投影指标的小波估计及应用
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摘要
合成孔径雷达(Synthetic Aperture Radar简称SAR)图像中的乘性噪声和统计的非正态性对传统的图像分割方法提出了挑战。投影寻踪方法是处理高维非下态数据的有效工具，而投影指标的估计又是投影寻踪方法中的核心问题。本文在研究投影指标的小波估计及其统计性质的基础上，对SAR图像的多尺度投影寻踪分割方法进行了探索性研究。
本文研究了投影指标的小波估计，给出了Cook投影指标族、Friedman投影指标、Hall投影指标和投影寻踪判别分析指标的小波估计。证明了Cook投影指标族、Friedman投影指标和Hall投影指标的小波估计的渐近无偏性和均方收敛性，投影寻踪判别分析指标(Projection Pursuit Dscriminant Analysis简称PPDA)的小波估计的渐近无偏性。
在估计投影指标时需要对密度函数进行估计，本文通过实例计算比较了密度函数的核估计和小波估计。其计算实例结果表明在样本量较大时，密度函数的小波估计较核估计好。
最后，本文给出了多尺度投影特征的提取方法和SAR图像的多尺度投影寻踪有监督分割方法。并分别对单波段机载SAR图像和多波段机载SAR图像作了实例分割。其结果表明该分割方法具有一定的实用性。
Nowadays, traditional methods of image segmentation have been challenged by the multiply noise and non-nomal data of SAR image. Projection pursuit is an effective method for dimension reduced. The estimation of projection index is the core of projection pursuit method. Based on the study of the wavelet estimator of projection index, we explore the segmentation of SAR image used multi-scale projection pursuit.
In this paper, we utilized the wavelet to estimate Cook's, Friedman's and Hall's projection index, and proved that the wavelet estimator of this projection pursuit indices are MSE convergence and asymptotically unbiased., and the wavelet estimator of the PPDA index is asymptotically unbiased.
Estimating the projection index required to estimate the density function of projection data. In this paper, we compared the density function of kernel estimator and the density function of the wavelet estimator through simulation. The result indicated that the wavelet estimator better than the kernel estimator in some conditions.
At last, we proposed a new character extraction method, so called Multi-scale Projection character. As well we give a way to segment SAR image by Multi-scale projection pursuit. From the segmentation results of the single-band and the Multi-band SAR image, we can see that this segmentation method is effective.
本文研究了投影指标的小波估计，给出了Cook投影指标族、Friedman投影指标、Hall投影指标和投影寻踪判别分析指标的小波估计。证明了Cook投影指标族、Friedman投影指标和Hall投影指标的小波估计的渐近无偏性和均方收敛性，投影寻踪判别分析指标(Projection Pursuit Dscriminant Analysis简称PPDA)的小波估计的渐近无偏性。
在估计投影指标时需要对密度函数进行估计，本文通过实例计算比较了密度函数的核估计和小波估计。其计算实例结果表明在样本量较大时，密度函数的小波估计较核估计好。
最后，本文给出了多尺度投影特征的提取方法和SAR图像的多尺度投影寻踪有监督分割方法。并分别对单波段机载SAR图像和多波段机载SAR图像作了实例分割。其结果表明该分割方法具有一定的实用性。
Nowadays, traditional methods of image segmentation have been challenged by the multiply noise and non-nomal data of SAR image. Projection pursuit is an effective method for dimension reduced. The estimation of projection index is the core of projection pursuit method. Based on the study of the wavelet estimator of projection index, we explore the segmentation of SAR image used multi-scale projection pursuit.
In this paper, we utilized the wavelet to estimate Cook's, Friedman's and Hall's projection index, and proved that the wavelet estimator of this projection pursuit indices are MSE convergence and asymptotically unbiased., and the wavelet estimator of the PPDA index is asymptotically unbiased.
Estimating the projection index required to estimate the density function of projection data. In this paper, we compared the density function of kernel estimator and the density function of the wavelet estimator through simulation. The result indicated that the wavelet estimator better than the kernel estimator in some conditions.
At last, we proposed a new character extraction method, so called Multi-scale Projection character. As well we give a way to segment SAR image by Multi-scale projection pursuit. From the segmentation results of the single-band and the Multi-band SAR image, we can see that this segmentation method is effective.
引文
[1] Kruskal, J. B. Linear transformation of multivariate data to reveal clustering. In Multidimensional Scaling: Theory and Application in the Behavioural Sciences, R. N. Shephard, A. K. Romney, and S. K. Nerlove, Eds. Seminar Press, New York, 1972, pp. 179—191.
[2] Friedman, J.H. & Tukey, J.W., A projection pursuit algorithm for exploratory data analysis. IEEE Trans. Computers C-23(1974), 881-889.
[3] Donoho, D., Huber, P.J The use of kinematic displays to represent high dimensional data. in Computer Science and Statistics, Proceedings of the 14th Annual Syrup on the Interface. 1981.
[4] Huber, EJ, Projection pursuit, Research Report PJH-6, Dept. of statistics, Harvard Univ. 1981.
[5] Huber, P.J Density estimation and projection pursuit methods. Research Report PJH-7, Dept. of statistics, Harvard Univ. 1981.
[6] Friedman, J.H and Stuetzle,W., Projection Pursuit Regression. J.Am. Statist. Assoc. Vol.76 pp.817-823, 1981.
[7] Friedman.J.H et al., Projection Pursuit density estimation. J.Am. Statist. Assoc. Vol.79 pp. 599-608, 1984.
[8] Donoho, D., Minimum entroy deconvolution, in Applied Time Series Ⅱ, ed. D.Findlev 1981.
[9] Donoho, D., Breakdown properties of multivaritate location estimation Ph.D. Qualifying Paper, Dept. of Statistics, Harvard Univ. 1982.
[10] 陈忠琏，李国英，Robust principal components and dispersion matrics via projection pursuit. Research Report PJH-8 Dept. of Statistics, Harvard Univ. 1981.
[11] 李国英，陈忠琏，Projection pursuit approach to robust dispersion matrices and principal components. Primary theory and Monte-Carlo. JASA 80 (1985) 759-766.
[12] Hurber, P.H., Projection pursuit. Ann. Statist. 13(1985), 435-475.
[13] 成平，吴键福，Discussion of Huber's paper "Projection pursuit", Ann. Statist. 13(1985), 490-493.
[14] Friedman, J.H. Exploratory projection pursuit. J.Am. Statist. Ass., 82(397): 249-266, March 1987.
[15] Hall, P., On polynomial-based projection indices for exploratory projection pursuit. Ann. Statist., 17(2):589-605, 1989.
[16] Cook, D., Buja,A. and Cabrera, J. Projection pursuit indics based on expansions with orthonormal functions. Journal of Computational and Graphical Statistics, 1993.
[17] Jones, M. C. and Sibson, R. What is projection pursuit? J. R. Statist. Soc. A, 150:1-36, 1987.
[18] Posse, C. An effective two-dimensional projection pursuit algorithm. Comm. Statist.
Simul. Comput., 19(4): 1143-1146, 1990.
[19] Yenyukov, I. S. Indics for projection pursuit. In Diday, editor, Data Analysis Learning Symbolic and Numeric Knowledge, Pages 181-188. Nova Science Publishers, New York, 1989.
[20] Nason, G. P. Deign and choice of projection indices. PhD thesis, University of Bath. 1992.
[21] Nason, G. P. Robust projection indices. J. R. Statist. Series, B, 63,551-567, 2001.
[22] Nason, G. P. and Sibson, R. Using projection pursuit in multispectral image analysis. In Elaine M.Keramidas editor, Proceedings of the 23rd Symposium on the Interface, pages 579-582, P. O. Box 7460, Fairfax station, VA 22039-7460, 1991.
[23] Leng-Neng Hwang, et. al. Regression Modeling in Back-Propagation and Projection Pursuit Learning, IEEE TRANSACTIONS ON NEURAL NETWORKS, Vol. 5 NO. 3, MAY 1994.
[24] C. M. Bachmann, et. al. Dynamically reconfigurable Projection pursuit ensemble for cloud detection in AVIRIS Imagery. IEEE1995.
[25] Trizna, D. B., Bachmann, C. et al, Projection Pursuit Classification of Multiband Polarimetric SAR Land Image, IEEE Trans. Geosci. Remote Sensing, Vol.39, PP.2380-2386, 2001.
[26] C. M. Bachmann, Novel Projection pursuit Indices for feature Extraction and Classification: An Inter-comparision in a Remote Sensing Application, 1997.
[27] M.M.Van Hulle, Nonparametric regression analysis achieved with topographic Maps developed in combination with projection pursuit leaning. IEEE Trans.Signal Processing, Vol.4, NO.5, pp. 117-120, 1997.
[28] S. S. Chiang et. al. Unsupervised Target Detection in Hyperspectrai Images Using Projection Pursuit, IEEE Trans. On Geoscience and Remote Sensing, Vol. 39 NO. 7 July 2001.
[29] C. M. Bachmann, Effects of Time Series on Automated Classification of Costal Wetland Environments Using Projection Pursuit Methods, 2001.
[30] 魏钟铨等著，合成孔径雷达卫星，科学出版社，2001．
[31] Brani Vidakovic, Statistics Modeling by Wavelets, Duke University,1999.
[32] 成平 等著，参数估计，上海科学技术出版社1985．
[33] Dennis B.Trizna, Charles Bachmannn, et. al., Projection Pursuit Classification of Multiband Polarimetric SAR Land Image, IEEE Trans. Geosci. Remote Sensing, Vol. 39, PP. 2380-2386, 2001.
[34] T. Gasser and H-G. Muller. Estimation regression function and their derivatives by the kernel method. Scand. J. Statist. 11 : 171-185,1984.
[35] S.Chitroub, A.Houacine and B.Sansal, Statistical Characterisation and Modeling of SAR Images, Signal Processing, 82: 69-92, 2002.
[36] 章毓晋著，图像分割，科学出版社，2001．
[37] J.K.Jao, Amplitude Distribution of composite Terrain Radar Clutter and K-distribution, IEEE Transactions Antenna Propagation, 32(10): 1049-1062, 1984.
[38] R.Joughin et al, Probability Density Function for Multilook Polarmetric Signatures, IEEE Transaction Geosci. Remote Sensing, 32(30:562-574, 1994.
[39] M.Basserille, A.Berveniste, and A Willsky, Multiscale Auto-regressive Processes, Part Ⅰ: Schur-Levinson Parameterizations, IEEE Transactions on signal Processing, 40: 1915-1934, 1992.
[40] M.Basserille, A.Berveniste, and A Willsky, Multiscale Auto-regressive Processes, Part Ⅱ: Lattice structures for Whitening and Modeling, IEEE Transaction on signal Processing, 40: 1935-1944, 1992.
[41] M.Basserille, A.Berveniste,K.chou, and A.Willsky, Modeling and Estimation of Multiresolution Stochastic Process, IEEE Transactions on Information Theory, 38: 766-784, 1992.
[2] Friedman, J.H. & Tukey, J.W., A projection pursuit algorithm for exploratory data analysis. IEEE Trans. Computers C-23(1974), 881-889.
[3] Donoho, D., Huber, P.J The use of kinematic displays to represent high dimensional data. in Computer Science and Statistics, Proceedings of the 14th Annual Syrup on the Interface. 1981.
[4] Huber, EJ, Projection pursuit, Research Report PJH-6, Dept. of statistics, Harvard Univ. 1981.
[5] Huber, P.J Density estimation and projection pursuit methods. Research Report PJH-7, Dept. of statistics, Harvard Univ. 1981.
[6] Friedman, J.H and Stuetzle,W., Projection Pursuit Regression. J.Am. Statist. Assoc. Vol.76 pp.817-823, 1981.
[7] Friedman.J.H et al., Projection Pursuit density estimation. J.Am. Statist. Assoc. Vol.79 pp. 599-608, 1984.
[8] Donoho, D., Minimum entroy deconvolution, in Applied Time Series Ⅱ, ed. D.Findlev 1981.
[9] Donoho, D., Breakdown properties of multivaritate location estimation Ph.D. Qualifying Paper, Dept. of Statistics, Harvard Univ. 1982.
[10] 陈忠琏，李国英，Robust principal components and dispersion matrics via projection pursuit. Research Report PJH-8 Dept. of Statistics, Harvard Univ. 1981.
[11] 李国英，陈忠琏，Projection pursuit approach to robust dispersion matrices and principal components. Primary theory and Monte-Carlo. JASA 80 (1985) 759-766.
[12] Hurber, P.H., Projection pursuit. Ann. Statist. 13(1985), 435-475.
[13] 成平，吴键福，Discussion of Huber's paper "Projection pursuit", Ann. Statist. 13(1985), 490-493.
[14] Friedman, J.H. Exploratory projection pursuit. J.Am. Statist. Ass., 82(397): 249-266, March 1987.
[15] Hall, P., On polynomial-based projection indices for exploratory projection pursuit. Ann. Statist., 17(2):589-605, 1989.
[16] Cook, D., Buja,A. and Cabrera, J. Projection pursuit indics based on expansions with orthonormal functions. Journal of Computational and Graphical Statistics, 1993.
[17] Jones, M. C. and Sibson, R. What is projection pursuit? J. R. Statist. Soc. A, 150:1-36, 1987.
[18] Posse, C. An effective two-dimensional projection pursuit algorithm. Comm. Statist.
Simul. Comput., 19(4): 1143-1146, 1990.
[19] Yenyukov, I. S. Indics for projection pursuit. In Diday, editor, Data Analysis Learning Symbolic and Numeric Knowledge, Pages 181-188. Nova Science Publishers, New York, 1989.
[20] Nason, G. P. Deign and choice of projection indices. PhD thesis, University of Bath. 1992.
[21] Nason, G. P. Robust projection indices. J. R. Statist. Series, B, 63,551-567, 2001.
[22] Nason, G. P. and Sibson, R. Using projection pursuit in multispectral image analysis. In Elaine M.Keramidas editor, Proceedings of the 23rd Symposium on the Interface, pages 579-582, P. O. Box 7460, Fairfax station, VA 22039-7460, 1991.
[23] Leng-Neng Hwang, et. al. Regression Modeling in Back-Propagation and Projection Pursuit Learning, IEEE TRANSACTIONS ON NEURAL NETWORKS, Vol. 5 NO. 3, MAY 1994.
[24] C. M. Bachmann, et. al. Dynamically reconfigurable Projection pursuit ensemble for cloud detection in AVIRIS Imagery. IEEE1995.
[25] Trizna, D. B., Bachmann, C. et al, Projection Pursuit Classification of Multiband Polarimetric SAR Land Image, IEEE Trans. Geosci. Remote Sensing, Vol.39, PP.2380-2386, 2001.
[26] C. M. Bachmann, Novel Projection pursuit Indices for feature Extraction and Classification: An Inter-comparision in a Remote Sensing Application, 1997.
[27] M.M.Van Hulle, Nonparametric regression analysis achieved with topographic Maps developed in combination with projection pursuit leaning. IEEE Trans.Signal Processing, Vol.4, NO.5, pp. 117-120, 1997.
[28] S. S. Chiang et. al. Unsupervised Target Detection in Hyperspectrai Images Using Projection Pursuit, IEEE Trans. On Geoscience and Remote Sensing, Vol. 39 NO. 7 July 2001.
[29] C. M. Bachmann, Effects of Time Series on Automated Classification of Costal Wetland Environments Using Projection Pursuit Methods, 2001.
[30] 魏钟铨等著，合成孔径雷达卫星，科学出版社，2001．
[31] Brani Vidakovic, Statistics Modeling by Wavelets, Duke University,1999.
[32] 成平 等著，参数估计，上海科学技术出版社1985．
[33] Dennis B.Trizna, Charles Bachmannn, et. al., Projection Pursuit Classification of Multiband Polarimetric SAR Land Image, IEEE Trans. Geosci. Remote Sensing, Vol. 39, PP. 2380-2386, 2001.
[34] T. Gasser and H-G. Muller. Estimation regression function and their derivatives by the kernel method. Scand. J. Statist. 11 : 171-185,1984.
[35] S.Chitroub, A.Houacine and B.Sansal, Statistical Characterisation and Modeling of SAR Images, Signal Processing, 82: 69-92, 2002.
[36] 章毓晋著，图像分割，科学出版社，2001．
[37] J.K.Jao, Amplitude Distribution of composite Terrain Radar Clutter and K-distribution, IEEE Transactions Antenna Propagation, 32(10): 1049-1062, 1984.
[38] R.Joughin et al, Probability Density Function for Multilook Polarmetric Signatures, IEEE Transaction Geosci. Remote Sensing, 32(30:562-574, 1994.
[39] M.Basserille, A.Berveniste, and A Willsky, Multiscale Auto-regressive Processes, Part Ⅰ: Schur-Levinson Parameterizations, IEEE Transactions on signal Processing, 40: 1915-1934, 1992.
[40] M.Basserille, A.Berveniste, and A Willsky, Multiscale Auto-regressive Processes, Part Ⅱ: Lattice structures for Whitening and Modeling, IEEE Transaction on signal Processing, 40: 1935-1944, 1992.
[41] M.Basserille, A.Berveniste,K.chou, and A.Willsky, Modeling and Estimation of Multiresolution Stochastic Process, IEEE Transactions on Information Theory, 38: 766-784, 1992.