InSAR相位解缠算法研究
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摘要
干涉合成孔径雷达(InSAR)三维成像技术是新近发展起来的空间观测技术,它利用合成孔径雷达数据的相位信息提取地面目标的三维信息,具有全天时、全天候、大范围、空间分辨率高等优点,可以广泛应用于军事领域和国民经济建设当中。
本文在总结InSAR技术发展历程的基础上,分析了InSAR系统的成像原理,介绍了系统的数据处理流程。二维相位解缠是InSAR数据处理流程中重要步骤之一,也是主要误差来源,无论是获取数字高程模型还是获取地表形变信息,其精确程度都高度依赖于有效的相位解缠。为了获得高精度的InSAR干涉测量结果,必须实现准确有效的相位解缠,所以有必要对InSAR相位解缠算法进行研究。
本文深入研究了相位解缠的原理,在此基础上分析了实际解缠过程中存在的难点。在详细分析枝切法和区域增长法相位解缠的处理流程及优缺点的基础上,提出了一种基于支持向量机(SVM)的相位解缠算法。算法综合考虑像元与相邻像元的相位差、相位导数方差等属性特征,利用支持向量机优异的分类能力对干涉图像元进行准确的分类,对不同区域分别采取Itoh方法和区域增长的方法进行解缠。基于真实和模拟数据分别进行了实验,实验结果表明该方法与传统方法相比能够避免低质量区域解缠失败,高质量区域更准确,具有更好的鲁棒性,可以应用于大量数据的相位解缠。
针对最小二乘算法能够有效实现全局解缠,但是存在对误差抑制不够导致整体解缠效果下降的问题,本文提出了一种质量图和残差点相结合的InSAR相位解缠算法。算法以干涉图中残差点的分布为依据,优化质量阈值的确定方法,将干涉相位图划分为高低质量区域,更好的指导加权最小二乘法权重的设置,实现相位解缠的顺利进行。实验结果表明,本文提出的方法减少了干涉图中有效信息的流失,并且指标明确,人工干预少,解缠结果准确有效。
本文最后研究了使用网络规划理论进行相位解缠的原理,实现了基于最小网络费用流的相位解缠算法。该算法对干涉相位图建立网络模型,将相位解缠的最小化问题转化为求网络最小费用问题,利用网络规划理论中成熟高效的算法求解。实验结果表明该算法获得了理想的解缠效果。
Interferometric Synthetic Aperture Radar(InSAR)is an advanced space observation technique that have been developed in the last two decades or so. It can be used to extract 3D information of the earth surface or detect the deformation extent of the earth surface from the phase information of SAR data. InSAR has quite a lot of advantages compared with traditional remote sensing technique. For example, it can work without the limitation of time or weather, and it can observe a huge area very quickly.
At the beginning of this paper, we look back the history of how InSAR developed in the last two decades, analyze some fundamental theory and give some description of the data processing procedure. Two-dimensional phase unwrapping is one of the most important steps of this flow, and also the main source of error. When you want to fetch Digital Elevation Model or the deformation of earth surface, you will find that their precision depends on successful phase unwrapping. So it is quite important to investigate InSAR phase unwrapping algorithms.
Then we analyze the principals of phase unwrapping, introduced several traditional methods. From the bases of investigating the advantages and disadvantages of some path-following algorithms, such as Branch-Cut and Regional Growth, we bring forward a algorithm that utilize Support Vector Machine (SVM). In this algorithm, the phase-related information including phased difference, residue and phase derivate variance are utilized. All the pixels are first classified by means of SVM into two types: residue pixels and non-residue pixels. The non-residue region is then unwrapped by the modified Itoh method, while the residue region is unwrapped by a region-growth strategy. The algorithm is finally validated with the real and simulated interferograms. The experiment results show the proposed algorithm is more effective and reliable than some existing algorithms.
As for the Weighted Least-squares algorithm, the extraction of the quality map and the determining of the quality threshold is a very important step. In this paper, we analyses the quality map and its usage, offer an algorithm which correlates quality map and residues. The algorithm optimizes the way to determine quality threshold based on the distribution of the residues. This method will determine the threshold mainly depending on the statistic of residues. According to the experiment results, this method is fast and efficient.
At the end of this paper, we investigate a relatively new phase unwrapping algorithm: minimum cost network flow algorithm. This method changes the phase unwrapping problem into a kind of searching the minimum cost of a network which has mature and efficient methods in network theory. We proved the validation of this algorithm through experiments.
本文在总结InSAR技术发展历程的基础上,分析了InSAR系统的成像原理,介绍了系统的数据处理流程。二维相位解缠是InSAR数据处理流程中重要步骤之一,也是主要误差来源,无论是获取数字高程模型还是获取地表形变信息,其精确程度都高度依赖于有效的相位解缠。为了获得高精度的InSAR干涉测量结果,必须实现准确有效的相位解缠,所以有必要对InSAR相位解缠算法进行研究。
本文深入研究了相位解缠的原理,在此基础上分析了实际解缠过程中存在的难点。在详细分析枝切法和区域增长法相位解缠的处理流程及优缺点的基础上,提出了一种基于支持向量机(SVM)的相位解缠算法。算法综合考虑像元与相邻像元的相位差、相位导数方差等属性特征,利用支持向量机优异的分类能力对干涉图像元进行准确的分类,对不同区域分别采取Itoh方法和区域增长的方法进行解缠。基于真实和模拟数据分别进行了实验,实验结果表明该方法与传统方法相比能够避免低质量区域解缠失败,高质量区域更准确,具有更好的鲁棒性,可以应用于大量数据的相位解缠。
针对最小二乘算法能够有效实现全局解缠,但是存在对误差抑制不够导致整体解缠效果下降的问题,本文提出了一种质量图和残差点相结合的InSAR相位解缠算法。算法以干涉图中残差点的分布为依据,优化质量阈值的确定方法,将干涉相位图划分为高低质量区域,更好的指导加权最小二乘法权重的设置,实现相位解缠的顺利进行。实验结果表明,本文提出的方法减少了干涉图中有效信息的流失,并且指标明确,人工干预少,解缠结果准确有效。
本文最后研究了使用网络规划理论进行相位解缠的原理,实现了基于最小网络费用流的相位解缠算法。该算法对干涉相位图建立网络模型,将相位解缠的最小化问题转化为求网络最小费用问题,利用网络规划理论中成熟高效的算法求解。实验结果表明该算法获得了理想的解缠效果。
Interferometric Synthetic Aperture Radar(InSAR)is an advanced space observation technique that have been developed in the last two decades or so. It can be used to extract 3D information of the earth surface or detect the deformation extent of the earth surface from the phase information of SAR data. InSAR has quite a lot of advantages compared with traditional remote sensing technique. For example, it can work without the limitation of time or weather, and it can observe a huge area very quickly.
At the beginning of this paper, we look back the history of how InSAR developed in the last two decades, analyze some fundamental theory and give some description of the data processing procedure. Two-dimensional phase unwrapping is one of the most important steps of this flow, and also the main source of error. When you want to fetch Digital Elevation Model or the deformation of earth surface, you will find that their precision depends on successful phase unwrapping. So it is quite important to investigate InSAR phase unwrapping algorithms.
Then we analyze the principals of phase unwrapping, introduced several traditional methods. From the bases of investigating the advantages and disadvantages of some path-following algorithms, such as Branch-Cut and Regional Growth, we bring forward a algorithm that utilize Support Vector Machine (SVM). In this algorithm, the phase-related information including phased difference, residue and phase derivate variance are utilized. All the pixels are first classified by means of SVM into two types: residue pixels and non-residue pixels. The non-residue region is then unwrapped by the modified Itoh method, while the residue region is unwrapped by a region-growth strategy. The algorithm is finally validated with the real and simulated interferograms. The experiment results show the proposed algorithm is more effective and reliable than some existing algorithms.
As for the Weighted Least-squares algorithm, the extraction of the quality map and the determining of the quality threshold is a very important step. In this paper, we analyses the quality map and its usage, offer an algorithm which correlates quality map and residues. The algorithm optimizes the way to determine quality threshold based on the distribution of the residues. This method will determine the threshold mainly depending on the statistic of residues. According to the experiment results, this method is fast and efficient.
At the end of this paper, we investigate a relatively new phase unwrapping algorithm: minimum cost network flow algorithm. This method changes the phase unwrapping problem into a kind of searching the minimum cost of a network which has mature and efficient methods in network theory. We proved the validation of this algorithm through experiments.
引文
[1] H. Zebker, R. Goldstein. Topographic mapping from interferometric SAR observations. Journal of Geophysical Research,1986, 9(1):4993-4999
[2] Rogers A. E., Ingalls R. P. Venus: Mapping the Surface Reflectivity by Radar Interfero- metry. Science, 1969, 165:797-799
[3] Zisk S. H. A New Earth-Besed Radar Technique for the Measurement of Lunar Topogra- phy[J]. Moon, 1972, 4:296-300
[4] L. C. Graham. Synthetic Interferometer Radar for Topographic Mapping. Proceedings of the IEEE, 1974, 62:763-768
[5] Gabriel A K, Goldstein R M. Crossed Orbit Interferometry: Theory and Experimental Restlts from SIR-B[J]. Int. J. Remote Sensing, 1988, 9: 857-872.
[6] Li F K, Goldstein R M. Studies of Multibaseline Spacebome Interferometric Synthetic Aperture Radars[J]. IEEE Trans. on Geosci. Remote Sensing, 1990, 28: 88-97.
[7] Didier Massonnet, Kurt L. Feigl. Discrimination of Geophysical Phenomena in Satellite Radar Interferograms. GEOPHYSICAL RESEARCH LETTERS, 1995, 22(12):1537– 1540.
[8] NASA/JPL官方网站 http://southport.jpl.nasa.gov/intpic.html
[9] Rao Y.S., Rao P.V.N., Venkataratnam, L. ERS-1 and JERS-1 SAR data analysis for soil moisture and land cover studies. In: Geoscience and Remote Sensing Symposium, 1996. IGARSS '96. 'Remote Sensing for a Sustainable Future.', International, 1996, 1: 27-31
[10] D. Geudtner, P. W. Vachon, K. E. Mattar, et al. RADARSAT repeat-pass SAR interfer- ometry. Proc. of IGARSS’98. Seattle, 1998, 14(4): 27-31
[11] A. L. Gray, K. E. Mattar and P. W. Vachon. RADARSAT interferometric SAR results obtained over Antartic terrain. In: CEOS workshop. Noordwijk, 1998
[12] J. I. H. Askne, P. B. G. Dammert, L. M. H. Ulander, et al. C-band repeat-pass interferom- etric SAR observations of the forest. IEEE Trans. Geosci. Remote Sensing, 1997, 35(1): 25-35
[13] R. N. Treuhaft, S. N. Madsen, M. Moghaddam, et al. Vegetation characteristics and underlying topography from interferometric radar. Radio Science, 1997, 31(6): 1449-1485
[14] C. Reigber, Y. Xia, G. W. Michel, et al. The antofagasta 1995 earthquake: crustal defor- mation pattern as observed by GPS and D-INSAR. In: Proc. 3rd ERS Symp. Florence, 1997
[15] H. A. Zebker, P. A. Rosen and S. Hensley. Atmospheric effects in interferometric synt- hetic aperture radar surface deformation and topographic maps. J. Geophys. Res., 1997, 102(B4): 7547-7563
[16] Rosen P A,Hensley S,Joughin I R,et al. Synthetic Aperture Radar Interferometry [J]. Proc. IEEE, 2000, 88:333-382.
[17] Massonnet D. Capabilities and limitations of the interferometric cartwheel, IEEE Trans, on Geosciencc and Remote Sensing, 2001, 39(3): 506-520.
[18] Ramongasie S., Phalippou L., Thouvenot E., Massonnet D.. Preliminary design of the payload for the interferometric Car Wheel[C] Proc. Of IGARSS, Honolulu, 2000; 9(3): 24-28.
[19] Suess M., Riegger S., Pitz W., et al . TerraSAR-X: Design andPerformance [C] . The 4th European Conference on Synthetic Aperture Radar , Cologne , Germany , 2002.
[20] Krieger G., Moreira A., Fiedler H., et al . TanDEM-X: a Satellite Formation for High Resolution SAR Interferometry [C] .FRINGE 2005 Workshop, ESA ESRIN , Frascati , Italy , 2005
[21] Hajnsek I., Moreira A. A TanDEM-X: Mission and Science Exploration During the Phase A Study[C] . The 6th European Conference on Synthetic Aperture Radar , Dresden , Germany , 2006
[22] Alberto Moreira, Gerhard Krieger. TanDEM-X: A TerraSAR-X Add-On Satellite for Single-Pass SAR Interferometry. www.dlr.de/hr/tdmx/TanDEM-X-IGAR--SS-2004.pdf
[23] Lengert W. ERS and Envisat missions status[C]. FRINGE 2005 Workshop, ESA ESRIN , Frascati , Italy , 2005
[24] Furuta R., Shimada M., Tadono T., et al . Interferometric Capabilities of ALOS PALSAR and Its Utilization[C] . FRINGE 2005
[25] 郭华东, 徐冠华. 星载雷达应用研究. 北京: 中国科学技术出版社, 1996
[26] 王超. 利用航天飞机成象雷达干涉数据提取数字高程模型. 遥感学报, 1997, 1(1): 46-49
[27] 毛建旭, 王耀南, 夏耶. 合成孔径雷达干涉成像技术及其应用. 系统工程与电子技术, 2003, 1:112-114
[28] 王超,张红,刘智. 星载合成孔径雷达干涉测量. 科学出版社, 2002
[29] 保铮,邢孟道,王彤. 雷达成像技术. 北京:电子工业出版社,2005
[30] Dennis C. Ghiglia Mark D. Pritt, Two-dimensional Phase Unwrapping, JOHN WILEY & SONS, INC., 1998
[31] Goldstein R M, Zebker H A, Werner C L. Satellite radar interferometry: two-dim- ensional phase unwrapping. Radio Science,1988,23:713-720
[32] Wei Xu, Ian Cumming. A region-growing algorithm for InSAR phase unwrapping. IEEE Transaction on Geoscience and Remote Sensing, 1999, 37(1):124-134
[33] Flynn T J. “Consistent 2-D phase unwrapping guided by a quality map,” Proceedings of the 1996 International Geoscience and Remote Sensing Symposium, Lincoln, NE, May 27-31, 1996, IEEE, Piscataway, NJ, 1996,.2057-2059
[34] Flynn T J. Two-dimensional phase unwrapping with minimum weighted discontinuity. Journal of the Optical Society of America, 1997, 14(10):2692- 2701
[35] Ghiglia D. C. and Romero L. A., Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods[J], J. Optical Society of America, 1994, 11(1):107-117.
[36] Pritt M D. Phase unwrapping by means of multigrid techniques for interferometric SAR. IEEE Transaction on Geoscience and Remote Sensing, 1996, 34(3): 728-738
[37] Ghiglia D C, Romero L A. Minimum LP-norm two-dimensional phase unwrapping. Journal of the Optical Society of America,1996, 13(10):1-15
[38] Costantini M. A novel phase unwrapping method based on network programming. IEEE Transaction on Geoscience and Remote Sensing, 1998, 36(3):813-821
[39] A.Ferretti, "Multibaseline interferometric techniques and applications", Ph.D.dissertation, Tech.Univ.Milan, Milan, Italy, 1997
[40] Christopher J. C. Burges. "A Tutorial on Support Vector Machines for Pattern Recogni- tion". Data Mining and Knowledge Discovery , 1998,2:121 - 167
[41] KeHai Xin, ZhangXue Gong. Editing Support Vector Machine. In Proceedings of International Joint Conference on Neural Networks ,Washton ,USA ,2001 ,2:1464-1467.
[42] 孙龙, 胡茂林, 张长耀. InSAR相位解缠最小二乘算法的研究. 雷达科学与技术,2005,4(1):55-57.
[43] Pritt M. D., “Least-squares two-dimensional phase unwrapping using FFT's[J], IEEE Transaction on Geoscience and Remote Sensing, 1994,32(3): 706-708
[44] 张文生 科学计算中的偏微分方程有限差分法 高等教育出版社, 2006.06
[45] Carballo G F,Fieguth P W. Probabilistic Cost Functions for Network Flow Phase Unwrapping. IEEETrans.Geosci.Remote Sens.,2000,38(5):2192-22O1
[46] Chen C W,Zebker H A. Network Approaches to Two-dimensional Phase Unwrapping: Intractability and Two New Algorithms. Journal of the Optical Society of America A,2000,17(3):401-414
[47] Chen C W, Zebker H A.Two-dimensional Phase Unwrapping with Use of Statistical Models for Cost Functions in Nonlinear Optimization.Journal of the Optical Society of America A .2001,18(2):338-351
[2] Rogers A. E., Ingalls R. P. Venus: Mapping the Surface Reflectivity by Radar Interfero- metry. Science, 1969, 165:797-799
[3] Zisk S. H. A New Earth-Besed Radar Technique for the Measurement of Lunar Topogra- phy[J]. Moon, 1972, 4:296-300
[4] L. C. Graham. Synthetic Interferometer Radar for Topographic Mapping. Proceedings of the IEEE, 1974, 62:763-768
[5] Gabriel A K, Goldstein R M. Crossed Orbit Interferometry: Theory and Experimental Restlts from SIR-B[J]. Int. J. Remote Sensing, 1988, 9: 857-872.
[6] Li F K, Goldstein R M. Studies of Multibaseline Spacebome Interferometric Synthetic Aperture Radars[J]. IEEE Trans. on Geosci. Remote Sensing, 1990, 28: 88-97.
[7] Didier Massonnet, Kurt L. Feigl. Discrimination of Geophysical Phenomena in Satellite Radar Interferograms. GEOPHYSICAL RESEARCH LETTERS, 1995, 22(12):1537– 1540.
[8] NASA/JPL官方网站 http://southport.jpl.nasa.gov/intpic.html
[9] Rao Y.S., Rao P.V.N., Venkataratnam, L. ERS-1 and JERS-1 SAR data analysis for soil moisture and land cover studies. In: Geoscience and Remote Sensing Symposium, 1996. IGARSS '96. 'Remote Sensing for a Sustainable Future.', International, 1996, 1: 27-31
[10] D. Geudtner, P. W. Vachon, K. E. Mattar, et al. RADARSAT repeat-pass SAR interfer- ometry. Proc. of IGARSS’98. Seattle, 1998, 14(4): 27-31
[11] A. L. Gray, K. E. Mattar and P. W. Vachon. RADARSAT interferometric SAR results obtained over Antartic terrain. In: CEOS workshop. Noordwijk, 1998
[12] J. I. H. Askne, P. B. G. Dammert, L. M. H. Ulander, et al. C-band repeat-pass interferom- etric SAR observations of the forest. IEEE Trans. Geosci. Remote Sensing, 1997, 35(1): 25-35
[13] R. N. Treuhaft, S. N. Madsen, M. Moghaddam, et al. Vegetation characteristics and underlying topography from interferometric radar. Radio Science, 1997, 31(6): 1449-1485
[14] C. Reigber, Y. Xia, G. W. Michel, et al. The antofagasta 1995 earthquake: crustal defor- mation pattern as observed by GPS and D-INSAR. In: Proc. 3rd ERS Symp. Florence, 1997
[15] H. A. Zebker, P. A. Rosen and S. Hensley. Atmospheric effects in interferometric synt- hetic aperture radar surface deformation and topographic maps. J. Geophys. Res., 1997, 102(B4): 7547-7563
[16] Rosen P A,Hensley S,Joughin I R,et al. Synthetic Aperture Radar Interferometry [J]. Proc. IEEE, 2000, 88:333-382.
[17] Massonnet D. Capabilities and limitations of the interferometric cartwheel, IEEE Trans, on Geosciencc and Remote Sensing, 2001, 39(3): 506-520.
[18] Ramongasie S., Phalippou L., Thouvenot E., Massonnet D.. Preliminary design of the payload for the interferometric Car Wheel[C] Proc. Of IGARSS, Honolulu, 2000; 9(3): 24-28.
[19] Suess M., Riegger S., Pitz W., et al . TerraSAR-X: Design andPerformance [C] . The 4th European Conference on Synthetic Aperture Radar , Cologne , Germany , 2002.
[20] Krieger G., Moreira A., Fiedler H., et al . TanDEM-X: a Satellite Formation for High Resolution SAR Interferometry [C] .FRINGE 2005 Workshop, ESA ESRIN , Frascati , Italy , 2005
[21] Hajnsek I., Moreira A. A TanDEM-X: Mission and Science Exploration During the Phase A Study[C] . The 6th European Conference on Synthetic Aperture Radar , Dresden , Germany , 2006
[22] Alberto Moreira, Gerhard Krieger. TanDEM-X: A TerraSAR-X Add-On Satellite for Single-Pass SAR Interferometry. www.dlr.de/hr/tdmx/TanDEM-X-IGAR--SS-2004.pdf
[23] Lengert W. ERS and Envisat missions status[C]. FRINGE 2005 Workshop, ESA ESRIN , Frascati , Italy , 2005
[24] Furuta R., Shimada M., Tadono T., et al . Interferometric Capabilities of ALOS PALSAR and Its Utilization[C] . FRINGE 2005
[25] 郭华东, 徐冠华. 星载雷达应用研究. 北京: 中国科学技术出版社, 1996
[26] 王超. 利用航天飞机成象雷达干涉数据提取数字高程模型. 遥感学报, 1997, 1(1): 46-49
[27] 毛建旭, 王耀南, 夏耶. 合成孔径雷达干涉成像技术及其应用. 系统工程与电子技术, 2003, 1:112-114
[28] 王超,张红,刘智. 星载合成孔径雷达干涉测量. 科学出版社, 2002
[29] 保铮,邢孟道,王彤. 雷达成像技术. 北京:电子工业出版社,2005
[30] Dennis C. Ghiglia Mark D. Pritt, Two-dimensional Phase Unwrapping, JOHN WILEY & SONS, INC., 1998
[31] Goldstein R M, Zebker H A, Werner C L. Satellite radar interferometry: two-dim- ensional phase unwrapping. Radio Science,1988,23:713-720
[32] Wei Xu, Ian Cumming. A region-growing algorithm for InSAR phase unwrapping. IEEE Transaction on Geoscience and Remote Sensing, 1999, 37(1):124-134
[33] Flynn T J. “Consistent 2-D phase unwrapping guided by a quality map,” Proceedings of the 1996 International Geoscience and Remote Sensing Symposium, Lincoln, NE, May 27-31, 1996, IEEE, Piscataway, NJ, 1996,.2057-2059
[34] Flynn T J. Two-dimensional phase unwrapping with minimum weighted discontinuity. Journal of the Optical Society of America, 1997, 14(10):2692- 2701
[35] Ghiglia D. C. and Romero L. A., Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods[J], J. Optical Society of America, 1994, 11(1):107-117.
[36] Pritt M D. Phase unwrapping by means of multigrid techniques for interferometric SAR. IEEE Transaction on Geoscience and Remote Sensing, 1996, 34(3): 728-738
[37] Ghiglia D C, Romero L A. Minimum LP-norm two-dimensional phase unwrapping. Journal of the Optical Society of America,1996, 13(10):1-15
[38] Costantini M. A novel phase unwrapping method based on network programming. IEEE Transaction on Geoscience and Remote Sensing, 1998, 36(3):813-821
[39] A.Ferretti, "Multibaseline interferometric techniques and applications", Ph.D.dissertation, Tech.Univ.Milan, Milan, Italy, 1997
[40] Christopher J. C. Burges. "A Tutorial on Support Vector Machines for Pattern Recogni- tion". Data Mining and Knowledge Discovery , 1998,2:121 - 167
[41] KeHai Xin, ZhangXue Gong. Editing Support Vector Machine. In Proceedings of International Joint Conference on Neural Networks ,Washton ,USA ,2001 ,2:1464-1467.
[42] 孙龙, 胡茂林, 张长耀. InSAR相位解缠最小二乘算法的研究. 雷达科学与技术,2005,4(1):55-57.
[43] Pritt M. D., “Least-squares two-dimensional phase unwrapping using FFT's[J], IEEE Transaction on Geoscience and Remote Sensing, 1994,32(3): 706-708
[44] 张文生 科学计算中的偏微分方程有限差分法 高等教育出版社, 2006.06
[45] Carballo G F,Fieguth P W. Probabilistic Cost Functions for Network Flow Phase Unwrapping. IEEETrans.Geosci.Remote Sens.,2000,38(5):2192-22O1
[46] Chen C W,Zebker H A. Network Approaches to Two-dimensional Phase Unwrapping: Intractability and Two New Algorithms. Journal of the Optical Society of America A,2000,17(3):401-414
[47] Chen C W, Zebker H A.Two-dimensional Phase Unwrapping with Use of Statistical Models for Cost Functions in Nonlinear Optimization.Journal of the Optical Society of America A .2001,18(2):338-351