小波基的一对Hilbert变换
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摘要
本文考虑了小波形成一对Hilbert变换的一对小波基的设计。该推导是基于无限乘积公式定义的极限函数。结果发现,尺度滤波器应该相互抵消(偏移)一个半采样。这给出了Kingsbury结果的替代推导和解释,即当尺度滤波器满足相同的偏移时,双树复小波变换是几乎平移不变的。
This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an alternative derivation and explanation for the result by Kingsbury,that the dual-tree wavelet transform is nearly shift-invariant when the scaling filters satisfy the same offset.
This paper considers the design of pairs of wavelet bases where the wavelets form a Hilbert transform pair. The derivation is based on the limit functions defined by the infinite product formula. It is found that the scaling filters should be offset from one another by a half sample. This gives an alternative derivation and explanation for the result by Kingsbury,that the dual-tree wavelet transform is nearly shift-invariant when the scaling filters satisfy the same offset.
引文
[1]吐尔洪江·阿布都克力木.小波信号处理基础[M].北京:北京邮电大学出版社,2014.
[2] Jean-Marc Lina.Image Processing with ComplexDaubechies Wavelet[J]. Journal of MathematicalImaging and Vision,1997,7(3):211-223.
[3] Xueqiao Wang,Qiuqi Rua Yi,Jin Gaoyun. Anthree-dimensional face recognition based on dual-tree complex wavelet transform and weighted spher-ical face representation.978-1-4673-2197-6/12/IEEE.Beijing Key Laboratory of Advanced Informa-tion Science and Network Technology Institution ofinformation Science[D].Beijing Jiaotong University,2012.
[4] E Ozturk,O Kucur,and G Atkin,”Waveformencoding of binary signals using a wavelet and itsHilbert transform”in Proc. IEEE Int. Conf[C]//Acoust. Speech, Signal Processing(ICASSP),Istanbul,Turkey,2000.
[5] W T Freeman and E H Adelson.The design and useof steerable filters[J]. IEEE Trans. Pattern Anal.Machine Intell,1991(13):891-906.
[6] G Beylkin and B Torresani. Implementation ofoperators via filter banks:Autocorrelation shell andhardy wavelets. Appl. Comput. Harmon. Anal,1996(3):164-185.
[7]成礼智,王红霞,罗永,等.小波的理论与应用[M].北京:科学出版社,2004.
[8]苗晟,王威廉,姚绍文,Hilbert-Huang变换发展历程及其应用[J].电子测量与仪器学报,2014,28(8):812-818.
[9]徐晓刚,徐冠雷,王孝通,等.二维Hilbert变换研究[J],通信技术,2016,49(10):1265-1270.
[10]R YU,H Ozkaramanli. Hilbert transform Pairs ofbiorthogonal wavelet bases[J]. IEEE Trans. SignalProcess,2016,54(6):2119-2125.
[11]Selvaraaju Murugesan,David B H Tay. A newclass of almost symmetric orthogonal Hilbert pairof wavelets. Signal Processing,2014(95):76-87.
[12]刘文涛,陈红,蔡晓霞,等.基于双树复小波变换的信号去噪算法[J].火力与指挥控制,2014,39(12):84-87.
[13]陈彬强,张周锁,郭婷,等.双树复小波在齿轮系装配间隙检测的应用[J].西安交通大学学报,2013,47(3):7-12.
[14]石宏理,胡波,双树复小波变换及其应用综述[J].信息与电子工程,2007(3):229-234.
[15]赵杰.基于Hilbert曲线和小波变换的图像分割[J],吉林工程技术师范学院学报,2013,29(1):77-80.
[16]李万社,田丽伟.互为Hilbert变换对的对偶树复小波的构造[J].云南师范大学学报,2011,31(1):4-8.
[17]薛婷婷,刘秀平,张凯兵,等.基于学习的Gabor滤波器多样式布匹瑕疵检测[J].西安工程大学学报,2018(6):775-780.
[18]李建伟.基于FPGA的高效可调频率滤波器[J].现代电子技术,2017(23):93-96.
[2] Jean-Marc Lina.Image Processing with ComplexDaubechies Wavelet[J]. Journal of MathematicalImaging and Vision,1997,7(3):211-223.
[3] Xueqiao Wang,Qiuqi Rua Yi,Jin Gaoyun. Anthree-dimensional face recognition based on dual-tree complex wavelet transform and weighted spher-ical face representation.978-1-4673-2197-6/12/IEEE.Beijing Key Laboratory of Advanced Informa-tion Science and Network Technology Institution ofinformation Science[D].Beijing Jiaotong University,2012.
[4] E Ozturk,O Kucur,and G Atkin,”Waveformencoding of binary signals using a wavelet and itsHilbert transform”in Proc. IEEE Int. Conf[C]//Acoust. Speech, Signal Processing(ICASSP),Istanbul,Turkey,2000.
[5] W T Freeman and E H Adelson.The design and useof steerable filters[J]. IEEE Trans. Pattern Anal.Machine Intell,1991(13):891-906.
[6] G Beylkin and B Torresani. Implementation ofoperators via filter banks:Autocorrelation shell andhardy wavelets. Appl. Comput. Harmon. Anal,1996(3):164-185.
[7]成礼智,王红霞,罗永,等.小波的理论与应用[M].北京:科学出版社,2004.
[8]苗晟,王威廉,姚绍文,Hilbert-Huang变换发展历程及其应用[J].电子测量与仪器学报,2014,28(8):812-818.
[9]徐晓刚,徐冠雷,王孝通,等.二维Hilbert变换研究[J],通信技术,2016,49(10):1265-1270.
[10]R YU,H Ozkaramanli. Hilbert transform Pairs ofbiorthogonal wavelet bases[J]. IEEE Trans. SignalProcess,2016,54(6):2119-2125.
[11]Selvaraaju Murugesan,David B H Tay. A newclass of almost symmetric orthogonal Hilbert pairof wavelets. Signal Processing,2014(95):76-87.
[12]刘文涛,陈红,蔡晓霞,等.基于双树复小波变换的信号去噪算法[J].火力与指挥控制,2014,39(12):84-87.
[13]陈彬强,张周锁,郭婷,等.双树复小波在齿轮系装配间隙检测的应用[J].西安交通大学学报,2013,47(3):7-12.
[14]石宏理,胡波,双树复小波变换及其应用综述[J].信息与电子工程,2007(3):229-234.
[15]赵杰.基于Hilbert曲线和小波变换的图像分割[J],吉林工程技术师范学院学报,2013,29(1):77-80.
[16]李万社,田丽伟.互为Hilbert变换对的对偶树复小波的构造[J].云南师范大学学报,2011,31(1):4-8.
[17]薛婷婷,刘秀平,张凯兵,等.基于学习的Gabor滤波器多样式布匹瑕疵检测[J].西安工程大学学报,2018(6):775-780.
[18]李建伟.基于FPGA的高效可调频率滤波器[J].现代电子技术,2017(23):93-96.