then, we propose a general recipe for constructing biorthogonal balanced multiwavelets in any dimension, for any lattice with any multiplicity and any order of vanishing moment. A series of biorthogonal multiwavelet with some desired properties were obtained. To our knowledge, there is no formal publication on such systematic construction.
3) We firstly define a new predictor named as Hermite-Neville filter, and from which a general design framework is developed for building Hermite interpolation filter banks of any derivative order for any lattice in any dimension with any number of primal and dual vanishing moments. And almost all of the existed multivariate interpolation wavelets become a special case of our algorithm. Moreover, based on lifting scheme, a novel pre-filter design for arbitrary dimensional Hermite interpolation multiwavelet with arbitrary multiplicity is proposed.
引文
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[2]. J.E. Littlewood and R.E.A.C. Paley, Theorems on Fourier series and power series. Part I, J. London Math. Soc. 1931,6: 230-233; Part II, Proc. London Math. Soc. 1937, 42: 52-89; Part III, ibid. 1937, 43:105-126. 1937.
[3]. D. Gabor, Theory of communication. Journal of the Institute of Electrical Engineers, 1946, 93(3): 429-457.
[4]. J. Str?mberg. A modified Haar system and higher order spline systems on Rn as unconditional bases for Hardy Space. in Conference inharmonic analysis in honor of Antoni Zygmund II. 1981, Belmont, California. 475-493.
[5]. A. Grossman and J. Morlet, Decomposition of Hardy Functions into Square Integable Wavelets of Constant Shape. SIAM. J. Math. Anal, 1984, 15(4): 723-736.
[6]. J. Morlet, et al., Wave Propagation and Sampling Theory. Geophysics, 1982, 47: 203-236.
[7]. Y. Meyer, Principe d'incertitude, bases hilbertiennes et algebres d'operateurs. Bourbaki seminar, 1985-1986, 662.
[8]. I. Daubechies, Orthonormal Bases of Compactly Supported Wavelets. Comm. Pure Appl. Math., 1988, 41: 909-996.
[9]. S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1989, 11(7): 674-693.
[10]. S. Mallat, Multifrequency channel decompositions of images and wavelet models. IEEE Trans. Acoustics, Speech and Signal Processing, 1989, 37(12): 2091-2110.
[11]. A. Cohen, I. Daubechies, and J. Feauveau, Bi-orthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math, 1992, 45: 485-560.
[12]. T.N.T. Goodman and S.L. Lee, Wavelets of multiplicity r. Transactions on American Mathematics Society, 1994, 342(1): 307-324.
[13]. J.S. Geronimo, D.P. Hardin, and P.R. Massopust, Fractal functions and wavelet expansions based on several scaling functions. Journal of Approximation Theory, 1994, 78: 373-401.
[14]. C.K. Chui and J.-A. Lian, A study of orthonormal multi-wavelets. Applied Numerical Mathematics, 1996, 20(3): 273-298.
[15]. I.W. Selesnick, Interpolating multiwavelets bases and the sampling theorem. IEEE Transactions on Signal Processing, 1999, 47(6): 1615-1621.
[16]. S.D. Riemenschneider and Z. Shen, Box splines, cardinal series, and wavelets, in Approximation Theory and Functional Analysis, C.K.C. eds., Editor. 1991, Academic Press: New York. 133-149.
[17]. S.D. Riemenschneider and Z. Shen, Construction of compactly supported biorthogonal wavelets in L2(Rd), in Physics and modern topics in mechanical and electrical engineering, N.E. Mastorakis, Editor. 1999, World Scientific and Engineering Society Press. 201-206.
[18]. S.D. Riemenschneider and Z. Shen. Construction of compactly supported biorthogonal wavelets in L2(Rd) II. in Proceedings of SPIE. 1999. 264-272.
[19]. W. He and M. Lai, Construction of Bivariate Compactly Supported Biorthogonal Box Spline Wavelets with Arbitrarily High Regularities. Applied and Computational Harmonic Analysis, 1999, 6(1): 53-74.
[20]. W. He and M. Lai, Construction of trivariate compactly supported biorthogonal box spline wavelets. Journal of Approximation Theory, 2003, 120(1): 1-19.
[21]. J. Kova?evi? and M. Vetterli, Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for Rn. IEEE Transactions on Information Theory, 1992, 38(2): 533-555.
[22]. A. Cohen and I. Daubechies, Nonseparable bidimensional wavelet bases. Rev. Mat. Iberoamericana, 1993, 9: 51-137.
[23]. A. Karoui and R. Vaillancourt, McClellan transformation and the construction of biorthogonal wavelet bases of L2(R2). Computers & Mathematics with Applications, 1995, 29(12): 13.
[24]. K. Abderrazek, A Technique for the Construction of Compactly Supported Biorthogonal Wavelets of L2(Rn). Journal of Mathematical Analysis and Applications, 2000, 249(2): 367-392.
[25]. E. Belogay and Y. Wang, Arbitrary smooth orthogonal nonseparable wavelets in R2. SIAM Journal of Mathematical Analysis, 1999, 30: 678-697.
[26]. D.-R. Chen, B. Han, and S.D. Riemenschneider, Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments. Advances in Computational Mathematics, 2000, 13(2): 131-165.
[27]. B. Han, Analysis and construction of optimal multivariate biorthogonal wavelets with compact support. SIAM Journal on Mathematical Analysis, 2000, 31(2): 274-304.
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