关于框架小波的一些研究
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摘要
作为一种比较理想的时频分析工具,小波分析是在Fourier分析的基础上产生和发展起来的.它有效的弥补了Fourier分析的不足,被看作是多元调和分析半个世纪以来的工作结晶.因其具有良好的自适应性以及“数学显微镜”特性,小波分析被广泛地应用于信号与图像处理、智能计算、量子理论、网络信息安全等众多领域.目前,有关小波分析的理论研究在进一步的深入,其应用范围也在不断扩大.
小波变换是一种时频局部化的工具,包括连续小波变换和离散小波变换,而小波框架理论就构成了离散小波变换的主体部分.1952年Duffin和Schaef-fer在研究非调和Fourier分析时首次提出了Hilbert空间中框架的概念,之后Daubechies、Meyer等人将小波变换引入到框架理论中,提出了小波框架的概念.小波框架是小波基的一种推广和扩展,降低了小波基对正交性的要求,引入了冗余性,使得小波框架的设计具有较大的自由度.同时冗余性在表示的稳定性、对噪声的鲁棒性方面都优于正交基.由于小波框架兼有小波和框架的优良特性,现在已成为国内外学者共同研究的热点,在图像处理、边缘检测、去噪等方面表现出巨大的应用潜能.虽然有关小波框架的讨论已涉及到各个方面,其理论研究还有待进一步完善,有良好性质的框架小波还有待于我们的设计.
本文着重讨论了小波框架的相关性质,研究了对偶小波框架以及Parseval小波框架的构造,并刻画了具有高阶消失矩的小波框架,得出一些结论,有助于小波框架理论的进一步发展.全文共由四部分构成：
第1章：绪论.简要介绍了小波分析和小波框架理论的发展历程以及最新研究成果.
第2章：小波框架及相关性质.首先给出了框架的定义,讨论了框架和Riesz基的关系,通过框架算子定义了对偶框架,给出了信号的重构公式,最后讨论了框架多分辨分析的一些性质.
第3章：对偶小波框架的构造.利用框架多分辨分析,讨论了对偶小波框架和Parseval小波框架的构造,给出了具体的矩阵扩充方法.
第4章：具有高阶消失矩的小波框架的刻画.讨论了框架小波的消失矩特性,利用斜扩张原则构造出具有高阶消失矩的对偶框架小波,给出了构造算法.
As an ideal time-frequency analysis tool, wavelet analysis is based on Fourier analysis and development together. It effectively makes up for the short comings of Fourier analysis and is seen as the work of multiharmonic analysis in half a century. Besides,because of its good adaptability and mathematical microscope feature, wavelet analysis is widely used in signal or image processing, intelligent computing, quantum theory, network information security and other lots of fields. At present, the study about wavelet analysis theory is getting more and more deep and its application range is getting more and more wide.
Wavelet transform is a time-frequency localization tools, including continu-ous wavelet transform and discrete wavelet transform,and wavelet frame theory is composed as the main part of to constitute the main part of the discrete wavelet transform. Duffin and Schaeffer was proposed the concept of the frame of Hilbert space for the first time in 1952 when then studied non-harmonic Fourier analysis. Then wavelet transform was introduced into the frame theory and the concept of wavelet frame was proposed by Daubechies, Meyer and others. Wavelet frame is a generalization and expansion of wavelet bases, reducing the requirements in terms of orthogonality of wavelet bases and introducing the redundancy, which makes the wavelet frame design having a large degree of freedom. At the same time,its re-dundancy is better than the property of the stability and robustness about noise. Because the wavelet frame has good characteristics of both wavelets and frames, it has become a focus of attention of scholars both at home and abroad and shown great potential in image processing, edge detection, noise reduction and so on. Al-though the discussion about wavelet frame has been involved in lots of aspects,the theory study still needs further improvement and the wavelet frame that has good property needs us to design.
This article primarily studies the relevant properties of the wavelet frame, the structure of dual wavelet frames and Parseval wavelet frames.In addition, the wavelet frame with high vanishing moments is characterized and several conclusions are given,which is helpful for the further development of wavelet frame theory.The paper consists of four parts.
The first chapter is the introduction.The development process and the latest research results about wavelet analysis and wavelet frame is described in this part.
The second chapter is about wavelet frame and associated properties. Firstly, the definition of frame is given and the relationship between frames and Riesz basis is discussed.Then,the dual frame is defined through frame operator and the signal reconstruction formula is given.Finally,some properties about the multi-resolution analysis of frame are discussed.
The third chapter mainly studies the construction of dual wavelet frames.The construction of dual wavelet frames and Parseval wavelet frames are discussed by the multi-resolution analysis.Moreover, a specific matrix expansion method is given.
The four chapter is about the characterization about the wavelet frame with higher vanishing moments. Firstly,the high vanishing moment properties of wavelets frame is studied.Then the dual frame wavelet with high vanishing moments is con-structed by the OEP.Besides,the construction algorithm of dual frame wavelet is given in this paper.
小波变换是一种时频局部化的工具,包括连续小波变换和离散小波变换,而小波框架理论就构成了离散小波变换的主体部分.1952年Duffin和Schaef-fer在研究非调和Fourier分析时首次提出了Hilbert空间中框架的概念,之后Daubechies、Meyer等人将小波变换引入到框架理论中,提出了小波框架的概念.小波框架是小波基的一种推广和扩展,降低了小波基对正交性的要求,引入了冗余性,使得小波框架的设计具有较大的自由度.同时冗余性在表示的稳定性、对噪声的鲁棒性方面都优于正交基.由于小波框架兼有小波和框架的优良特性,现在已成为国内外学者共同研究的热点,在图像处理、边缘检测、去噪等方面表现出巨大的应用潜能.虽然有关小波框架的讨论已涉及到各个方面,其理论研究还有待进一步完善,有良好性质的框架小波还有待于我们的设计.
本文着重讨论了小波框架的相关性质,研究了对偶小波框架以及Parseval小波框架的构造,并刻画了具有高阶消失矩的小波框架,得出一些结论,有助于小波框架理论的进一步发展.全文共由四部分构成：
第1章：绪论.简要介绍了小波分析和小波框架理论的发展历程以及最新研究成果.
第2章：小波框架及相关性质.首先给出了框架的定义,讨论了框架和Riesz基的关系,通过框架算子定义了对偶框架,给出了信号的重构公式,最后讨论了框架多分辨分析的一些性质.
第3章：对偶小波框架的构造.利用框架多分辨分析,讨论了对偶小波框架和Parseval小波框架的构造,给出了具体的矩阵扩充方法.
第4章：具有高阶消失矩的小波框架的刻画.讨论了框架小波的消失矩特性,利用斜扩张原则构造出具有高阶消失矩的对偶框架小波,给出了构造算法.
As an ideal time-frequency analysis tool, wavelet analysis is based on Fourier analysis and development together. It effectively makes up for the short comings of Fourier analysis and is seen as the work of multiharmonic analysis in half a century. Besides,because of its good adaptability and mathematical microscope feature, wavelet analysis is widely used in signal or image processing, intelligent computing, quantum theory, network information security and other lots of fields. At present, the study about wavelet analysis theory is getting more and more deep and its application range is getting more and more wide.
Wavelet transform is a time-frequency localization tools, including continu-ous wavelet transform and discrete wavelet transform,and wavelet frame theory is composed as the main part of to constitute the main part of the discrete wavelet transform. Duffin and Schaeffer was proposed the concept of the frame of Hilbert space for the first time in 1952 when then studied non-harmonic Fourier analysis. Then wavelet transform was introduced into the frame theory and the concept of wavelet frame was proposed by Daubechies, Meyer and others. Wavelet frame is a generalization and expansion of wavelet bases, reducing the requirements in terms of orthogonality of wavelet bases and introducing the redundancy, which makes the wavelet frame design having a large degree of freedom. At the same time,its re-dundancy is better than the property of the stability and robustness about noise. Because the wavelet frame has good characteristics of both wavelets and frames, it has become a focus of attention of scholars both at home and abroad and shown great potential in image processing, edge detection, noise reduction and so on. Al-though the discussion about wavelet frame has been involved in lots of aspects,the theory study still needs further improvement and the wavelet frame that has good property needs us to design.
This article primarily studies the relevant properties of the wavelet frame, the structure of dual wavelet frames and Parseval wavelet frames.In addition, the wavelet frame with high vanishing moments is characterized and several conclusions are given,which is helpful for the further development of wavelet frame theory.The paper consists of four parts.
The first chapter is the introduction.The development process and the latest research results about wavelet analysis and wavelet frame is described in this part.
The second chapter is about wavelet frame and associated properties. Firstly, the definition of frame is given and the relationship between frames and Riesz basis is discussed.Then,the dual frame is defined through frame operator and the signal reconstruction formula is given.Finally,some properties about the multi-resolution analysis of frame are discussed.
The third chapter mainly studies the construction of dual wavelet frames.The construction of dual wavelet frames and Parseval wavelet frames are discussed by the multi-resolution analysis.Moreover, a specific matrix expansion method is given.
The four chapter is about the characterization about the wavelet frame with higher vanishing moments. Firstly,the high vanishing moment properties of wavelets frame is studied.Then the dual frame wavelet with high vanishing moments is con-structed by the OEP.Besides,the construction algorithm of dual frame wavelet is given in this paper.
引文
[1]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.
[2]程正兴,杨守志,冯晓霞.小波分析的理论、算法、进展和应用[M].北京：国防工业出版社,2007.
[3]邸继征.小波分析原理[M].北京：科学出版社,2010.
[4]冯象初,甘小冰,宋国乡.数值泛函与小波理论[M].西安：西安电子科技大学出版社,2003.
[5]A.Boggess, F.J.Narcowich. A First Course in Wavelets with Fourier Analysis [M]. Prentice Hall:Pearson Education,2001.
[6]许天舟,黄春光.小波分析理论、算法及其应用[M].北京：国防工业出版社,2007.
[7]李建平.小波分析信息传输基础[M].北京：国防工业出版社,2004.
[8]胡广书.数字信号处理理论、算法与实现(第二版)[M].北京：清华大学出版社,2003.
[9]程正兴,林勇平.小波分析在图像处理中的应用[J].工程数学学报,2001,18(S1)：57-86.
[10]M. Bownik, Z.Rzeszotnik, D.Speegle. A characterization of dimension function of orthogonal wavelets [J]. Applied and Computation Harmonic Analysis,2001,10(1): 71-92.
[11]P.J.Burt, E.H.Adelson. The Laplacian pyramid as a compact image code [J]. IEEE Communications,1983,31(4):532-540.
[12]H.Kaneko, D.R.Noren, B.Novaprateep. Wavelet application to the Petrov-Galerkin method for Hamme-Rstein equations [J]. Applied and Numerical Mathematics,2003, 45(2):255-273.
[13]M.Vetterli. Wavelets, Approximation and Compression [J]. IEEE Signal Processing Magazine,2001,18(5):59-73.
[14]B.Han, Z.Shen. Wavelets from the loop scheme [J]. Fourier Analysis Application. 2005,11(6):615-637.
[15]S.Grgic, K.Kers, M.Grgic. Image Compression Using Wavelets [C]. Bled. Slovenia. 1999(1):99-104.
[16]S.Mallat. A wavelets tour of signal processing [M]. Academic Press,1999.
[17]C.K.Chui, J.A.Liana. A study on Orthogonal Multiwavelets [J]. Applied Numerical Mathematics,1996,20(3):273-298.
[18]C.K.Chui. An Introduction to wavelets [M]. New York:Academic press Inc,1992.
[19]A.Cohen, I.Daubechies. A new technique to estimate the regularity of refinable functions [J]. Revista Matematica Iberoamericana,1996,12(2):527-591.
[20]A.Cohen, I.Daubechies, J.C. Feauveau. Biorthogonal bases of compactly supported wavelets [J]. Communications on pure and Applied Mathematics,1992,45(5):485-560.
[21]I.Daubechies. Ten lectures on wavelets [M]. CBMS-NSF series in applied math-emitics, Philadelphia:SIAM press,1992.
[22]O.Christensen. An introduction to frames and Riesz bases[M]. Boston:Birkhauser, 2002.
[23]曲长文,何友,刘卫华,李楠.框架理论及应用[M].北京：国防工业出版社,2009.
[24]J.J.Benedetto, S.Li. The theory of multi-resolution analysis frames and application to filter banks[J]. Applied and Computation Harmonic Analysis,1998,5(4):389-427.
[25]Ferreira, P.J.S.G. Mathematics for Multimedia Signal Processing II:Discrete Finite Frames and Signal Reconstruction [C]. Signal Pocessing for Multimedia. IOS Press, 1999.
[26]O.Christensen. Moment problems for frames and applications to irregular Sampling and Gabor frames [J]. Applied and Computation Harmonic Analysis,1996,3(1):82-86.
[27]辛友明.框架多分辨分析谱的研究[J].华东师范大学学报,2006(5)：59-65.
[28]H.O.Kim, R.Y.Kim, J.K.Lim. On the spectrums of frame multiresolution analysis [J]. J. Math. Anal. Appl.,2005,305(2):528-545.
[29]Z.W.Liu, G.E.Hu, G.C.Wu. Frame scaling function sets and frame wavelet sets in Rd [J]. Chaos, Solitons and Fractals,2009,40(15):2483-2490.
[30]Z.W.Liu, G.E.Hu, G.C.Wu, B.Jing. Semi-orthogonal frame wavelets and Parseval frame wavelets associated with GMRA [J]. Chaos. Solitons and Fractals,2008,38(5): 1449-1456.
[31]D.Bakic, I.Krishtal, E.N.Wilson. Parseval frames wavelets with En2-dilations [J]. Applied and Computation Harmonic Analysis,2005,19(3):386-431.
[32]G.C.Wu, Z.X.Cheng, D.F.Li, F.J.Zhang. Parseval frame wavelets associated with A-FMRA [J]. Chaos, Solitons and Fractals,2008,37(4):1233-1243.
[33]M.Skopina. On construction of multivariate wavelet with vanishing moments [J]. Applied and Computation Harmonic Analysis,2006,20(3):375-390.
[34]M.Skopina. On construction of multivariate wavelet frames [J]. Applied and Com-putation Harmonic Analysis,2009,27(1):55-72.
[35]M.J.Lai, J.Stockler. Construction of multivariate compactly supported tight wavelet frames [J]. Applied and Computation Harmonic Analysis,2006,21(3):324-348.
[36]M.Ehler. B.Han. Wavelet bi-frames with few generators from multivariate refinablc functions [J]. Applied and Computation Harmonic Analysis,2008,25(3):407-414.
37] M.Charina, C.K.Chui, W.J.He. Tight frames of compactly supported multivari-ate multi-wavelets [J]. Journal of Computational and Applied Mathematics,2010, 233(8):2044-2061.
[38]M.Ehler. On Multivariate Compactly Supported Bi-Frames [J]. Journal Fourier Analysis and Applications,2007,13(5):512-532.
[39]M.Bownik. A characterization of affine dual frames in L2(Rn) [J]. Applied and Computation Harmonic Analysis,2000,8(2):203-221.
[40]I.Daubechies, B.Han, A.Ron, Z.Shen. Framelets:MRA-based constructions of wavelet frames [J]. Applied and Computation Harmonic Analysis,2003,14(1):1-46.
[41]G.Bhatt, B.D.Johnson, E.Weber. Orthogonal wavelet frames and vector-valued wavelet transforms [J]. Applied and Computation Harmonic Analysis,2007,23(2): 215-234.
[42]B.Han, Q.Mo. Symmetric MRA tight wavelet frames with three generators and high vanishing moments [J]. Applied and Computation Harmonic Analysis,2005,18(1): 67-93.
[43]Z.H.Zhang, N.Saito. Constructions of periodic wavelet frames using extension prici-ples [J]. Applied and Computation Harmonic Analysis,2009,27(1):12-23.
[44]S.Li. A theory of generalized multiresolution structure and pseudo-frames of trans-lates [J]. Journal of Fourier Analysis Applications,2001,7(1):23-40.
[45]A.Ron. Z.Shen. Affine systems in L2(Rd):The analysis of the analysis operator [J]. Functional Analysis in Clinical Treatment,1997,148(2):408-447.
[46]M.Ehler. On Multivariate Compactly Supported Bi-Frames [J]. Journal of Fourier Analysis Applications,2007,13(5):511-532.
[47]B.Han, Q.Mo. Multiwavelet frames from refinable function vectors [J]. Advances Computational Mathematics,2003,18(2):211-245.
[2]程正兴,杨守志,冯晓霞.小波分析的理论、算法、进展和应用[M].北京：国防工业出版社,2007.
[3]邸继征.小波分析原理[M].北京：科学出版社,2010.
[4]冯象初,甘小冰,宋国乡.数值泛函与小波理论[M].西安：西安电子科技大学出版社,2003.
[5]A.Boggess, F.J.Narcowich. A First Course in Wavelets with Fourier Analysis [M]. Prentice Hall:Pearson Education,2001.
[6]许天舟,黄春光.小波分析理论、算法及其应用[M].北京：国防工业出版社,2007.
[7]李建平.小波分析信息传输基础[M].北京：国防工业出版社,2004.
[8]胡广书.数字信号处理理论、算法与实现(第二版)[M].北京：清华大学出版社,2003.
[9]程正兴,林勇平.小波分析在图像处理中的应用[J].工程数学学报,2001,18(S1)：57-86.
[10]M. Bownik, Z.Rzeszotnik, D.Speegle. A characterization of dimension function of orthogonal wavelets [J]. Applied and Computation Harmonic Analysis,2001,10(1): 71-92.
[11]P.J.Burt, E.H.Adelson. The Laplacian pyramid as a compact image code [J]. IEEE Communications,1983,31(4):532-540.
[12]H.Kaneko, D.R.Noren, B.Novaprateep. Wavelet application to the Petrov-Galerkin method for Hamme-Rstein equations [J]. Applied and Numerical Mathematics,2003, 45(2):255-273.
[13]M.Vetterli. Wavelets, Approximation and Compression [J]. IEEE Signal Processing Magazine,2001,18(5):59-73.
[14]B.Han, Z.Shen. Wavelets from the loop scheme [J]. Fourier Analysis Application. 2005,11(6):615-637.
[15]S.Grgic, K.Kers, M.Grgic. Image Compression Using Wavelets [C]. Bled. Slovenia. 1999(1):99-104.
[16]S.Mallat. A wavelets tour of signal processing [M]. Academic Press,1999.
[17]C.K.Chui, J.A.Liana. A study on Orthogonal Multiwavelets [J]. Applied Numerical Mathematics,1996,20(3):273-298.
[18]C.K.Chui. An Introduction to wavelets [M]. New York:Academic press Inc,1992.
[19]A.Cohen, I.Daubechies. A new technique to estimate the regularity of refinable functions [J]. Revista Matematica Iberoamericana,1996,12(2):527-591.
[20]A.Cohen, I.Daubechies, J.C. Feauveau. Biorthogonal bases of compactly supported wavelets [J]. Communications on pure and Applied Mathematics,1992,45(5):485-560.
[21]I.Daubechies. Ten lectures on wavelets [M]. CBMS-NSF series in applied math-emitics, Philadelphia:SIAM press,1992.
[22]O.Christensen. An introduction to frames and Riesz bases[M]. Boston:Birkhauser, 2002.
[23]曲长文,何友,刘卫华,李楠.框架理论及应用[M].北京：国防工业出版社,2009.
[24]J.J.Benedetto, S.Li. The theory of multi-resolution analysis frames and application to filter banks[J]. Applied and Computation Harmonic Analysis,1998,5(4):389-427.
[25]Ferreira, P.J.S.G. Mathematics for Multimedia Signal Processing II:Discrete Finite Frames and Signal Reconstruction [C]. Signal Pocessing for Multimedia. IOS Press, 1999.
[26]O.Christensen. Moment problems for frames and applications to irregular Sampling and Gabor frames [J]. Applied and Computation Harmonic Analysis,1996,3(1):82-86.
[27]辛友明.框架多分辨分析谱的研究[J].华东师范大学学报,2006(5)：59-65.
[28]H.O.Kim, R.Y.Kim, J.K.Lim. On the spectrums of frame multiresolution analysis [J]. J. Math. Anal. Appl.,2005,305(2):528-545.
[29]Z.W.Liu, G.E.Hu, G.C.Wu. Frame scaling function sets and frame wavelet sets in Rd [J]. Chaos, Solitons and Fractals,2009,40(15):2483-2490.
[30]Z.W.Liu, G.E.Hu, G.C.Wu, B.Jing. Semi-orthogonal frame wavelets and Parseval frame wavelets associated with GMRA [J]. Chaos. Solitons and Fractals,2008,38(5): 1449-1456.
[31]D.Bakic, I.Krishtal, E.N.Wilson. Parseval frames wavelets with En2-dilations [J]. Applied and Computation Harmonic Analysis,2005,19(3):386-431.
[32]G.C.Wu, Z.X.Cheng, D.F.Li, F.J.Zhang. Parseval frame wavelets associated with A-FMRA [J]. Chaos, Solitons and Fractals,2008,37(4):1233-1243.
[33]M.Skopina. On construction of multivariate wavelet with vanishing moments [J]. Applied and Computation Harmonic Analysis,2006,20(3):375-390.
[34]M.Skopina. On construction of multivariate wavelet frames [J]. Applied and Com-putation Harmonic Analysis,2009,27(1):55-72.
[35]M.J.Lai, J.Stockler. Construction of multivariate compactly supported tight wavelet frames [J]. Applied and Computation Harmonic Analysis,2006,21(3):324-348.
[36]M.Ehler. B.Han. Wavelet bi-frames with few generators from multivariate refinablc functions [J]. Applied and Computation Harmonic Analysis,2008,25(3):407-414.
37] M.Charina, C.K.Chui, W.J.He. Tight frames of compactly supported multivari-ate multi-wavelets [J]. Journal of Computational and Applied Mathematics,2010, 233(8):2044-2061.
[38]M.Ehler. On Multivariate Compactly Supported Bi-Frames [J]. Journal Fourier Analysis and Applications,2007,13(5):512-532.
[39]M.Bownik. A characterization of affine dual frames in L2(Rn) [J]. Applied and Computation Harmonic Analysis,2000,8(2):203-221.
[40]I.Daubechies, B.Han, A.Ron, Z.Shen. Framelets:MRA-based constructions of wavelet frames [J]. Applied and Computation Harmonic Analysis,2003,14(1):1-46.
[41]G.Bhatt, B.D.Johnson, E.Weber. Orthogonal wavelet frames and vector-valued wavelet transforms [J]. Applied and Computation Harmonic Analysis,2007,23(2): 215-234.
[42]B.Han, Q.Mo. Symmetric MRA tight wavelet frames with three generators and high vanishing moments [J]. Applied and Computation Harmonic Analysis,2005,18(1): 67-93.
[43]Z.H.Zhang, N.Saito. Constructions of periodic wavelet frames using extension prici-ples [J]. Applied and Computation Harmonic Analysis,2009,27(1):12-23.
[44]S.Li. A theory of generalized multiresolution structure and pseudo-frames of trans-lates [J]. Journal of Fourier Analysis Applications,2001,7(1):23-40.
[45]A.Ron. Z.Shen. Affine systems in L2(Rd):The analysis of the analysis operator [J]. Functional Analysis in Clinical Treatment,1997,148(2):408-447.
[46]M.Ehler. On Multivariate Compactly Supported Bi-Frames [J]. Journal of Fourier Analysis Applications,2007,13(5):511-532.
[47]B.Han, Q.Mo. Multiwavelet frames from refinable function vectors [J]. Advances Computational Mathematics,2003,18(2):211-245.