小波变换在图像边缘检测和降噪中的应用
摘要
图像边缘包含了一幅图像的绝大部分主要信息,边缘的提取在图像处理和机器视觉中占据着非常重要的作用。在图像的获取、传输和存储过程中往往会因各种原因引入噪声,因此,如何改进这些图像的质量,就成为数字图像处理中的一个重要任务。小波分析是一种有效的分析工具,近年来随着小波理论的不断发展完善,小波理论己经被应用到图像处理的几乎所有的分支,如:图像降噪、边缘检测、图像压缩、图像分割等。本文主要研究其在图像边缘检测和降噪领域的应用。
本文对小波变换理论进行了系统的学习、研究与总结,介绍了连续小波变换、离散小波变换、多分辨分析、小波基构造和二进小波变换,并给出离散二进小波变换的快速分解与重构算法(Mallat算法)等。
本文介绍了传统的边缘检测算法并分析其优缺点;重点研究了基于Mallat算法和多孔算法下的小波边缘检测,在此基础上针对多孔算法边缘检测提出改进方案,大大降低了算法复杂度;本文还研究了数字形态学在边缘检测中应用,分别针对边缘检测算子和采取的结构元提出改进方案。最后,本文提出了一种新的小波与形态学相结合的边缘检测算法,实验证明该方法得到的边缘细节丰富,且抗噪性能较好。
针对传统的图像降噪方法,在去除噪声的同时往往会造成边缘的模糊的问题,本文提出基于边缘检测的图像降噪法,在检测出图像边缘之后,将图像分为“边缘区”和“噪声区”,针对它们的特点分别采用不同的方法进行处理,最后将两种小波系数相结合的方法,达到既保护了图像边缘又有效去除噪声的目的。
Image edge contains most of the important information of an image, detection of edge plays an important role in image processing and machine vision. Noise is added to the image during the acquirement、transmission and deposit for many reasons. So how to improve the quality of the image becomes a very important task in image processing. Wavelet is an useful kind of analytic tool. Along with the improvement and perfection of the theory of wavelet in recent years, it has been applied on almost all the embranchments of image processing, such as image denoising, edge detection, image compression and image division etc. This article mainly studies the application of wavelet on edge detection and image denoising.
This article systemically introduces and summarizes the theory of wavelet. Continuous Wavelet Transform, Discrete Wavelet Transform, Multi-Resolution Analysis, Wavelet Bases and Dyadic Wavelet Transform are introduced. The fast algorithm of Discrete Dyadic Wavelet Transform is also given.
The classic methods of edge-detection are introduced and analyzed. Algorithm of edge detection by modulus maximums based on Mallat and Trous is deeply studied. Based on the above analyses, a simplification modulus maximums edge detection algorithm based on Trous which reduces the complexity greatly is proposed. And it also studies the application of Mathematical Morphology on edge detection and proposes some improvements on the morphology edge examination operator and structure element. Then it presents a new edge detection algorithm combines wavelet and morphology together, which is proved to be efficient on edge precision, detail edge detection and restraining noise by experiments.
Traditional methods for denoising will result in dim edge while taking out the noise. This article proposes a method on image denoising based on edge detection which divides the image into”edge area”and”noise area”after edge detection. Apply different method for different area, then combine the two kinds of modulus together to achieve the gain that protect the edge while denoising.
本文对小波变换理论进行了系统的学习、研究与总结,介绍了连续小波变换、离散小波变换、多分辨分析、小波基构造和二进小波变换,并给出离散二进小波变换的快速分解与重构算法(Mallat算法)等。
本文介绍了传统的边缘检测算法并分析其优缺点;重点研究了基于Mallat算法和多孔算法下的小波边缘检测,在此基础上针对多孔算法边缘检测提出改进方案,大大降低了算法复杂度;本文还研究了数字形态学在边缘检测中应用,分别针对边缘检测算子和采取的结构元提出改进方案。最后,本文提出了一种新的小波与形态学相结合的边缘检测算法,实验证明该方法得到的边缘细节丰富,且抗噪性能较好。
针对传统的图像降噪方法,在去除噪声的同时往往会造成边缘的模糊的问题,本文提出基于边缘检测的图像降噪法,在检测出图像边缘之后,将图像分为“边缘区”和“噪声区”,针对它们的特点分别采用不同的方法进行处理,最后将两种小波系数相结合的方法,达到既保护了图像边缘又有效去除噪声的目的。
Image edge contains most of the important information of an image, detection of edge plays an important role in image processing and machine vision. Noise is added to the image during the acquirement、transmission and deposit for many reasons. So how to improve the quality of the image becomes a very important task in image processing. Wavelet is an useful kind of analytic tool. Along with the improvement and perfection of the theory of wavelet in recent years, it has been applied on almost all the embranchments of image processing, such as image denoising, edge detection, image compression and image division etc. This article mainly studies the application of wavelet on edge detection and image denoising.
This article systemically introduces and summarizes the theory of wavelet. Continuous Wavelet Transform, Discrete Wavelet Transform, Multi-Resolution Analysis, Wavelet Bases and Dyadic Wavelet Transform are introduced. The fast algorithm of Discrete Dyadic Wavelet Transform is also given.
The classic methods of edge-detection are introduced and analyzed. Algorithm of edge detection by modulus maximums based on Mallat and Trous is deeply studied. Based on the above analyses, a simplification modulus maximums edge detection algorithm based on Trous which reduces the complexity greatly is proposed. And it also studies the application of Mathematical Morphology on edge detection and proposes some improvements on the morphology edge examination operator and structure element. Then it presents a new edge detection algorithm combines wavelet and morphology together, which is proved to be efficient on edge precision, detail edge detection and restraining noise by experiments.
Traditional methods for denoising will result in dim edge while taking out the noise. This article proposes a method on image denoising based on edge detection which divides the image into”edge area”and”noise area”after edge detection. Apply different method for different area, then combine the two kinds of modulus together to achieve the gain that protect the edge while denoising.
引文
[1]崔锦泰,程正兴译,小波分析导论,西安:西安交通大学出版社,1995
[2]李世雄,小波变换的工程分析与应用,北京:高等教育出版社,1999
[3]彭玉华,小波变换与工程应用,北京:科学出版社,2000
[4]Grossman A., Morlet J. Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape. SIAM J. Math. Anal., 1984, 15(4): 723-736
[5]MeyerY. Wavelets and Operators. Cambridge. UK: Cambridge University Press, 1992
[6]Mallat S. Multiresolution Approximations and Wavelet Orthonormal bases of L2(R), Trans. Amer. Math. Soc., 1989, 315(9): 69-87
[7]Daubechies I. Orthonormal bases of Compactly Supported Wavelets. Comm. Pure Appl. Math., 1988, 41(11): 909-996
[8]Mallat S. A Theory for Multiresolution Signal Decomposition: the Wavelet Representation. IEEE Trans. on PAMI, 1989, 11 (7): 674-693
[9]王建忠,多尺度B样条小波边缘检测算子,中国科学(A辑),1995, 25(4): 426-437
[10]Coifiman R. and Wickerhauser M. Entropy based Algorithms for Best Basis Selection. IEEE Trans. on IT, 1992, 38(2): 713-718.
[11] Wickerhauser M.V. Adapted Wavelet Analysis from Theory to Software. IEEE Press, The Institute of Electrical and Electronics Engineers, Inc. New York, 1994
[12]李旭超.小波变换和马尔可夫随机场在图像降噪与分割中的应用研究:[博士毕业论文],浙江:浙江大学,2006
[13]Mallat, S. Zhong S. Characterization of Signals from Multiscale Edges. IEEE Transactions on patern analysis and machine intelligence, 1992, 14(7): 710-732
[14]李建平,小波分析与信号处理一理论、应用及软件实现,重庆:重庆出版社,1997, 12
[15]屈汉章,徐晨等,连续小波变换及其应用,北京:中国电子音像出版社,2002
[16] Goswami J. C., Chan A. K. and Chui C. K. On Solving First-kind Integral Equations using Wavelets on Abounded Interval. IEEE Trans. on AP., 1995, 43(6): 614-622
[17] Struzik Z. R. The Wavelet Transform in the Solution to the Inverse Fractal Problem. Fractals, 1995, 3(2): 329-350
[18] Rioul O. and Vetterli M. Wavelets and Signal Processing. IEEE Signal Processing Magzine, 1991, 8(4): 14-38
[19] Mallat S. A Theory for Multiresolution: Signal Decomposition: the Wavelet representation. IEEE Trans. on PAMI, 1989, 11(7): 674-693
[20] Antonini M., Barlaud M, Mathieu P. and et al. Image Coding using Wavelet Transform. IEEE Trans. on Image Processing, 1992, 1(2): 205-220
[21] Averbueh A., Lazar D. and Israeli M. Image Compression using Wavelet Transform and MultiresoIution Decomposition. IEEE Trans. on Image Processing, 1996, 5(1): 4-15
[22]赵瑞珍,小波理论及其在图像、信号处理中的算法研究:[博士学位论文],西安:西安电子科技大学,2001, 6
[23] Hartung F. and Kuter M. Multimedia: Watermarking Techniques. Proceedings of the IEEE, 1999, 87(7): 1079-1107
[24]Jun Zhang, et al. Wavelet Networks for Function Learning. IEEE Trans. on SP, 1995, 43(6): 1485-1496
[25]屈汉章,徐晨等,连续小波变换及其应用,北京:中国电子音像出版社,2002
[26]程正兴,小波分析算法与应用,西安:西安交通大学出版社,1998
[27] Daubechies. Ten Lectures on Wavelets. Siam, Philadelpha, 1992
[28]J. Prewit. Object Enhancement and Extraction. B. Lipkin and A. Rosenfeld, eds., Picture Processing and Psychopictoric Press, New York, 1970
[29] R.A.Kirsch. Computer Determination of the Constituent Structure of Biological Images. Computers in Biomedical research, 1971, 4: 315-328,
[30]L.G.Roberts. Machine Perception of Three-Dimensional Soliods. Optical and Electro-Optical Information Processing, MIT Press, Cambridge, MA, 1965: 159-197
[31] L.S.Davis. A Survey of Edge Detection Techniques. CGIP, 1975, 4: 248-270
[32]D.C.Marr and Hildreth. Theory of edge detection. Proc. Roy. Soc. London, 1980, 207, 187-217
[33] John Canny. A Computational Approach to Edge Detection. IEEE Transactions on Pattern Analysis. 1986, 8: 679-698
[34] Mallat S., Zhong S., Characterization of Signals from Multiscale Edges. IEEE Trans on pattern analysis and machine intelligence. 1992, 14(7): 710-732
[35]陈凡武等,彩色图像边缘检测的新方法-广义模糊算子法,中国科学(A辑), 1995, 25(2): 219
[36] Chan Minghua, Lee David. Residual analysis for Feature Detection. IEEE Trans on Patt and Mach Intelligence. 1991, 13(1): 30-40
[37]Stephance Mallat著,杨力华,戴道清,黄文良等译,信号处理的小波引导,北京:机械工业出版社,2002, 9
[38]李继军,小波分析及其在信号、图像降噪中的研究:[硕士学位论文],西安:西安电子科技大学,2006, 1
[39]袁华,张卫宁等,一种改进的小波去噪中模极大值处理的方法,系统仿真学报,2005, 3: 838～843
[40]柳薇,马争鸣,基于边缘检测的图象小波闽值去噪方法,中国图象图形学报,2002, 7 (8): 788-79
[41]田岩岩,齐国清,基于小波变换模极大值的边缘检测方法,大连海事大学学报,2007,33(1): 102-106
[42]袁野,欧宗瑛,侯建华,CT图像表面重建技术中的边缘检测和跟踪补偿,机械科学与技术,2001, 20(6): 962-965
[43]刘慧敏,舒宁,林卉,图像边缘信息分析中数学形态学的应用方法,武汉大学学报(信息科学版),2001, 26(4): 325-329
[44]龚炜,石青云,程德民,数字空间中的数学形态学-理论及应用,北京:科学出版社,1997
[45]崔屹,图像处理与分析-数学形态学方法及应用,北京:科学出版社,2000: 56~62
[46]李向吉,脉冲噪声污染图像中的数学形态学边缘检测器,中国图象图形学报,1998, 11: 903-905
[47]杨述斌,彭复员,多尺度自适应加权形态边缘检测方法,华中科技大学学报(自然科学版),2002, 30(10): 87-89
[48]Mallat S., Hwang W.L. Singularity Detection and Processing with Wavelets. IEEE Trans. on IT., 1992, 38(2): 617-643
[49]J. Lu, J. B. Weaver, D. M. Healy, et al. Noise Reduction with Multiscale Edge Representation and Perceptual Criteria. Proc. IEEE-SP Int. Symp. Time-Frequency and Time-Scale Analysis, Victoria, BC, 1992, 555-558
[50]Y.Xu, J.B. Weaver, D. M. Healy, et al. Wavelet Transform Domain Filters: A Spatially Selective Noise Filtration Technique. IEEE Trans. Image Processing, Nov. 1994, 3: 747-758
[51]Donoho D.L. Denoising by Soft-thresholding. IEEE. Trans. Inform. Theory, 1995, 41: 613-627
[52] Donoho D. L., I.M.Johnstone. Ideal Spatial Adaptation via Wavelet Shrinkage. Biometrika, 1994, 81: 425-455
[53]H.J.Larson, B.O.Shubert. Probabilistic Models in Engineering Sciences. New York: Wiley, 1979, 1
[54]Clsudio Rosito Jung, Jacob Scharcanski. Adaptive Image Denoising and Edge Enhancement Inscale-space using the Wavelet Transform. Pattern Recognition Letters. 2003, 956-971
[55]钟平,罗建书,基于小波变换的含噪图片的自适应边缘提取,中国空间科学技术,2003, 1
[56]闫敬文,数字图像处理,北京:国防工业出版社,2007
[57]Chang S.G, Yu.B, Vetterli M. Spaptive Wavelet Thresholding with Context Modeling for Image Denoising. IEEE trans. on image processing, 2000, 9(9): 1522-1531
[58]管聪慧,基于相关性及边缘检测的小波图像降噪方法研究:[硕士学位论文],青岛:青岛大学,2007, 5
[2]李世雄,小波变换的工程分析与应用,北京:高等教育出版社,1999
[3]彭玉华,小波变换与工程应用,北京:科学出版社,2000
[4]Grossman A., Morlet J. Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape. SIAM J. Math. Anal., 1984, 15(4): 723-736
[5]MeyerY. Wavelets and Operators. Cambridge. UK: Cambridge University Press, 1992
[6]Mallat S. Multiresolution Approximations and Wavelet Orthonormal bases of L2(R), Trans. Amer. Math. Soc., 1989, 315(9): 69-87
[7]Daubechies I. Orthonormal bases of Compactly Supported Wavelets. Comm. Pure Appl. Math., 1988, 41(11): 909-996
[8]Mallat S. A Theory for Multiresolution Signal Decomposition: the Wavelet Representation. IEEE Trans. on PAMI, 1989, 11 (7): 674-693
[9]王建忠,多尺度B样条小波边缘检测算子,中国科学(A辑),1995, 25(4): 426-437
[10]Coifiman R. and Wickerhauser M. Entropy based Algorithms for Best Basis Selection. IEEE Trans. on IT, 1992, 38(2): 713-718.
[11] Wickerhauser M.V. Adapted Wavelet Analysis from Theory to Software. IEEE Press, The Institute of Electrical and Electronics Engineers, Inc. New York, 1994
[12]李旭超.小波变换和马尔可夫随机场在图像降噪与分割中的应用研究:[博士毕业论文],浙江:浙江大学,2006
[13]Mallat, S. Zhong S. Characterization of Signals from Multiscale Edges. IEEE Transactions on patern analysis and machine intelligence, 1992, 14(7): 710-732
[14]李建平,小波分析与信号处理一理论、应用及软件实现,重庆:重庆出版社,1997, 12
[15]屈汉章,徐晨等,连续小波变换及其应用,北京:中国电子音像出版社,2002
[16] Goswami J. C., Chan A. K. and Chui C. K. On Solving First-kind Integral Equations using Wavelets on Abounded Interval. IEEE Trans. on AP., 1995, 43(6): 614-622
[17] Struzik Z. R. The Wavelet Transform in the Solution to the Inverse Fractal Problem. Fractals, 1995, 3(2): 329-350
[18] Rioul O. and Vetterli M. Wavelets and Signal Processing. IEEE Signal Processing Magzine, 1991, 8(4): 14-38
[19] Mallat S. A Theory for Multiresolution: Signal Decomposition: the Wavelet representation. IEEE Trans. on PAMI, 1989, 11(7): 674-693
[20] Antonini M., Barlaud M, Mathieu P. and et al. Image Coding using Wavelet Transform. IEEE Trans. on Image Processing, 1992, 1(2): 205-220
[21] Averbueh A., Lazar D. and Israeli M. Image Compression using Wavelet Transform and MultiresoIution Decomposition. IEEE Trans. on Image Processing, 1996, 5(1): 4-15
[22]赵瑞珍,小波理论及其在图像、信号处理中的算法研究:[博士学位论文],西安:西安电子科技大学,2001, 6
[23] Hartung F. and Kuter M. Multimedia: Watermarking Techniques. Proceedings of the IEEE, 1999, 87(7): 1079-1107
[24]Jun Zhang, et al. Wavelet Networks for Function Learning. IEEE Trans. on SP, 1995, 43(6): 1485-1496
[25]屈汉章,徐晨等,连续小波变换及其应用,北京:中国电子音像出版社,2002
[26]程正兴,小波分析算法与应用,西安:西安交通大学出版社,1998
[27] Daubechies. Ten Lectures on Wavelets. Siam, Philadelpha, 1992
[28]J. Prewit. Object Enhancement and Extraction. B. Lipkin and A. Rosenfeld, eds., Picture Processing and Psychopictoric Press, New York, 1970
[29] R.A.Kirsch. Computer Determination of the Constituent Structure of Biological Images. Computers in Biomedical research, 1971, 4: 315-328,
[30]L.G.Roberts. Machine Perception of Three-Dimensional Soliods. Optical and Electro-Optical Information Processing, MIT Press, Cambridge, MA, 1965: 159-197
[31] L.S.Davis. A Survey of Edge Detection Techniques. CGIP, 1975, 4: 248-270
[32]D.C.Marr and Hildreth. Theory of edge detection. Proc. Roy. Soc. London, 1980, 207, 187-217
[33] John Canny. A Computational Approach to Edge Detection. IEEE Transactions on Pattern Analysis. 1986, 8: 679-698
[34] Mallat S., Zhong S., Characterization of Signals from Multiscale Edges. IEEE Trans on pattern analysis and machine intelligence. 1992, 14(7): 710-732
[35]陈凡武等,彩色图像边缘检测的新方法-广义模糊算子法,中国科学(A辑), 1995, 25(2): 219
[36] Chan Minghua, Lee David. Residual analysis for Feature Detection. IEEE Trans on Patt and Mach Intelligence. 1991, 13(1): 30-40
[37]Stephance Mallat著,杨力华,戴道清,黄文良等译,信号处理的小波引导,北京:机械工业出版社,2002, 9
[38]李继军,小波分析及其在信号、图像降噪中的研究:[硕士学位论文],西安:西安电子科技大学,2006, 1
[39]袁华,张卫宁等,一种改进的小波去噪中模极大值处理的方法,系统仿真学报,2005, 3: 838～843
[40]柳薇,马争鸣,基于边缘检测的图象小波闽值去噪方法,中国图象图形学报,2002, 7 (8): 788-79
[41]田岩岩,齐国清,基于小波变换模极大值的边缘检测方法,大连海事大学学报,2007,33(1): 102-106
[42]袁野,欧宗瑛,侯建华,CT图像表面重建技术中的边缘检测和跟踪补偿,机械科学与技术,2001, 20(6): 962-965
[43]刘慧敏,舒宁,林卉,图像边缘信息分析中数学形态学的应用方法,武汉大学学报(信息科学版),2001, 26(4): 325-329
[44]龚炜,石青云,程德民,数字空间中的数学形态学-理论及应用,北京:科学出版社,1997
[45]崔屹,图像处理与分析-数学形态学方法及应用,北京:科学出版社,2000: 56~62
[46]李向吉,脉冲噪声污染图像中的数学形态学边缘检测器,中国图象图形学报,1998, 11: 903-905
[47]杨述斌,彭复员,多尺度自适应加权形态边缘检测方法,华中科技大学学报(自然科学版),2002, 30(10): 87-89
[48]Mallat S., Hwang W.L. Singularity Detection and Processing with Wavelets. IEEE Trans. on IT., 1992, 38(2): 617-643
[49]J. Lu, J. B. Weaver, D. M. Healy, et al. Noise Reduction with Multiscale Edge Representation and Perceptual Criteria. Proc. IEEE-SP Int. Symp. Time-Frequency and Time-Scale Analysis, Victoria, BC, 1992, 555-558
[50]Y.Xu, J.B. Weaver, D. M. Healy, et al. Wavelet Transform Domain Filters: A Spatially Selective Noise Filtration Technique. IEEE Trans. Image Processing, Nov. 1994, 3: 747-758
[51]Donoho D.L. Denoising by Soft-thresholding. IEEE. Trans. Inform. Theory, 1995, 41: 613-627
[52] Donoho D. L., I.M.Johnstone. Ideal Spatial Adaptation via Wavelet Shrinkage. Biometrika, 1994, 81: 425-455
[53]H.J.Larson, B.O.Shubert. Probabilistic Models in Engineering Sciences. New York: Wiley, 1979, 1
[54]Clsudio Rosito Jung, Jacob Scharcanski. Adaptive Image Denoising and Edge Enhancement Inscale-space using the Wavelet Transform. Pattern Recognition Letters. 2003, 956-971
[55]钟平,罗建书,基于小波变换的含噪图片的自适应边缘提取,中国空间科学技术,2003, 1
[56]闫敬文,数字图像处理,北京:国防工业出版社,2007
[57]Chang S.G, Yu.B, Vetterli M. Spaptive Wavelet Thresholding with Context Modeling for Image Denoising. IEEE trans. on image processing, 2000, 9(9): 1522-1531
[58]管聪慧,基于相关性及边缘检测的小波图像降噪方法研究:[硕士学位论文],青岛:青岛大学,2007, 5