关于MRA超小波的研究
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摘要
小波分析应用到信号处理和图象处理已是众所周知,其成功之处在于空间L~2(R)的多分辨分析(MRA)的引入,正因为有MRA结构,我们可以给出实现小波分析的快速算法.经典的MRA小波是一种重要的正交小波,在应用中许多著名例子及小波都属于正交小波类。
超小波是指希尔伯特空间L~2(R)~((m))=L~2(R)(?)…(?)L~2(R)中的小波,是基于小波分析基础上的信号分析方法。超小波的独特之处在于,它对解决信号的多重性问题有小波无法比拟的优势。可构造L~2(R)~((m))空间的紧支撑小波的正交基,理论上,这些基要比已知的小波基较好。在数据压缩方面,当有同样数据时,超小波可以同时压缩多个信号。类似地,研究空间L~2(R)~((m))中的MRA超小波有十分重要作用。在2000年,Han D.及Larson D.已证明了完全可以构造L~2(R)~((m))中的MRA超小波。于2005年,Bildea S.,Dutkay D E.及Galriel P.给出了构造空间L~2(R)~((m))中的MRA超小波的一种方法,并且构造了紧支撑小波的正交基。
建立在对规范紧框架小波分析的基础之上,本文给出了关于MRA超小波的判定定理,即一个超小波是MRA超小波的充分和必要条件。给出并证明了一个超小波是MRA结构当且仅当每个特殊线性子空间的维数是0或者1。特别地,当E_i是有界规范紧框架小波集时,一个超小波是MRA结构当且仅当E_i~s=∪_(k≥1)2~(-k)E_i是2π平移同余到[-π,π)的一个子集。除此之外还给出一个例子。
本文共分四部分:第一章,概述了小波分析及超小波的产生、发展发展过程。第二章,主要讨论了超小波的基本性质。规范紧框架小波扩展为超小波是超小波的一个重要的方面。因此,我们有必要在此介绍可扩展的一些性质。第三章,着重研究了超小波的构造方法。如何根据所处理的信号特征构造或者选择最佳的超小波函数应该成为我们要研究的问题之一。第四章,是本文的核心内容,讨论了直和希尔伯特空间L~2(R)~((m))中的MRA超小波的判定条件。
The applications of wavelet theory to signal processing and image processing are now well-know.Probably the main reason for the success of the wavelet theory was the introduction of the concept of multiresolution analysis(MRA),which provided the right framework to construct orthogonal wavelet bases with good localization properties.The classical MRA wavelets are probably the most important class of orthogonal wavelets.Many of the better known examples as well as those often used in applications belong to this class.
The super-wavelets is a wavelet in the space L~2(R)~((m)) = L~2(R)(?)···(?)L~2(R).It is a signal analysis method that bases on wavelet analysis.An advantage of superwavelet is solving multiple signals.We constucted orthonarmal bases of compactly supported wavelets for the space L~2(R)~((m)).Theoretically,these bases appear to be better than the know wavelets because,with the same amount of data,one cancompress not just one signal,but several.Similarly,It is important to study MRA super-wavelets in the space L~2(R)~((m)).Han D.and Larson D.have shown that the technique of multiresolution analysis breaks down for the super-wavelets in 2000.Bildea S.,Dutkay D E.and Picioraga G.gave a method of constructing MRA super-wavelet and constructed orthogonal bases of compactly supported wavelets for the space L~2(R)~((m)) in 2005.
Building up from a careful analysis of normalized tight frame wavelet,we developed a theory of the MRA super-wavelets,which is necessary and sufficient conditions of super-wavelets arises from MRA construction.It is shown that a superwavelet arises from our MRA construction if and only if the dimension of each particular linear space is either zero or one.In particular,when E_i is a bounded normalized tight frame wavelet set,it is shown that a super-wavelet arise from our MRA construction if and only if E_i~s =∪_(κ≥1)2~(-κ)E_i is 2πtranslation congruent to a subset of[—π,π).In addition,an example is given.
This paper is composed of four parts:The chapter 1 is an introduction which summarizes the emergence,develepment of wavelet analysis and super-wavelet theory. The chapter 2 presents the basic properties of super-wavelet.It is an important that the normalized tight frame wavelet extending to the super-wavelet.The chapter 3 studies the construction method of super-wavelet.It is one of our studing problem that how to choose better function according to the proceeding signals.The chapter 4 is the key of the paper,we dicuss the condition of a super-wavelets arises arises from MRA construction.
超小波是指希尔伯特空间L~2(R)~((m))=L~2(R)(?)…(?)L~2(R)中的小波,是基于小波分析基础上的信号分析方法。超小波的独特之处在于,它对解决信号的多重性问题有小波无法比拟的优势。可构造L~2(R)~((m))空间的紧支撑小波的正交基,理论上,这些基要比已知的小波基较好。在数据压缩方面,当有同样数据时,超小波可以同时压缩多个信号。类似地,研究空间L~2(R)~((m))中的MRA超小波有十分重要作用。在2000年,Han D.及Larson D.已证明了完全可以构造L~2(R)~((m))中的MRA超小波。于2005年,Bildea S.,Dutkay D E.及Galriel P.给出了构造空间L~2(R)~((m))中的MRA超小波的一种方法,并且构造了紧支撑小波的正交基。
建立在对规范紧框架小波分析的基础之上,本文给出了关于MRA超小波的判定定理,即一个超小波是MRA超小波的充分和必要条件。给出并证明了一个超小波是MRA结构当且仅当每个特殊线性子空间的维数是0或者1。特别地,当E_i是有界规范紧框架小波集时,一个超小波是MRA结构当且仅当E_i~s=∪_(k≥1)2~(-k)E_i是2π平移同余到[-π,π)的一个子集。除此之外还给出一个例子。
本文共分四部分:第一章,概述了小波分析及超小波的产生、发展发展过程。第二章,主要讨论了超小波的基本性质。规范紧框架小波扩展为超小波是超小波的一个重要的方面。因此,我们有必要在此介绍可扩展的一些性质。第三章,着重研究了超小波的构造方法。如何根据所处理的信号特征构造或者选择最佳的超小波函数应该成为我们要研究的问题之一。第四章,是本文的核心内容,讨论了直和希尔伯特空间L~2(R)~((m))中的MRA超小波的判定条件。
The applications of wavelet theory to signal processing and image processing are now well-know.Probably the main reason for the success of the wavelet theory was the introduction of the concept of multiresolution analysis(MRA),which provided the right framework to construct orthogonal wavelet bases with good localization properties.The classical MRA wavelets are probably the most important class of orthogonal wavelets.Many of the better known examples as well as those often used in applications belong to this class.
The super-wavelets is a wavelet in the space L~2(R)~((m)) = L~2(R)(?)···(?)L~2(R).It is a signal analysis method that bases on wavelet analysis.An advantage of superwavelet is solving multiple signals.We constucted orthonarmal bases of compactly supported wavelets for the space L~2(R)~((m)).Theoretically,these bases appear to be better than the know wavelets because,with the same amount of data,one cancompress not just one signal,but several.Similarly,It is important to study MRA super-wavelets in the space L~2(R)~((m)).Han D.and Larson D.have shown that the technique of multiresolution analysis breaks down for the super-wavelets in 2000.Bildea S.,Dutkay D E.and Picioraga G.gave a method of constructing MRA super-wavelet and constructed orthogonal bases of compactly supported wavelets for the space L~2(R)~((m)) in 2005.
Building up from a careful analysis of normalized tight frame wavelet,we developed a theory of the MRA super-wavelets,which is necessary and sufficient conditions of super-wavelets arises from MRA construction.It is shown that a superwavelet arises from our MRA construction if and only if the dimension of each particular linear space is either zero or one.In particular,when E_i is a bounded normalized tight frame wavelet set,it is shown that a super-wavelet arise from our MRA construction if and only if E_i~s =∪_(κ≥1)2~(-κ)E_i is 2πtranslation congruent to a subset of[—π,π).In addition,an example is given.
This paper is composed of four parts:The chapter 1 is an introduction which summarizes the emergence,develepment of wavelet analysis and super-wavelet theory. The chapter 2 presents the basic properties of super-wavelet.It is an important that the normalized tight frame wavelet extending to the super-wavelet.The chapter 3 studies the construction method of super-wavelet.It is one of our studing problem that how to choose better function according to the proceeding signals.The chapter 4 is the key of the paper,we dicuss the condition of a super-wavelets arises arises from MRA construction.
引文
[1]Gabor D..Theory of communications[J].Jour.Inst.Elec.Eng,1946,93(2):429-457.
[2]Daubechies I..Teu Lectures on Wavelets[J].in:CBMS Conf.Series in Appl.Math,1992,61,SIAM,Philadelphia.
[3]龙瑞麟.高维小波分析[M].北京:世界图书出版社,1995.
[4]Christensen O..An Introduction to Frames and Riesz Bases[M].Birkhauser,Boston,2003.
[5]Mallat S..A Wavelet Tour of Signal Processing[M].Academic Press,New York,1998.
[6]Bilan R..Density and redundancy of the noncoherent Weyl-Heisenberg superframe [J].Journal of Contemporary Mathematics,1999,247(1):29-41.
[7]Han D.,Larson D..Frames,bases and group representations[J].Proc.Amer.Math.Soc,2000,147(1):1-88.
[8]Bildea S.,Dutkay D E.,Gabriel P..MRA super-wavelets[J].New York J.Math,2005,11(6):1-19.
[9]Dutkay D.,Jorgensen P..Oversampling Generates Super-wavelets[J].Proceeding of the American Mathematical Society,2007,135(8):2219-2227.
[10]Gu Q.,Han D..Super-Wavelets and Decomposable Wavelet Frames[J].The Journal of Fourier Analysis and Applications,2005,11(6):684-696.
[11]Dai X.,Dido Y.,Gu Q..On Super-Wavelets[J].Operator Theory:Advances and Applications,2004,149(3):153-165.
[12]Paluszy'nski M.,Siki'c H..Weiss G.and Xiao S.Tight frame wavelet,their dimension functions MRA tight frame wavelets and connectivity properties[J].Advances in computational mathematics,2003,18(2):297-327.
[13]Consortium W..Basic properties of wavelets[J].Fourier Anal.Appl,1998,30(4):575-594.
[14]Frazier M.,Jawerth B..A discrete transform and decomposition of distribution spaces[J].Journal of Functional Analysis,1990,93(1):34-170.
[15]Hernandcz E.,Weiss G..A first course on wavelets[M].CRC Press,Bocaraton,1996.
[16]Larson D..Von Neumann Algebras and Wavelets[J].Operator Algebras and Applications,1997,495(4):267-312.
[17]Dutkay D E..The local trace function for super-wavelets[J].Journal of Contemporary Mathematics,345(2):115-136.
[18]Weber E..Orthogonal frames of translates[J].Applied and Computational Harmonic Analysis,2004,17(1):69-90.
[19]李万社,刘志国.一种有效的小波紧框架设计[J].西安电子科技大学学报,2006.339(4):665-669.
[20]李万社.郭晓旋.框架和Riesz基的稳定性[J].陕西师范大学学报,自然科学版,2006,34(4):7-12.
[21]Daubeehies I..Ten lectures on wavelets小波十讲[M].李建平,杨万年,译.北京:国防工业出版社,2004.
[22]程蓉,李万社.利用两尺度相似变换提高有限元多尺度函数的逼近阶[J].陕西师范大学学报,(自然科学版),2005,32(2):22-25.
[23]成礼智,王红霞,罗永.小波的理论与应用[M].北京:科学出版社,2004:76-79.
[24]李建华,李万社.小波理论及其应用(综述)[J].河西学院学报,2005,22(2):1-7.
[25]Dai X.,Diao Y.,Gu Q..Frame wavelet sets in R[J].Proceeding of the American Mathematical Society,2002,129(7):2045-2055.
[26]Bownik M..The structure of shift-invariant subspaces of L~2(R~n)[J].Journal of Functional Analysis,2000,177(1):282-309.
[27]Dai X.,Diao Y.,Gu Q.,Han D..Frame wavelets in subspaces of L~2(R~d)[J].Proceeding of the American Mathematical Society,2002,130(6):1661-1667.
[28]Mallat S..A wavelets tour Of signal processing信号处理的小波导引(第二版)[M].杨力华,戴道清等译.北京:机械工业出版社,2002.
[29]Dai X.,Larson D..Wavelet sets in R~n,[J]Mem.Amer.Math.Soc,2000,147(2):1121-1322.
[30]姚喜妍.小波分析中的框架理论发展综述[J].运城学院学报,2005,23(5):7-9.
[31]曹怀信,赵建伟.小波发展综述[J].咸阳师范学院学报,2002,17(6):5-8.
[32]Zou Q..Biorthogonal wavelet filters spectral analysis by discrete super-wavelet transforms[J].(Chinese)Math Theory Applied.2005,25(1):56-58.
[33]彭玉华.小波变换与工程应用[M].北京:科学出版社,2003:39-40.
[34]Dutkay D..Harmonic analysis of signed Ruelle transfer operators[J].Mathematics Analysis applied,2002,273(2):590-617.
[35]Dutkay D..The wavelet Galerkin operator[J].Operator Theory,2004,51(1):49-70.
[36]Lawton W..Necessary and sufficient conditions for constructing orthonormal wavelet bases[J].Mathematical Physics,1991,32(1):57-61.
[37]Kim H.,Lim J..On frame wavelets associated with frame multiresolution analysis[J].Applied and Computational Harmonic Analysis,2001,10(1):61-70.
[38]Mallat S..A theory of multiresolution approximations and wavelet orthonormal basis of L~2(R)[J].Transactions of the American Mathematical Society 1989,315(1):69-167.
[39]Benedetto J.,Li S..The theorem of multiresolution analysis frames and application to filter banks[J].Applied and Computational Harmonic Analysis,1998,5(2):398-427.
[40]Papadakis M.,Siki'c H.,Weiss G..The characterization of low pass filters and some basic properties of wavelets,scaling functions and related concepts[J].Fourier Anal.Appl,1999,29(5):495-521.
[41]Paluszy'nski M.,Siki'c H.,Weiss G..and Xiao S.Generalized low pass filters and MRA frame wavelets[J].Geom.Anal.2001,11(2):311-342.
[2]Daubechies I..Teu Lectures on Wavelets[J].in:CBMS Conf.Series in Appl.Math,1992,61,SIAM,Philadelphia.
[3]龙瑞麟.高维小波分析[M].北京:世界图书出版社,1995.
[4]Christensen O..An Introduction to Frames and Riesz Bases[M].Birkhauser,Boston,2003.
[5]Mallat S..A Wavelet Tour of Signal Processing[M].Academic Press,New York,1998.
[6]Bilan R..Density and redundancy of the noncoherent Weyl-Heisenberg superframe [J].Journal of Contemporary Mathematics,1999,247(1):29-41.
[7]Han D.,Larson D..Frames,bases and group representations[J].Proc.Amer.Math.Soc,2000,147(1):1-88.
[8]Bildea S.,Dutkay D E.,Gabriel P..MRA super-wavelets[J].New York J.Math,2005,11(6):1-19.
[9]Dutkay D.,Jorgensen P..Oversampling Generates Super-wavelets[J].Proceeding of the American Mathematical Society,2007,135(8):2219-2227.
[10]Gu Q.,Han D..Super-Wavelets and Decomposable Wavelet Frames[J].The Journal of Fourier Analysis and Applications,2005,11(6):684-696.
[11]Dai X.,Dido Y.,Gu Q..On Super-Wavelets[J].Operator Theory:Advances and Applications,2004,149(3):153-165.
[12]Paluszy'nski M.,Siki'c H..Weiss G.and Xiao S.Tight frame wavelet,their dimension functions MRA tight frame wavelets and connectivity properties[J].Advances in computational mathematics,2003,18(2):297-327.
[13]Consortium W..Basic properties of wavelets[J].Fourier Anal.Appl,1998,30(4):575-594.
[14]Frazier M.,Jawerth B..A discrete transform and decomposition of distribution spaces[J].Journal of Functional Analysis,1990,93(1):34-170.
[15]Hernandcz E.,Weiss G..A first course on wavelets[M].CRC Press,Bocaraton,1996.
[16]Larson D..Von Neumann Algebras and Wavelets[J].Operator Algebras and Applications,1997,495(4):267-312.
[17]Dutkay D E..The local trace function for super-wavelets[J].Journal of Contemporary Mathematics,345(2):115-136.
[18]Weber E..Orthogonal frames of translates[J].Applied and Computational Harmonic Analysis,2004,17(1):69-90.
[19]李万社,刘志国.一种有效的小波紧框架设计[J].西安电子科技大学学报,2006.339(4):665-669.
[20]李万社.郭晓旋.框架和Riesz基的稳定性[J].陕西师范大学学报,自然科学版,2006,34(4):7-12.
[21]Daubeehies I..Ten lectures on wavelets小波十讲[M].李建平,杨万年,译.北京:国防工业出版社,2004.
[22]程蓉,李万社.利用两尺度相似变换提高有限元多尺度函数的逼近阶[J].陕西师范大学学报,(自然科学版),2005,32(2):22-25.
[23]成礼智,王红霞,罗永.小波的理论与应用[M].北京:科学出版社,2004:76-79.
[24]李建华,李万社.小波理论及其应用(综述)[J].河西学院学报,2005,22(2):1-7.
[25]Dai X.,Diao Y.,Gu Q..Frame wavelet sets in R[J].Proceeding of the American Mathematical Society,2002,129(7):2045-2055.
[26]Bownik M..The structure of shift-invariant subspaces of L~2(R~n)[J].Journal of Functional Analysis,2000,177(1):282-309.
[27]Dai X.,Diao Y.,Gu Q.,Han D..Frame wavelets in subspaces of L~2(R~d)[J].Proceeding of the American Mathematical Society,2002,130(6):1661-1667.
[28]Mallat S..A wavelets tour Of signal processing信号处理的小波导引(第二版)[M].杨力华,戴道清等译.北京:机械工业出版社,2002.
[29]Dai X.,Larson D..Wavelet sets in R~n,[J]Mem.Amer.Math.Soc,2000,147(2):1121-1322.
[30]姚喜妍.小波分析中的框架理论发展综述[J].运城学院学报,2005,23(5):7-9.
[31]曹怀信,赵建伟.小波发展综述[J].咸阳师范学院学报,2002,17(6):5-8.
[32]Zou Q..Biorthogonal wavelet filters spectral analysis by discrete super-wavelet transforms[J].(Chinese)Math Theory Applied.2005,25(1):56-58.
[33]彭玉华.小波变换与工程应用[M].北京:科学出版社,2003:39-40.
[34]Dutkay D..Harmonic analysis of signed Ruelle transfer operators[J].Mathematics Analysis applied,2002,273(2):590-617.
[35]Dutkay D..The wavelet Galerkin operator[J].Operator Theory,2004,51(1):49-70.
[36]Lawton W..Necessary and sufficient conditions for constructing orthonormal wavelet bases[J].Mathematical Physics,1991,32(1):57-61.
[37]Kim H.,Lim J..On frame wavelets associated with frame multiresolution analysis[J].Applied and Computational Harmonic Analysis,2001,10(1):61-70.
[38]Mallat S..A theory of multiresolution approximations and wavelet orthonormal basis of L~2(R)[J].Transactions of the American Mathematical Society 1989,315(1):69-167.
[39]Benedetto J.,Li S..The theorem of multiresolution analysis frames and application to filter banks[J].Applied and Computational Harmonic Analysis,1998,5(2):398-427.
[40]Papadakis M.,Siki'c H.,Weiss G..The characterization of low pass filters and some basic properties of wavelets,scaling functions and related concepts[J].Fourier Anal.Appl,1999,29(5):495-521.
[41]Paluszy'nski M.,Siki'c H.,Weiss G..and Xiao S.Generalized low pass filters and MRA frame wavelets[J].Geom.Anal.2001,11(2):311-342.