二维经验模式分解（BEMD）在图像处理中的应用
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摘要
经验模式分解(EMD)方法是1998年Huang提出的一种新的信号处理方法,在非平稳信号分析方面有良好的性能。该方法能将复杂的非平稳信号分解成若干具有不同特征尺度的平稳的数据分量与趋势项的叠加,具有自适应性,适于处理非平稳信息。
鉴于EMD方法在一维信号处理方面获得的巨大成功,国内外学者将它推广到二维,提出了二维经验模式分解(BEMD)方法,并应用于二维信号处理。由于二维信号的复杂性,一般的BEMD方法还存在许多问题有待研究。
首先详细论述了一维经验模式分解和二维经验模式分解的模型和原理。然后给出了二维经验模式分解方法的实现流程。并详细介绍了筛选过程中利用数学形态学方法选取图像局部极值点,平面散乱点集delaunay三角剖分以及通过BB样条插值法求包络曲面等关键技术。另外,根据内禀模式函数的定义提出了一种新的结束条件判别方法,提高了分解速度。
将二维经验模式分解(BEMD)方法应用于图像处理,可将图像分解为一系列细节信息和趋势信息。由图像的趋势信息重建图像,可达到去除图像噪声的目的。实验结果表明,与均值滤波、中值滤波和维纳滤波相比,该方法对去除乘性噪声具有较优的效果,图像峰值信噪比(PSNR)得到明显地提高。
The Empirical Mode Decomposition (EMD), which has been recently introduced by Huang in 1998, is a new method of signal processing and has superior performance in non-stationary signal analysis. This method can be applied to decompose the non-stationary data into a series of data layers with different character-scale and the trend data. It is adaptive, and, therefore, suitable for processing nonlinear and non-stationary data.
Because of the superior performance in one-dimensional signal processing, the EMD is extended to Bidimensional EMD (BEMD) and then applied to process bidimensional signal. On account of the complexity of bidimensional signal, the general BEMD has some shortages and need to be improved in future.
In this thesis, the theories of the empirical mode decomposition (EMD) and its extension Bidimensional EMD are introduced first. Then we describe the processing flow of the BEMD. The sifting process is realized using mathematical morphology operators to detect regional extrema and thanks to delaunay triangulation and Bernstein-Bezier spline interpolation based on triangular meshes for surface interpolation. In addition, according to the definition of the intrinsic mode function, we present a new condition to end the sifting process, and it is proved that the speed of decomposition increased.
By the BEMD method, an image with noise is decomposed into a series of detail data and the trend data. The noise can be removed by the image reconstruction with the trend data. Experimental results demonstrate that comparing with mean filter, median filter, wiener filter, this method has more notable performance to get rid of the multiplicative noise and the value of PSNR markedly increased.
鉴于EMD方法在一维信号处理方面获得的巨大成功,国内外学者将它推广到二维,提出了二维经验模式分解(BEMD)方法,并应用于二维信号处理。由于二维信号的复杂性,一般的BEMD方法还存在许多问题有待研究。
首先详细论述了一维经验模式分解和二维经验模式分解的模型和原理。然后给出了二维经验模式分解方法的实现流程。并详细介绍了筛选过程中利用数学形态学方法选取图像局部极值点,平面散乱点集delaunay三角剖分以及通过BB样条插值法求包络曲面等关键技术。另外,根据内禀模式函数的定义提出了一种新的结束条件判别方法,提高了分解速度。
将二维经验模式分解(BEMD)方法应用于图像处理,可将图像分解为一系列细节信息和趋势信息。由图像的趋势信息重建图像,可达到去除图像噪声的目的。实验结果表明,与均值滤波、中值滤波和维纳滤波相比,该方法对去除乘性噪声具有较优的效果,图像峰值信噪比(PSNR)得到明显地提高。
The Empirical Mode Decomposition (EMD), which has been recently introduced by Huang in 1998, is a new method of signal processing and has superior performance in non-stationary signal analysis. This method can be applied to decompose the non-stationary data into a series of data layers with different character-scale and the trend data. It is adaptive, and, therefore, suitable for processing nonlinear and non-stationary data.
Because of the superior performance in one-dimensional signal processing, the EMD is extended to Bidimensional EMD (BEMD) and then applied to process bidimensional signal. On account of the complexity of bidimensional signal, the general BEMD has some shortages and need to be improved in future.
In this thesis, the theories of the empirical mode decomposition (EMD) and its extension Bidimensional EMD are introduced first. Then we describe the processing flow of the BEMD. The sifting process is realized using mathematical morphology operators to detect regional extrema and thanks to delaunay triangulation and Bernstein-Bezier spline interpolation based on triangular meshes for surface interpolation. In addition, according to the definition of the intrinsic mode function, we present a new condition to end the sifting process, and it is proved that the speed of decomposition increased.
By the BEMD method, an image with noise is decomposed into a series of detail data and the trend data. The noise can be removed by the image reconstruction with the trend data. Experimental results demonstrate that comparing with mean filter, median filter, wiener filter, this method has more notable performance to get rid of the multiplicative noise and the value of PSNR markedly increased.
引文
[1] Huang N E, Shen Z, Long S R. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis [J]. Proc. R. Soc Lond. A, 1998, 454: 903-995
[2]杨志华,齐东旭,杨力华等.一种基于HHT的信号周期性分析方法及应用[J].中山大学学报(自然科学版), 2005(3): 14-18
[3]于德介,程军圣,杨宇. Hilbert-Huang变换在齿轮故障诊断中的应用[J].机械工程学报, 2005(6): 102-107
[4]宋平舰,张杰.二维经验模分解在海洋遥感图像信息分离中的应用[J].高技术通讯, 2001(9): 62-67
[5] J. C. Nunes, Y. Bouaoune, E. Delechelle. Texture analysis based on the bidimensional empirical mode decomposition with gray-level co-occurrence models [J]. Mach. Vision App. 2003 : 633-635
[6] J. C. Nunes, Y. Bouaoune, E. Delechelle. Image analysis by bidimensional empirical mode decomposition[J]. Image and Vision Computing, 2003(21): 1019-1026
[7] J. C. Nunes, S. Guyot, E. Delechelle. Texture analysis based on local analysis of the bidimensional empirical mode decomposition[J]. Machine Vision and Applications, 2005(16): 177-188
[8] J. C. Nunes, Y. Bouaoune, E. Delechelle. Bidimensional empirical mode decomposition modified for texture analysis[C]. SCIA 2003 13th Scandinavian Conference on Image Analysis, June 29–July 2 2003 : 171-177
[9]秦勃,时鹏.经验模式多尺度图像分解[J].工程图学学报, 2002(4): 73-78
[10]韩春明,郭华东,王长林.利用经验模态分解方法抑制SAR斑点噪声[J].遥感学报, 2002(4): 266-271
[11]张蓬,彭思龙.基于结构的纹理合成[J].计算机辅助设计与图形学学报, 2004(3): 290-296
[12]崔峰,沈滨,彭思龙.基于EMD细化四元数谱的纹理分割[J].计算机应用, 2005(3): 573-576
[13]沈滨,崔峰,彭思龙.二维EMD的纹理分析及图像瞬时频率估计[J].计算机辅助设计与图形学学报, 2005(10): 2345-2352
[14]刘忠轩,彭思龙.方向EMD分解与其在纹理分割中的应用[J].中国科学E辑, 2005(2): 113-123
[15] S. Sinclair. Empirical mode decomposition in 2-D space and time: a tool for space-time rainfall analysis and nowcasting[C]. Hydrol. Earth Syst. Sci. Discuss. 2005, 2: 289-318.
[16]杨志华,齐东旭,杨力华等.基于经验模式分解的汉字字体识别方法[J].软件学报, 2005(8): 1438-1444
[17] Christophe Damerval. A fast algorithm for bidimensional EMD[J]. Signal Processing Letters, IEEE, 2005, 12(10): 701-704
[18] Yan Tian, Ying Huang, Yongjiang Li. Image Zooming Method Using 2D EMD[C]. Technique Intelligent Control and Automation, 2006. WCICA 2006. The Sixth World Congress on Volume 2, 21-23 June 2006 Page(s): 10036-10040
[19]邓拥军. EMD方法及Hilbert变化中边界问题的处理[J].科学通报, 2001, 46(3): 257-263
[20]黄大吉,赵进平.希尔伯特-黄变化的端点延拓[J].海洋学报, 2003, 1: 1-11
[21]张郁山,梁建文.应用自回归模型处理EMD方法中的边界问题[J].自然科学进展, 2003 10: 1054-1059
[22]张贤达等.非平稳信号分析与处理[M].北京:国防工业出版社, 1998: 134-160
[23]张海勇,马孝江,盖强.一种新的时频分析方法[J].火力与指挥控制, 2000, 25(1): 39-42
[24] S. Olhede, A. T. Walden. The Hilbert spectrum via wavelet projections[J]. The Royal Society, 2004, 955-975
[25] Ming-Jun Lai. Scattered data interpolation and approximation using bivariate C1 piecewise cubic polynomials[J]. Computer Aided Geometric Design , 1996, 13: 81-88
[26]张彩明,孙德法,汪嘉业.散乱数据点的三次多项式插值.计算机辅助设计与图形学学报, 1998, 10(5): 416-423
[27] M. J. D. Powell. The theory of radial basis functions approximation[M]. Oxford Univ. Press, London, 1992: 105-210
[28] Yingcai Xiao, J. P. Ziebarth. FEM-based scattered data modeling and visualization[J]. Computers & Graphics, 2000, 24(5): 775-789
[29]王树文,闫成新,张天序等.数学形态学在图像处理中的应用[J].计算机工程与应用, 2004(32): 89-92
[30]崔屹.图象处理与分析——数学形态学方法及应用[M].北京:科学出版社, 2002
[31] J. Serra. Image analysis and mathematical morphology[M]. London: Academic Press, 1982
[32]罗军辉,冯平,哈力旦?A等. MATLAB7. 0在图像处理中的应用[M].北京:机械工业出版社,2005
[33]张永春,达飞鹏,宋文忠.三维散乱点集的曲面三角剖分[J].中国图象图形学报, 2003, 8(A)(12): 1379-1388
[34]谢伟松,李继钢. Delaunay三角剖分在有限元预处理中的应用.海洋技术, 2004,. 23( 4): 140-142
[35]闵卫东,唐泽圣.二维任意域内点集的Delaunay三角划分的研究[J].计算机学报, 1995, 18(5): 357-364
[36]周培德.平面散乱点线集三角剖分的算法[J]. .计算机辅助设计与图形学学报[J].计算机辅助设计与图形学学报, 2003. 9,15: 1141-1144
[37]陈学工,潘懋.平面散乱点集约束Delaunay三角形剖分切割算法[J].计算机工程与应用, 2001, 15: 96-104
[38]宋平舰.二维EMD方法及其在图象处理中的应用[D].青岛:国家海洋局第一研究所, 2001,6
[39]孙玉文,王晓明,刘健.密集散乱数据的三角形网格曲面逼近方法[J].计算机辅助设计与图形学学报, 2000, 12(4): 281-285
[40] Feng Renzhong, Wang Renhong. Smooth Spline Surfaces over Arbitrary Topological Triangular Meshes[J]. Journal of Software, 2003, 4: 830-837
[41]李薇.三维散乱数据的光滑插值曲面[J].高等学校计算数学学报, 1991, 9(3): 197-206
[42] Mark Tabb, Narendra Ahuja.Multiscale Image Segmentation by Intergrated Edge and Region Detection[J]. IEEE Transaction on Image Processing, 1997, 6(5): 642-1235
[2]杨志华,齐东旭,杨力华等.一种基于HHT的信号周期性分析方法及应用[J].中山大学学报(自然科学版), 2005(3): 14-18
[3]于德介,程军圣,杨宇. Hilbert-Huang变换在齿轮故障诊断中的应用[J].机械工程学报, 2005(6): 102-107
[4]宋平舰,张杰.二维经验模分解在海洋遥感图像信息分离中的应用[J].高技术通讯, 2001(9): 62-67
[5] J. C. Nunes, Y. Bouaoune, E. Delechelle. Texture analysis based on the bidimensional empirical mode decomposition with gray-level co-occurrence models [J]. Mach. Vision App. 2003 : 633-635
[6] J. C. Nunes, Y. Bouaoune, E. Delechelle. Image analysis by bidimensional empirical mode decomposition[J]. Image and Vision Computing, 2003(21): 1019-1026
[7] J. C. Nunes, S. Guyot, E. Delechelle. Texture analysis based on local analysis of the bidimensional empirical mode decomposition[J]. Machine Vision and Applications, 2005(16): 177-188
[8] J. C. Nunes, Y. Bouaoune, E. Delechelle. Bidimensional empirical mode decomposition modified for texture analysis[C]. SCIA 2003 13th Scandinavian Conference on Image Analysis, June 29–July 2 2003 : 171-177
[9]秦勃,时鹏.经验模式多尺度图像分解[J].工程图学学报, 2002(4): 73-78
[10]韩春明,郭华东,王长林.利用经验模态分解方法抑制SAR斑点噪声[J].遥感学报, 2002(4): 266-271
[11]张蓬,彭思龙.基于结构的纹理合成[J].计算机辅助设计与图形学学报, 2004(3): 290-296
[12]崔峰,沈滨,彭思龙.基于EMD细化四元数谱的纹理分割[J].计算机应用, 2005(3): 573-576
[13]沈滨,崔峰,彭思龙.二维EMD的纹理分析及图像瞬时频率估计[J].计算机辅助设计与图形学学报, 2005(10): 2345-2352
[14]刘忠轩,彭思龙.方向EMD分解与其在纹理分割中的应用[J].中国科学E辑, 2005(2): 113-123
[15] S. Sinclair. Empirical mode decomposition in 2-D space and time: a tool for space-time rainfall analysis and nowcasting[C]. Hydrol. Earth Syst. Sci. Discuss. 2005, 2: 289-318.
[16]杨志华,齐东旭,杨力华等.基于经验模式分解的汉字字体识别方法[J].软件学报, 2005(8): 1438-1444
[17] Christophe Damerval. A fast algorithm for bidimensional EMD[J]. Signal Processing Letters, IEEE, 2005, 12(10): 701-704
[18] Yan Tian, Ying Huang, Yongjiang Li. Image Zooming Method Using 2D EMD[C]. Technique Intelligent Control and Automation, 2006. WCICA 2006. The Sixth World Congress on Volume 2, 21-23 June 2006 Page(s): 10036-10040
[19]邓拥军. EMD方法及Hilbert变化中边界问题的处理[J].科学通报, 2001, 46(3): 257-263
[20]黄大吉,赵进平.希尔伯特-黄变化的端点延拓[J].海洋学报, 2003, 1: 1-11
[21]张郁山,梁建文.应用自回归模型处理EMD方法中的边界问题[J].自然科学进展, 2003 10: 1054-1059
[22]张贤达等.非平稳信号分析与处理[M].北京:国防工业出版社, 1998: 134-160
[23]张海勇,马孝江,盖强.一种新的时频分析方法[J].火力与指挥控制, 2000, 25(1): 39-42
[24] S. Olhede, A. T. Walden. The Hilbert spectrum via wavelet projections[J]. The Royal Society, 2004, 955-975
[25] Ming-Jun Lai. Scattered data interpolation and approximation using bivariate C1 piecewise cubic polynomials[J]. Computer Aided Geometric Design , 1996, 13: 81-88
[26]张彩明,孙德法,汪嘉业.散乱数据点的三次多项式插值.计算机辅助设计与图形学学报, 1998, 10(5): 416-423
[27] M. J. D. Powell. The theory of radial basis functions approximation[M]. Oxford Univ. Press, London, 1992: 105-210
[28] Yingcai Xiao, J. P. Ziebarth. FEM-based scattered data modeling and visualization[J]. Computers & Graphics, 2000, 24(5): 775-789
[29]王树文,闫成新,张天序等.数学形态学在图像处理中的应用[J].计算机工程与应用, 2004(32): 89-92
[30]崔屹.图象处理与分析——数学形态学方法及应用[M].北京:科学出版社, 2002
[31] J. Serra. Image analysis and mathematical morphology[M]. London: Academic Press, 1982
[32]罗军辉,冯平,哈力旦?A等. MATLAB7. 0在图像处理中的应用[M].北京:机械工业出版社,2005
[33]张永春,达飞鹏,宋文忠.三维散乱点集的曲面三角剖分[J].中国图象图形学报, 2003, 8(A)(12): 1379-1388
[34]谢伟松,李继钢. Delaunay三角剖分在有限元预处理中的应用.海洋技术, 2004,. 23( 4): 140-142
[35]闵卫东,唐泽圣.二维任意域内点集的Delaunay三角划分的研究[J].计算机学报, 1995, 18(5): 357-364
[36]周培德.平面散乱点线集三角剖分的算法[J]. .计算机辅助设计与图形学学报[J].计算机辅助设计与图形学学报, 2003. 9,15: 1141-1144
[37]陈学工,潘懋.平面散乱点集约束Delaunay三角形剖分切割算法[J].计算机工程与应用, 2001, 15: 96-104
[38]宋平舰.二维EMD方法及其在图象处理中的应用[D].青岛:国家海洋局第一研究所, 2001,6
[39]孙玉文,王晓明,刘健.密集散乱数据的三角形网格曲面逼近方法[J].计算机辅助设计与图形学学报, 2000, 12(4): 281-285
[40] Feng Renzhong, Wang Renhong. Smooth Spline Surfaces over Arbitrary Topological Triangular Meshes[J]. Journal of Software, 2003, 4: 830-837
[41]李薇.三维散乱数据的光滑插值曲面[J].高等学校计算数学学报, 1991, 9(3): 197-206
[42] Mark Tabb, Narendra Ahuja.Multiscale Image Segmentation by Intergrated Edge and Region Detection[J]. IEEE Transaction on Image Processing, 1997, 6(5): 642-1235