二维非可分MRA Parseval框架小波的构造
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摘要
设A是一个2×2的模2(如果矩阵行列式的绝对值是2,则称该矩阵是模2的)整数膨胀矩阵(也即它的特征值的模大于1,且其阵元都是整数)。设ψ(t)∈L2(R2),如果{Aj/2ψ(Ajt-k)：j∈Z,k∈z}是L2)的框架界为1的紧框架,则称ψ(t)是L2(R2)中的A-扩张Parseval框架小波。本文的主要工作是对L2(R2)中模2的具有多分辨分析(MRA)的非可分Parseval框架小波的构造进行研究。利用Parseval框架小波乘子给出了构造二维非可分MRA Parseval框架小波的构造方法,而且对二维空间中的Parseval框架小波乘子进行系统刻画。另外我们给出了构造二维非可分紧支集MRA框架小波的显示算法,并且利用实例对比了二维非可分小波框架与可分小波框架在图象处理中的优越性。全文由六部分组成,其中第五、六部分是本文的主体部分。第五部分对L2(R2)中模2的具有多分辨分析(MRA)的非可分Parseval框架小波的构造以及Parseval框架小波乘子进行系统研究。第六部分我们给出了二维非可分MRA紧支集Parseval框架小波的构造算法,并且与可分离框架小波对比了在图象分析中的优越性。
Let A be a 2×2 integer expand matrix (its eigenvalue's modulus is more than 1 and its entries are integers) and its determinant's absolute value is 2. Letψ(t)∈L2(R2),ψ(t) is called an A-dilation Parseval Frame wavelet if﹛A2/1(A1t-k):j∈Z,k∈Z2} is an A-dilation Parseval Frame for L2(R2). In this paper, we discuss construction methods of A-dilation nonseparable Parseval MRA Frame wavelets and character A-dilation Parseval Frame wavelets multipliers for L2(R2). There are six parts in the paper. The main results in the last two parts. We give a detail discussion on bidimensional nonseparable MRA Frame wavelets in the fifth part and give a numerical algorithm of bidimensional nonseparable MRA Frame wavelets with compact support in the last part. We also show the superiority of nonseparable Frame wavelets with compact support by contrasting them with separable frames in image processing.
Let A be a 2×2 integer expand matrix (its eigenvalue's modulus is more than 1 and its entries are integers) and its determinant's absolute value is 2. Letψ(t)∈L2(R2),ψ(t) is called an A-dilation Parseval Frame wavelet if﹛A2/1(A1t-k):j∈Z,k∈Z2} is an A-dilation Parseval Frame for L2(R2). In this paper, we discuss construction methods of A-dilation nonseparable Parseval MRA Frame wavelets and character A-dilation Parseval Frame wavelets multipliers for L2(R2). There are six parts in the paper. The main results in the last two parts. We give a detail discussion on bidimensional nonseparable MRA Frame wavelets in the fifth part and give a numerical algorithm of bidimensional nonseparable MRA Frame wavelets with compact support in the last part. We also show the superiority of nonseparable Frame wavelets with compact support by contrasting them with separable frames in image processing.
引文
[1]E.Belogay,Y.Wang,Arbitrarily smooth orthogonal nonseparable wavel ets in R2,SIAMJ.Math.Anal.1999,30(3):678-697.
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[15]胡昌华,张军波,夏军,张伟.基于MATLAB的系统分析与设计-小波分析.西安电子科技大学出版社.2001,9.
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[17]张志涌等.精通MATLAB6.5版.北京航空航天大学出版社.2006,3.
[18]纪跃波.多分辨时频分析理论与多功能时频分析系统的研究.重庆大学博士学位论文.2002,10(4).
[19]Y.Li.On a class of bidimensional nonseparable wavelet multipliers.J. Math.Anal.Appl.270 2002:543-560.
[20]徐长发,李国宽.实用小波方法.华中科技大学出版社.2001,7.
[21]杨建国.小波分析及其工程应用.机械工业出版社.2005.
[22]王公宝,马伟明,吴旭升.连续小波变换及小波框架算子的一些性质. 应用泛函分析学报.2003,3,vol.5(1)：64-68.
[23]Lothar Berg,Gerlind Plonka.Compactly supporter solutions of two-sc-ale difference equations.Linear Algebra and its Applications.1998:49-75.
[24]张红岩,张登攀.图像边缘检测二维小波算法研究与实现.中国工程科学.2003,4,vol.5(4)：61-64.
[25]章春娥,裘正定.二维各向同性不可分小波变换特性分析.北京交通大学学报.2006,10,vol.30(5)：24-28.
[26]龙瑞麟.高维小波分析.世界图书出版公司.1995.
[27]C.K.Chui.Wavelets:A mathematical tool for signal analysis.SIAM,Ph-iladelphia.1997.
[28]The wutam consortium basic properties of wavelets. J.Fourier Anal Appl.1998,vol.4:575-594.
[29]Silong.Peng.Construction of two-dimensional compactly supported orthogonal wavelets filters with linear phase[J].Acta Mathematica Sinica.2002,vol.18:719-726.
[30]W.He,M.J.Lai.Examples of bivariate non-separable continuous comp-actly supported orthonormal wavelets.Proceedings of SPIE.1997,vol. 3169.
[31]成礼智,罗永,王红霞.小波的理论及应用.科学出版社.2004.
[32]J.Kovacevic,M.Vetterli.Nonseparable multidimentional perfect recon-struction filter-banks IEEE Trans Inform Theory 1992,vol.38:533-555.
[33]S.Mallat.Multifrequency channel decomposition of images and wave-let models.IEEE Trans on Patter Analysis and Machine Intelligence. 1989,vol.12(37):2091-2110.
[34]杨美香,丁宜浩.关于多分辨分析的定义.广西科学.2006,13(3)：199-202.
[35]Papadalcis,M.Sikic.H,Weiss.G.The Characterizetion of lowpass filters and related concepts.J.Fourier Anal Appl.1999,vol.5:495-521.
[36]Lizhong.Peng,Haihui.Wang.Construction for a class of smooth tight waveletframes.Sciences in China(Ser.F.English).46:6(2003):445-458.
[37]李弼程,罗建书.小波分析及其应用.电子工业出版社.2003.
[38]Qiuhui.Chen,Charles.A.Micchelli,Silong.Peng,Yuesheng.Xu.Multiva-riate filter banks having a matrix factorization.SIAM Journal on Matr-ix Analysis and its Application.2003,vol.2(25):517-531.
[39]Ning.Bi,Bin.Han,Zuowei.Shen.Examples of refinable componentwise polynomials.Applied and Computational Harmonic Analysis.2007,vol. 22:368-373.
[40]陈清江,李杰.二元小波紧框架存在的充分条件.云南大学学报.2008,30(3)：217-223.
[2]A.Cohen,I.Daubechies.Non-separable bidimensional wavelet bases. Rev. Mat.Iberoamericana.1993,vol.9:51-137.
[3]Zhongyan Li,X.Dai,Y.Diao,J.Xin. Multipliers Phases and Connectivity of MRA Wavelets in L2(R2) J Fourier and Anal Appl.2010,16: 155-176.
[4]Hernandez.E,Weiss,G.A first course on wavelets.CRC Press LLC,Boca Raton,FL.1996.
[5]W.Lawton.Tight frames of compactly supported affine wavelets. J. Mat h.Phys.1990,vol.31:1898-1901.
[6]B.Han.On dual wavelet tight frames,Appl.Comput.Harmonic Anal. 1997,vol.4:380-413.
[7]杨守志,张可村,杨建伟.二维小波函数值的快速算法.西安交通大学学报.2001,6,vo1.35(6)：637-639.
[8]张平文.小波函数值的计算.计算数学.1995,5,vo1.2：173-185.
[9]栾秀珍,王百锁.紧支小波的研究.大连海事大学学报.1999,11,vo1.25(4)：91-93.
[10]程正兴.小波分析及应用实例.西安交通大学出版社.2006.
[11]I.Daubechies.Ten lectures on wavelets.Society for Industrial and Applied Mathematics,Philadelphia,1992.
[12]Joseph O.Chapa,Raghuveer M.Rao.Algorithms for designing wavelets to match a specified signal.IEEE Transactions on signal processing. 2000,12,vol.48(12):3395-3406.
[13]C.K.Chui.An introduction to wavelets.Academic Press,Boston.1992.
[14]孙彦奎.小波分析及其应用.机械工业出版社.2004.
[15]胡昌华,张军波,夏军,张伟.基于MATLAB的系统分析与设计-小波分析.西安电子科技大学出版社.2001,9.
[16]李建平等.从傅立叶分析到小波分析：回顾与发展.计算机科学,1999,26(12).
[17]张志涌等.精通MATLAB6.5版.北京航空航天大学出版社.2006,3.
[18]纪跃波.多分辨时频分析理论与多功能时频分析系统的研究.重庆大学博士学位论文.2002,10(4).
[19]Y.Li.On a class of bidimensional nonseparable wavelet multipliers.J. Math.Anal.Appl.270 2002:543-560.
[20]徐长发,李国宽.实用小波方法.华中科技大学出版社.2001,7.
[21]杨建国.小波分析及其工程应用.机械工业出版社.2005.
[22]王公宝,马伟明,吴旭升.连续小波变换及小波框架算子的一些性质. 应用泛函分析学报.2003,3,vol.5(1)：64-68.
[23]Lothar Berg,Gerlind Plonka.Compactly supporter solutions of two-sc-ale difference equations.Linear Algebra and its Applications.1998:49-75.
[24]张红岩,张登攀.图像边缘检测二维小波算法研究与实现.中国工程科学.2003,4,vol.5(4)：61-64.
[25]章春娥,裘正定.二维各向同性不可分小波变换特性分析.北京交通大学学报.2006,10,vol.30(5)：24-28.
[26]龙瑞麟.高维小波分析.世界图书出版公司.1995.
[27]C.K.Chui.Wavelets:A mathematical tool for signal analysis.SIAM,Ph-iladelphia.1997.
[28]The wutam consortium basic properties of wavelets. J.Fourier Anal Appl.1998,vol.4:575-594.
[29]Silong.Peng.Construction of two-dimensional compactly supported orthogonal wavelets filters with linear phase[J].Acta Mathematica Sinica.2002,vol.18:719-726.
[30]W.He,M.J.Lai.Examples of bivariate non-separable continuous comp-actly supported orthonormal wavelets.Proceedings of SPIE.1997,vol. 3169.
[31]成礼智,罗永,王红霞.小波的理论及应用.科学出版社.2004.
[32]J.Kovacevic,M.Vetterli.Nonseparable multidimentional perfect recon-struction filter-banks IEEE Trans Inform Theory 1992,vol.38:533-555.
[33]S.Mallat.Multifrequency channel decomposition of images and wave-let models.IEEE Trans on Patter Analysis and Machine Intelligence. 1989,vol.12(37):2091-2110.
[34]杨美香,丁宜浩.关于多分辨分析的定义.广西科学.2006,13(3)：199-202.
[35]Papadalcis,M.Sikic.H,Weiss.G.The Characterizetion of lowpass filters and related concepts.J.Fourier Anal Appl.1999,vol.5:495-521.
[36]Lizhong.Peng,Haihui.Wang.Construction for a class of smooth tight waveletframes.Sciences in China(Ser.F.English).46:6(2003):445-458.
[37]李弼程,罗建书.小波分析及其应用.电子工业出版社.2003.
[38]Qiuhui.Chen,Charles.A.Micchelli,Silong.Peng,Yuesheng.Xu.Multiva-riate filter banks having a matrix factorization.SIAM Journal on Matr-ix Analysis and its Application.2003,vol.2(25):517-531.
[39]Ning.Bi,Bin.Han,Zuowei.Shen.Examples of refinable componentwise polynomials.Applied and Computational Harmonic Analysis.2007,vol. 22:368-373.
[40]陈清江,李杰.二元小波紧框架存在的充分条件.云南大学学报.2008,30(3)：217-223.