紧支撑对称—反对称正交多小波的构造
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摘要
小波分析是傅里叶分析发展170多年来对其最辉煌的继承、总结和发展,对分析工具起着承前启后、继往开来的重要作用。小波分析的理论研究是与小波分析的应用紧密的结合在一起的。多小波理论是在小波理论基础上发展起来的一种新的小波构造理论,相对于单小波具有独特的优势,从而更好的应用于信号以及图像处理。现今,小波分析已经成为科学技术研究工作的重要内容,它已经把信息工业和信息技术推向了一个新的时代。本文分别从多小波的构造理论和小波变换的应用两方面进行了研究。
在理论上,基于多小波具有单小波不能同时满足对称性、正交性、紧支撑性、高阶消失矩的优势,在多小波构造理论的基础上,利用紧支撑对称正交单小波函数,通过平移后与一个正交矩阵相乘构造一个二重紧支撑对称—反对称正交多小波,并将其推广到r重情形,从而构造出r重紧支撑对称—反对称正交多小波函数。由此构造的多小波不但具有紧支性和对称—反对称性,而且具有正交性。
在应用上,基于Canny算法和小波变换算法,提出一种图像融合的边缘检测方法。对于传统的小波变换在边缘不连续和抑制噪声能力弱的问题上进行改进,给出一种改进的小波变换方法。对原图像分别采用改进的小波变换和Canny算子两种方法进行边缘提取,再将两种方法的检测结果进行图像融合。该融合方法结合了两种边缘检测方法的优点,达到了图像边缘检测完整和定位准确的效果。
Wavelet analysis is the most glorious inheritance, summary and development of the Fourier analysis in the latest 170 years. It plays an important role in inheriting traditions and breaking new grounds for the future. Theoretical study of wavelet analysis combined closely with its applications. Multiwavelet theory is a new wavelet construction theory which developed on the basis of wavelet theory, it has a unique advantage as opposed to single wavelet, so the theory can be applied to the process of signal and image processing. Now the wavelet analysis is an important content of scientific and technological research work, it has brought the information industry and information technology to a new era. The two aspects of the structure theory of multiwavelet and application of wavelet transform are studied in this paper.
Multiwavelet has the disadvantages that the single wavelet can not simultaneously satisfy the symmetry, orthogonality, compact support and higher-order vanishing moments in theory. On the basis of multiwavelets constructed theory, the single compactly supported symmetric orthogonal wavelet function is used to construct bivariate compactly supported symmetric-antisymmetric orthogonal multiwavelets through multipling with an orthogonal matrix after translation and this method is extended to the circumstances of r multiplicity, thus compactly supported symmetric-antisymmetric orthogonal multi-wavelet function is reconstructed in this way. The constructed multiwavelets not only have the nature of compact support and symmetry-antisymmetry, but also orthogonality.
Based on the Canny algorithm and wavelet transform algorithm, an edge detection method that based on the image fusion in application is proposed in this paper. It has improved the question of the traditional wavelet transform discontinuity at the edge and the weak ability to suppress noise and gaven an improved method of wavelet transform. Two methods, wavelet transform and Canny operator, are used to extract the edge separately, and the results of the two methods are fused together. The fusion method combined the advantages of the two kinds of edge detection method to achieve complete and accurate positioning results.
在理论上,基于多小波具有单小波不能同时满足对称性、正交性、紧支撑性、高阶消失矩的优势,在多小波构造理论的基础上,利用紧支撑对称正交单小波函数,通过平移后与一个正交矩阵相乘构造一个二重紧支撑对称—反对称正交多小波,并将其推广到r重情形,从而构造出r重紧支撑对称—反对称正交多小波函数。由此构造的多小波不但具有紧支性和对称—反对称性,而且具有正交性。
在应用上,基于Canny算法和小波变换算法,提出一种图像融合的边缘检测方法。对于传统的小波变换在边缘不连续和抑制噪声能力弱的问题上进行改进,给出一种改进的小波变换方法。对原图像分别采用改进的小波变换和Canny算子两种方法进行边缘提取,再将两种方法的检测结果进行图像融合。该融合方法结合了两种边缘检测方法的优点,达到了图像边缘检测完整和定位准确的效果。
Wavelet analysis is the most glorious inheritance, summary and development of the Fourier analysis in the latest 170 years. It plays an important role in inheriting traditions and breaking new grounds for the future. Theoretical study of wavelet analysis combined closely with its applications. Multiwavelet theory is a new wavelet construction theory which developed on the basis of wavelet theory, it has a unique advantage as opposed to single wavelet, so the theory can be applied to the process of signal and image processing. Now the wavelet analysis is an important content of scientific and technological research work, it has brought the information industry and information technology to a new era. The two aspects of the structure theory of multiwavelet and application of wavelet transform are studied in this paper.
Multiwavelet has the disadvantages that the single wavelet can not simultaneously satisfy the symmetry, orthogonality, compact support and higher-order vanishing moments in theory. On the basis of multiwavelets constructed theory, the single compactly supported symmetric orthogonal wavelet function is used to construct bivariate compactly supported symmetric-antisymmetric orthogonal multiwavelets through multipling with an orthogonal matrix after translation and this method is extended to the circumstances of r multiplicity, thus compactly supported symmetric-antisymmetric orthogonal multi-wavelet function is reconstructed in this way. The constructed multiwavelets not only have the nature of compact support and symmetry-antisymmetry, but also orthogonality.
Based on the Canny algorithm and wavelet transform algorithm, an edge detection method that based on the image fusion in application is proposed in this paper. It has improved the question of the traditional wavelet transform discontinuity at the edge and the weak ability to suppress noise and gaven an improved method of wavelet transform. Two methods, wavelet transform and Canny operator, are used to extract the edge separately, and the results of the two methods are fused together. The fusion method combined the advantages of the two kinds of edge detection method to achieve complete and accurate positioning results.
引文
[1] MEYER Y. Ondelettes et Operatewrs[M]. Paris: Hermann, 1990: 187-199.
[2] DENG CAI XIA. MRA in the Space H 1[0,1][J]. Thai Journal of Mathematics, 2003, 1(1): 69-78.
[3]崔锦泰.小波分析导论[M].西安:西安交通大学出版社,1997:83-103.
[4] LOKENATH DEBNATH. Wavelet Transforms and Their Aapplications[M]. Boston: Birkhauser, 2002: 87-99.
[5] GOODMAN T N, LEE S L. Wavelets of Multiplicity r[J]. Trans. on Amer. Math. Soc., 1994, 338(2): 639-654.
[6] GERONIMO J S, HARDIN D F, MASSOPUST P R. Fractal Functions and Wavelet Expansions Based on Several Scaling Functions[J]. J. Approx. Theory, 1994, 78(14): 371-401.
[7] GERONIMO J S, HARDIN D F, MASSOPUST P R. Construction of Orthogonal Wavelets Using Fractal Interpolation Functions[J]. SIAM J. Math. Anal., 1996, 27(12): 1158-1192.
[8] STRELA. Multi-wavelet: Theory and Applications[D]. Boston: MIT, 1996: 108-210.
[9] PLONKA G, STRELA V. Construction of Multi-Scaling Functions With Approximation and Symmetry[J]. SIAM J. Math. Anal., 1998, 29(2): 481-510.
[10] LANTON W, LEE S L, SHEN Z. An Algorithm for Matrix Extension and Wavelets Constuction[J]. Math. Comp, 1996, 65(6): 723-737.
[11] MARIANTONIA COTRONEI, LAURA B, MONTEFUSCO. Multi-Wavelet Analysis and Signal Processing[J] . IEEE Trans.: on Circuits and Systems-Ⅱ: Analog and Digital Signal Processing, 1998, 45(8): 970-987.
[12]杨建伟,程正兴.正交多小波的构造[J].高等学校计算数学学报,2003,25(2):110-115.
[13] SHEN LI XIN, HWEE HUAT TAN. On a Family of Orthonormal Scalar Wavelets and Related Balanced Multi-Wavelets[J]. IEEE Trans. on Signal Processing, 2001, 49(7): 1447-1453.
[14]崔丽洪,程正兴.由给定尺度函数构造短支撑正交多小波的一种算法[J].西安交通大学学报,2003,37(2):211-214.
[15]张钦礼,吴勃英,何春江.Hermite多尺度小波[J].哈尔滨工业大学学报,2004,36(6):787-789.
[16] YANG SHOU ZHI, PENG LI ZHONG. Construction of High Order Balanced MultiScaling Functions via PTST[J]. Science in China(Series F: Information Sciences), 2006, 49(4): 504-515.
[17]蒋勇,邓彩霞,王旭.由β函数构造一种紧支撑小波[J].哈尔滨理工大学学报,2007,12(6):69-71.
[18]罗萍,郭洪林,方勤华.多重向量值双正交小波的存在性及构造[J].河北大学学报:自然科学版,2009,29(1):13-18.
[19] ROBERTS L G. Machine Perception of Three-Dimension Solids[J]. MIT Press, 1965, 39(10): 159-197.
[20] PREWITT J. Object Enhancement and Extraction, Picture Process[J]. International Journal of Computer Vision, 1970, 23(5): 75-149.
[21] SOBEL L. Camera Models and Machine Perception[J]. IEEE Transactions on Electronic Computers, 1970, 18(12): 104-197.
[22] KIRSCH R. Computer Determination of the Constituent Structure of Biological Images[J]. Computer and Biomedical Research, 1971, 18(1): 113-125.
[23] FREI W, CHEN C. Fast Boundary Detection: A Generalization and a New Algorithm[J]. IEEE Transactions on Electronic Computers, 1977, 26(2): 346-458.
[24]姜杭毅,蔡元龙.采用正交多项式曲面拟合法的边缘检测[J].自动化学报,1990,16(3):202-209.
[25] MARR D C, HILDRETH E. Theory of Edge Detection[J]. Proc. Roy. Soc. London, 1980, 27(2): 187-217.
[26]赵志刚,万娇娜.一种基于梯度和零交叉点的图像边缘检测新方法[J].仪器仪表学报,2006,27(8):821-824.
[27]濮群,余桂.用线性模型检测图像边缘[J].清华大学学报:自然科学[J].西安交通大学学报,2003,37(2):211-214.
[15]张钦礼,吴勃英,何春江.Hermite多尺度小波[J].哈尔滨工业大学学报,2004,36(6):787-789.
[16] YANG SHOU ZHI, PENG LI ZHONG. Construction of High Order Balanced MultiScaling Functions via PTST[J]. Science in China(Series F: Information Sciences), 2006, 49(4): 504-515.
[17]蒋勇,邓彩霞,王旭.由β函数构造一种紧支撑小波[J].哈尔滨理工大学学报,2007,12(6):69-71.
[18]罗萍,郭洪林,方勤华.多重向量值双正交小波的存在性及构造[J].河北大学学报:自然科学版,2009,29(1):13-18.
[19] ROBERTS L G. Machine Perception of Three-Dimension Solids[J]. MIT Press, 1965, 39(10): 159-197.
[20] PREWITT J. Object Enhancement and Extraction, Picture Process[J]. International Journal of Computer Vision, 1970, 23(5): 75-149.
[21] SOBEL L. Camera Models and Machine Perception[J]. IEEE Transactions on Electronic Computers, 1970, 18(12): 104-197.
[22] KIRSCH R. Computer Determination of the Constituent Structure of Biological Images[J]. Computer and Biomedical Research, 1971, 18(1): 113-125.
[23] FREI W, CHEN C. Fast Boundary Detection: A Generalization and a New Algorithm[J]. IEEE Transactions on Electronic Computers, 1977, 26(2): 346-458.
[24]姜杭毅,蔡元龙.采用正交多项式曲面拟合法的边缘检测[J].自动化学报,1990,16(3):202-209.
[25] MARR D C, HILDRETH E. Theory of Edge Detection[J]. Proc. Roy. Soc. London, 1980, 27(2): 187-217.
[26]赵志刚,万娇娜.一种基于梯度和零交叉点的图像边缘检测新方法[J].仪器仪表学报,2006,27(8):821-824.
[27]濮群,余桂.用线性模型检测图像边缘[J].清华大学学报:自然科学
[40]陈亮,郭雷,王雅萍.一种基于结构张量的MAS边缘检测算法[J].计算机科学,2009,36(1):131-133.
[41]徐建东,蒋野,孙迎春.基于改进的形态学算子的灰度图像边缘检测[J].佳木斯大学学报:自然科学版,2009,27(6):857-859.
[42] JOHN CANNY. A Computational Approach to Edge Detection[J]. IEEE Trans., Pattern Analysis and Machine Intelligence, 1986, 8(1): 679-697.
[43] HEMANT TAGARE D. On the Localization Performance Measure and Optimal Edge Detection[J]. IEEE Trans., Pattern Analysis and Machine Intelligence, 1990, 12(2): 1186-1190.
[44] SPACER L A. Edge Detection and Motion Detection[J]. Image Vision Comput., 1986, 4(2): 43-44.
[45] DERICHE R. Using Canny's Criteria to Derive a Recursively Implemented Optimal Edge Detector[J]. Int. J. Comput. Vision, 1987, 1(2): 56-101.
[46]刘建伟,郭平.一种基于边缘检测的图像融合新方法[J].北京师范大学学报:自然科学版,2007,43(5):518-521.
[47]程正兴.小波分析算法与应用[M].西安:交通常大学出版社,1997:3-4.
[48]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,2007:6-10,77-80,356-358.
[49]孙延奎.小波分析及其应用[M].北京:机械工业出版社,2005:40-41,177-178,199-212.
[50]樊启斌.小波分析[M].武汉:武汉大学出版社,2008:246-266.
[51]徐长发,李国宽.实用小波方法[M].武汉:华中科技大学出版社,2001:57-79.
[52]张浩,蔡晋辉,周泽魁.DS证据理论在SAR图像边缘检测中的应用[J].武汉大学学报:信息科学版,2008,33(1):105-108.
[53]周洁,蒋晓华,程流泉.基于正交T-Snake模型的冠状动脉边缘检测[J].清华大学学报:自然科学版,2009,49(1):41-44.
[54]王大凯,彭进业.小波分析及其在信号处理中的应用[M].北京:电子工业出版社,2006,8(2):148-160.
[55]李小雄,牛双国.平衡多小波的Grobner基构造及其应用[J].河南大学学报,2007,37(6):646-648.
[56] LUAN DAN, DING XUAN HAO. The Construction of a B-Spline Wavelet[J]. Journal of Guangxi Acaderny of Sciences, 2007, 23(3): 138-143.
[57] LIU JIE. Regularity and Orthogonality of a New Family of Bivariate Wavelets[J]. Journal of Lanzhou University of Technology, 2008, 34(4): 250-255.
[58]林怡,阵鹰.一种线性提升小波的方向断面边缘检测法[J].同济大学学报:自然科学版,2007,35(12):1699-1703.
[59]江力,吕勇.一个紧支撑插值正交多小波的平衡性定理[J].中山大学学报,2008,47(2):14-17.
[60] LUO HUI, DENG CAI XIA, ZHU JIAN LI. An Method of Constructing Bivarate Box-Spline[C]. Tang Yuan Yan. Proceedings of 2008 International Conference on Wavelet Analysis and Pattern Recognition, Hong Kong, 2008, 2(2): 530-534.
[2] DENG CAI XIA. MRA in the Space H 1[0,1][J]. Thai Journal of Mathematics, 2003, 1(1): 69-78.
[3]崔锦泰.小波分析导论[M].西安:西安交通大学出版社,1997:83-103.
[4] LOKENATH DEBNATH. Wavelet Transforms and Their Aapplications[M]. Boston: Birkhauser, 2002: 87-99.
[5] GOODMAN T N, LEE S L. Wavelets of Multiplicity r[J]. Trans. on Amer. Math. Soc., 1994, 338(2): 639-654.
[6] GERONIMO J S, HARDIN D F, MASSOPUST P R. Fractal Functions and Wavelet Expansions Based on Several Scaling Functions[J]. J. Approx. Theory, 1994, 78(14): 371-401.
[7] GERONIMO J S, HARDIN D F, MASSOPUST P R. Construction of Orthogonal Wavelets Using Fractal Interpolation Functions[J]. SIAM J. Math. Anal., 1996, 27(12): 1158-1192.
[8] STRELA. Multi-wavelet: Theory and Applications[D]. Boston: MIT, 1996: 108-210.
[9] PLONKA G, STRELA V. Construction of Multi-Scaling Functions With Approximation and Symmetry[J]. SIAM J. Math. Anal., 1998, 29(2): 481-510.
[10] LANTON W, LEE S L, SHEN Z. An Algorithm for Matrix Extension and Wavelets Constuction[J]. Math. Comp, 1996, 65(6): 723-737.
[11] MARIANTONIA COTRONEI, LAURA B, MONTEFUSCO. Multi-Wavelet Analysis and Signal Processing[J] . IEEE Trans.: on Circuits and Systems-Ⅱ: Analog and Digital Signal Processing, 1998, 45(8): 970-987.
[12]杨建伟,程正兴.正交多小波的构造[J].高等学校计算数学学报,2003,25(2):110-115.
[13] SHEN LI XIN, HWEE HUAT TAN. On a Family of Orthonormal Scalar Wavelets and Related Balanced Multi-Wavelets[J]. IEEE Trans. on Signal Processing, 2001, 49(7): 1447-1453.
[14]崔丽洪,程正兴.由给定尺度函数构造短支撑正交多小波的一种算法[J].西安交通大学学报,2003,37(2):211-214.
[15]张钦礼,吴勃英,何春江.Hermite多尺度小波[J].哈尔滨工业大学学报,2004,36(6):787-789.
[16] YANG SHOU ZHI, PENG LI ZHONG. Construction of High Order Balanced MultiScaling Functions via PTST[J]. Science in China(Series F: Information Sciences), 2006, 49(4): 504-515.
[17]蒋勇,邓彩霞,王旭.由β函数构造一种紧支撑小波[J].哈尔滨理工大学学报,2007,12(6):69-71.
[18]罗萍,郭洪林,方勤华.多重向量值双正交小波的存在性及构造[J].河北大学学报:自然科学版,2009,29(1):13-18.
[19] ROBERTS L G. Machine Perception of Three-Dimension Solids[J]. MIT Press, 1965, 39(10): 159-197.
[20] PREWITT J. Object Enhancement and Extraction, Picture Process[J]. International Journal of Computer Vision, 1970, 23(5): 75-149.
[21] SOBEL L. Camera Models and Machine Perception[J]. IEEE Transactions on Electronic Computers, 1970, 18(12): 104-197.
[22] KIRSCH R. Computer Determination of the Constituent Structure of Biological Images[J]. Computer and Biomedical Research, 1971, 18(1): 113-125.
[23] FREI W, CHEN C. Fast Boundary Detection: A Generalization and a New Algorithm[J]. IEEE Transactions on Electronic Computers, 1977, 26(2): 346-458.
[24]姜杭毅,蔡元龙.采用正交多项式曲面拟合法的边缘检测[J].自动化学报,1990,16(3):202-209.
[25] MARR D C, HILDRETH E. Theory of Edge Detection[J]. Proc. Roy. Soc. London, 1980, 27(2): 187-217.
[26]赵志刚,万娇娜.一种基于梯度和零交叉点的图像边缘检测新方法[J].仪器仪表学报,2006,27(8):821-824.
[27]濮群,余桂.用线性模型检测图像边缘[J].清华大学学报:自然科学[J].西安交通大学学报,2003,37(2):211-214.
[15]张钦礼,吴勃英,何春江.Hermite多尺度小波[J].哈尔滨工业大学学报,2004,36(6):787-789.
[16] YANG SHOU ZHI, PENG LI ZHONG. Construction of High Order Balanced MultiScaling Functions via PTST[J]. Science in China(Series F: Information Sciences), 2006, 49(4): 504-515.
[17]蒋勇,邓彩霞,王旭.由β函数构造一种紧支撑小波[J].哈尔滨理工大学学报,2007,12(6):69-71.
[18]罗萍,郭洪林,方勤华.多重向量值双正交小波的存在性及构造[J].河北大学学报:自然科学版,2009,29(1):13-18.
[19] ROBERTS L G. Machine Perception of Three-Dimension Solids[J]. MIT Press, 1965, 39(10): 159-197.
[20] PREWITT J. Object Enhancement and Extraction, Picture Process[J]. International Journal of Computer Vision, 1970, 23(5): 75-149.
[21] SOBEL L. Camera Models and Machine Perception[J]. IEEE Transactions on Electronic Computers, 1970, 18(12): 104-197.
[22] KIRSCH R. Computer Determination of the Constituent Structure of Biological Images[J]. Computer and Biomedical Research, 1971, 18(1): 113-125.
[23] FREI W, CHEN C. Fast Boundary Detection: A Generalization and a New Algorithm[J]. IEEE Transactions on Electronic Computers, 1977, 26(2): 346-458.
[24]姜杭毅,蔡元龙.采用正交多项式曲面拟合法的边缘检测[J].自动化学报,1990,16(3):202-209.
[25] MARR D C, HILDRETH E. Theory of Edge Detection[J]. Proc. Roy. Soc. London, 1980, 27(2): 187-217.
[26]赵志刚,万娇娜.一种基于梯度和零交叉点的图像边缘检测新方法[J].仪器仪表学报,2006,27(8):821-824.
[27]濮群,余桂.用线性模型检测图像边缘[J].清华大学学报:自然科学
[40]陈亮,郭雷,王雅萍.一种基于结构张量的MAS边缘检测算法[J].计算机科学,2009,36(1):131-133.
[41]徐建东,蒋野,孙迎春.基于改进的形态学算子的灰度图像边缘检测[J].佳木斯大学学报:自然科学版,2009,27(6):857-859.
[42] JOHN CANNY. A Computational Approach to Edge Detection[J]. IEEE Trans., Pattern Analysis and Machine Intelligence, 1986, 8(1): 679-697.
[43] HEMANT TAGARE D. On the Localization Performance Measure and Optimal Edge Detection[J]. IEEE Trans., Pattern Analysis and Machine Intelligence, 1990, 12(2): 1186-1190.
[44] SPACER L A. Edge Detection and Motion Detection[J]. Image Vision Comput., 1986, 4(2): 43-44.
[45] DERICHE R. Using Canny's Criteria to Derive a Recursively Implemented Optimal Edge Detector[J]. Int. J. Comput. Vision, 1987, 1(2): 56-101.
[46]刘建伟,郭平.一种基于边缘检测的图像融合新方法[J].北京师范大学学报:自然科学版,2007,43(5):518-521.
[47]程正兴.小波分析算法与应用[M].西安:交通常大学出版社,1997:3-4.
[48]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,2007:6-10,77-80,356-358.
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