AB小波的MRA及双正交AB小波
摘要
20世纪80年代以来，小波分析在数学领域中迅速发展，这主要因为两方面原因：第一，它有着深刻的理论背景；第二，它在工程中的应用十分广泛，比如信号分析，图像处理等。但是在实践中我们也发现，小波分析在处理二维甚至更高维信号的突变处如边缘时，效果并不是十分理想，这主要是因为高维小波是一维小波通过张量积的方式得到的，方向性较差，而信号的边缘等突变处包含了高度各向异性的对象。为了克服小波分析的这种局限性，研究者们发现了许多具有方向性的小波。
2004年，Gitta Kutyniok, Demetrio Labate, Wang-Q Lim等等研究者们发现了复合伸缩小波（wavelets with composite dilations），由于该小波的构造基于控制伸缩及方向的矩阵A,B。所以称其为AB小波。AB小波中，有一类为正交小波，以二维情形为例，Demetrio Labate等建立了有三个母函数生成的的正交AB L2 (R 2 )多小波。由一个母函数生成的AB小波称为Shearlet。已有定理指出，Shearlet不能构成L2 (R 2 )的标准正交基。
AB小波不仅具有很好的方向性，能准确检测出信号突变处的方位，如边缘部分，而且相应的级数展开能有效表示信号的信息，因此，AB小波近年来受到很大的关注，有关这类小波的数学理论也在不断地建立与完善。
研究小波的一个重要工具是多尺度分析，即MRA，但是由于不同的AB小波由不同数量和类型的母函数生成，所以不能统一建立AB小波的多尺度分析理论。对于三个母函数生成的正交AB小波以及Shearlet，Gitta Kutyniok, Demetrio Labate等给出了相应的MRA，对于其它类型的AB小波，目前还没有建立MRA理论。正交小波的优越性是明显的，但如前所述，Shearlet缺少正交性，而由于前述正交AB小波有三个母函数生成，由此增加了其结构的复杂性。在不能建立正交小波的情况下，人们自然考虑建立双正交小波，因此建立双正交AB小波理论是非常有意义的。
本文研究了给出以下结果：(1)对一类非正交的AB小波建立了MRA结构；(2)研究了一类AB小波双正交理论，探索相应滤波器满足的关系；(3)建立该类AB小波双正交结构下的Mallat算法；(4)在上述的Mallat算法的基础上，为了能便于计算，研究了抽样值算法，并估计了误差；(5)研究了双正交AB小波的提升格式，并建立相应的函数分解与重构公式。
Since 1980s, wavelet analysis has been developing rapidly inmathematics, duing to its important role both in mathematical theory andapplications. It has been widely applied in many fields, such as signalprocessing, digital image processing and so on. But in practical applications,we find that it is not very effective in dealing with multidimensionalsignals containing distributed discontinuities such as edges, formultidimensional wavelet is formed by tensor product leading to its poordirectional sensitivity and the edges of signals are containing highdirectional objects of different traits. To overcome this limitation of waveletanalysis, many researchers have found many directional wavelets.
In 2004, Gitta Kutyniok, Demetrio Labate and Wang-Q Lim introducedwavelets with composite dilations based on scaling matrix A and shearmatrix B, so we call it AB-wavelet. Among AB wavelets, for twodimensions instance, there is one kind of AB wavelets established by Demetrio Labate etc. containing three generators forms the orthogonal basisof . We call the AB-wavelet constructed by one generator ShearletL2 (R 2 ) .And there is theorem pointing out that Shearlet can not form the orthogonalbasis of L2 (R 2 ) .
AB-wavelet not only has good directional sensitivity, detecting thelocation of discontinuities such as edges accurately, but also is efficient inrepresenting multidimensional signals. So in recent years, this kind ofwavelets is widely explored and getting complete.
One of the most important tools to study wavelet is multiscale analysis,i.e. MRA. Owing to the functions forming different AB-wavelets are ofdifferent amounts and traits, it is impossible to establish the multiscaleanalysis theory of AB-wavelet uniformly. For the orthogonal AB-waveletformed by three generators, there is corresponding MRA has been given byGitta Kutyniok, Demetrio Labate etc. But as to other types of AB-wavelets,there is not.
The advantage of orthogonal wavelets is obvious, and the abovediscussion show that shearlet is lack of orthogonality. ON AB-waveletformed by three generators increases complexity in the structure. When ONAB-wavelet is unavailable, it is significant for us to consider constructingbiorthogonal wavelets, also biorthogonal AB-wavelets.
The main results, obtained in this dissertation, may be summarized asfollows:(1) We construct MRA structure using one kind of nonorthogonal AB-wavelets;(2) We study the biorthogonal theory of one kind of ABwavelets,and explore the relation between the corresponding filters; (3) Weestablish the Mallat algorithm under the structure constructed above; (4) Westudy sampling algorithm based on Mallat algorithm;(5) We promote thebiorthogonal AB-wavelets, and establish the formulations fordecomposition and reconstruction.
2004年，Gitta Kutyniok, Demetrio Labate, Wang-Q Lim等等研究者们发现了复合伸缩小波（wavelets with composite dilations），由于该小波的构造基于控制伸缩及方向的矩阵A,B。所以称其为AB小波。AB小波中，有一类为正交小波，以二维情形为例，Demetrio Labate等建立了有三个母函数生成的的正交AB L2 (R 2 )多小波。由一个母函数生成的AB小波称为Shearlet。已有定理指出，Shearlet不能构成L2 (R 2 )的标准正交基。
AB小波不仅具有很好的方向性，能准确检测出信号突变处的方位，如边缘部分，而且相应的级数展开能有效表示信号的信息，因此，AB小波近年来受到很大的关注，有关这类小波的数学理论也在不断地建立与完善。
研究小波的一个重要工具是多尺度分析，即MRA，但是由于不同的AB小波由不同数量和类型的母函数生成，所以不能统一建立AB小波的多尺度分析理论。对于三个母函数生成的正交AB小波以及Shearlet，Gitta Kutyniok, Demetrio Labate等给出了相应的MRA，对于其它类型的AB小波，目前还没有建立MRA理论。正交小波的优越性是明显的，但如前所述，Shearlet缺少正交性，而由于前述正交AB小波有三个母函数生成，由此增加了其结构的复杂性。在不能建立正交小波的情况下，人们自然考虑建立双正交小波，因此建立双正交AB小波理论是非常有意义的。
本文研究了给出以下结果：(1)对一类非正交的AB小波建立了MRA结构；(2)研究了一类AB小波双正交理论，探索相应滤波器满足的关系；(3)建立该类AB小波双正交结构下的Mallat算法；(4)在上述的Mallat算法的基础上，为了能便于计算，研究了抽样值算法，并估计了误差；(5)研究了双正交AB小波的提升格式，并建立相应的函数分解与重构公式。
Since 1980s, wavelet analysis has been developing rapidly inmathematics, duing to its important role both in mathematical theory andapplications. It has been widely applied in many fields, such as signalprocessing, digital image processing and so on. But in practical applications,we find that it is not very effective in dealing with multidimensionalsignals containing distributed discontinuities such as edges, formultidimensional wavelet is formed by tensor product leading to its poordirectional sensitivity and the edges of signals are containing highdirectional objects of different traits. To overcome this limitation of waveletanalysis, many researchers have found many directional wavelets.
In 2004, Gitta Kutyniok, Demetrio Labate and Wang-Q Lim introducedwavelets with composite dilations based on scaling matrix A and shearmatrix B, so we call it AB-wavelet. Among AB wavelets, for twodimensions instance, there is one kind of AB wavelets established by Demetrio Labate etc. containing three generators forms the orthogonal basisof . We call the AB-wavelet constructed by one generator ShearletL2 (R 2 ) .And there is theorem pointing out that Shearlet can not form the orthogonalbasis of L2 (R 2 ) .
AB-wavelet not only has good directional sensitivity, detecting thelocation of discontinuities such as edges accurately, but also is efficient inrepresenting multidimensional signals. So in recent years, this kind ofwavelets is widely explored and getting complete.
One of the most important tools to study wavelet is multiscale analysis,i.e. MRA. Owing to the functions forming different AB-wavelets are ofdifferent amounts and traits, it is impossible to establish the multiscaleanalysis theory of AB-wavelet uniformly. For the orthogonal AB-waveletformed by three generators, there is corresponding MRA has been given byGitta Kutyniok, Demetrio Labate etc. But as to other types of AB-wavelets,there is not.
The advantage of orthogonal wavelets is obvious, and the abovediscussion show that shearlet is lack of orthogonality. ON AB-waveletformed by three generators increases complexity in the structure. When ONAB-wavelet is unavailable, it is significant for us to consider constructingbiorthogonal wavelets, also biorthogonal AB-wavelets.
The main results, obtained in this dissertation, may be summarized asfollows:(1) We construct MRA structure using one kind of nonorthogonal AB-wavelets;(2) We study the biorthogonal theory of one kind of ABwavelets,and explore the relation between the corresponding filters; (3) Weestablish the Mallat algorithm under the structure constructed above; (4) Westudy sampling algorithm based on Mallat algorithm;(5) We promote thebiorthogonal AB-wavelets, and establish the formulations fordecomposition and reconstruction.
引文
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[2]郭懋正.实变函数与泛函分析[M].北京大学出版社,北京,2005.
[3]成礼智,王红霞,罗永.小波的理论与应用[M].科学出版社,北京, 2004.
[4]陈武凡.小波分析及其在图像处理中的应用[M].科学出版社,北京, 2002.
[5]王长清.近代解析应用数学基础[M].西安电子科技大学出版社,西安,2001.
[6]程正兴.小波分析算法与应用[M].西安交通大学出版社,西安, 1978.
[7]陈仲英,巫斌.小波分析[M].科学出版社,北京, 2007.
[8] Daubechies I.著,李建平,杨万年译.小波十讲[M].国防工业出版社,北京, 2004.
[9]关履泰.小波方法与应用[M].高等教育出版社,北京, 2007.
[10]李弼程,罗建书.小波分析及其应用[M].电子工业出版社,北京, 2003.
[11]龙瑞麟.高维小波分析[M].世界图书出版社,北京, 1995.
[12]李世雄.小波变换及其应用[M].高等教育大学出版社,北京, 1997.
[13]冉启文.小波分析方法及其应用[M].哈尔滨工业大学出版社,哈尔滨, 1995.
[14]冉启文.小波变换与分数傅里叶变换理论及其应用[M].哈尔滨工业大学出版社,哈尔滨, 2001.
[15]唐远炎,王玲.小波分析与文本文字识别[M].科学出版社,北京, 2004.
[16]徐佩霞,孙功宪.小波分析与应用实例[M].中国科学技术大学出版社,合肥,2001.
[17]朱长青.小波分析理论与影像分析[M].测绘出版社,北京, 1998.
[18]闫敬文,屈小波.超小波分析及应用[M].国防工业出版社,北京,2008.
[19]谢成俊.小波分析理论及工程应用[M].东北师范大学出版社,长春,2007.
[20]孙延奎.小波分析及其应用[M].机械工业出版社,北京,2005.
[21]邸继征.小波分析原理[M].科学出版社,北京,2009.
[22] J.P.Antoine, R.Murenzi, P.Vandergheynst, Directional wavelets revisited: Cauchywavelets and symmetry detection in patterns[J]. Applied and Computational Har-monic Analysis. 6(1999) 314-345.
[23] R.H.Bamberger, M.J.T.Smith, A filter bank for directional decomposition of im-ages:theory and design[J]. IEEE Transactions on Signal Processing.40(1992)882-893.
[24] E.J.Candes, D.L.Donoho, Ridgelets: a key to higher-dimensional intermittency?[J].Philosophical Transactions of The Royal Society of London A 357(1999),2495-2509.
[25] E.J.Candes, D.L.Donoho, New tight frames of curvelets and optimal representationsof objects with piecewise C2 singularities[J]. Communications on Pure and AppliedMathematics.56(2004)216-266.
[26] R.R.Coifman, F.G.Meyer, Brushlets: a tool for directional image analysis and imagecompression[J]. Applied and Computational Harmonic Analysis. 5(1997) 147-187.
[27] F.Colonna, G.R.Easley, Generalized discrete Radon transforms and their use in theridgelet transform[J]. Journal of Mathematical Imaging and Vision,23(2005) 145-165.
[28] M.N.Do, M.Vetterli, The contourlet transform: an efficient directional mul-tiresolution image representation[J]. IEEE Transactions on Image Processing.14(2005)2091-2106.
[29] D.L.Donoho, Sparse components of images and optimal atomic decomposition[J].Constructive Approximation. 17(2001)353-382.
[30] K.Guo, D.Labate, Optimally Sparse Multidimensional Representation using Shear-lets[J]. SIAM Journal on Mathematical Analysis. 39(2007)298-318.
[31] K.Guo, W-Q.Lim, D.Labate, G.Weiss, E.Wilson, Wavelets with composite dila-tions[J]. Electronic Research Announcements of The American Mathematical Soci-ety. 10(2004) 78-87.
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