﻿ 使用<i>L</i><sup>0</sup>范数代价函数的二维相位快速解缠方法
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 大地测量与地球动力学  2019, Vol. 39 Issue (1): 51-56  DOI: 10.14075/j.jgg.2019.01.010

### 引用本文

SHI Juan, GAO Jian, ZHANG Chengbin, et al. An Effective Two-Dimensional Phase Unwrapping Method Using Cost Function Model of L0 Norm[J]. Journal of Geodesy and Geodynamics, 2019, 39(1): 51-56.

### Foundation support

Open Fund of Key Laboratory of Precise Engineering and Industry Surveying, NASMG, No. PF2017-3.

### 第一作者简介

SHI Juan, PhD, lecturer, majors in photogrammerty and image processing, E-mail:shijuan39@126.com.

### 文章历史

1. 淮海工学院测绘与海洋信息学院, 江苏省连云港市苍梧路59号, 222005;
2. 南京邮电大学地理与生物信息学院，南京市文苑路9号, 210023;
3. 山东省国土测绘院，济南市经十东路2289号，250013

1 最小范数解缠方法的代价函数

 $\psi = W\left( \varphi \right)$

 $\mathop {\inf }\limits_\varphi \int\limits_\mathit{\Omega } {g\left( {\left| {\nabla u} \right|} \right){\rm{d}}x}$ (1)

 ${\rm{div}}\left( {\frac{{g'\left( {\left| {\nabla u} \right|} \right)}}{{\left| {\nabla u} \right|}}\nabla u} \right) = 0$ (2)

 $\nabla \varphi = W\left( {\nabla \psi } \right),\nabla u = \nabla \phi - \nabla \varphi = \nabla \phi - W\left( {\nabla \psi } \right)$

 ${\rm{div}}\left( {\frac{{g'\left( {\left| {\nabla \phi - W\left( {\nabla \psi } \right)} \right|} \right)}}{{\left| {\nabla \phi - W\left( {\nabla \psi } \right)} \right|}}\left| {\nabla \phi - W\left( {\nabla \psi } \right)} \right|} \right) = 0$ (3)

 图 1 代价函数的3种范数形式 Fig. 1 Cost function of three norm forms

 $\frac{{g'\left( {\left| {\nabla u} \right|} \right)}}{{\left| {\nabla u} \right|}}{u_{TT}} + g''\left( {\left| {\nabla u} \right|} \right){u_{NN}} = 0$

 $\begin{array}{*{20}{c}} {\mathop {\lim }\limits_{\left| {\nabla u} \right| \to {0^ + }} \frac{{g'\left( {\left| {\nabla u} \right|} \right)}}{{\left| {\nabla u} \right|}} = }\\ {\mathop {\lim }\limits_{\left| {\nabla u} \right| \to {0^ + }} g''\left( {\left| {\nabla u} \right|} \right) = g''\left( 0 \right) > 0} \end{array}$ (4)

 $\mathop {\lim }\limits_{\left| {\nabla u} \right| \to + \infty } \frac{{g'\left( {\left| {\nabla u} \right|} \right)}}{{\left| {\nabla u} \right|}} > 0,\mathop {\lim }\limits_{\left| {\nabla u} \right| \to + \infty } g''\left( {\left| {\nabla u} \right|} \right) = 0$

 $\begin{array}{l} \mathop {\lim }\limits_{\left| {\nabla u} \right| \to + \infty } \frac{{g'\left( {\left| {\nabla u} \right|} \right)}}{{\left| {\nabla u} \right|}} = 0\\ \mathop {\lim }\limits_{\left| {\nabla u} \right| \to + \infty } g''\left( {\left| {\nabla u} \right|} \right) = 0\\ \frac{{g'\left( {\left| {\nabla u} \right|} \right)}}{{\left| {\nabla u} \right|}} > g''\left( {\left| {\nabla u} \right|} \right) \end{array}$ (5)

2 基于L0范数准则的代价函数

 $g\left( {\left| {\nabla u} \right|} \right) = \frac{{{{\left| {\nabla u} \right|}^2}}}{{1 + {{\left| {\nabla u} \right|}^2}}}$ (6)

 $\frac{{g'\left( {\left| {\nabla u} \right|} \right)}}{{\left| {\nabla u} \right|}} = \frac{1}{{{{\left( {1 + {{\left| {\nabla u} \right|}^2}} \right)}^2}}}$
 $g''\left( {\left| {\nabla u} \right|} \right) = \frac{{1 - 3{{\left| {\nabla u} \right|}^2}}}{{{{\left( {1 + {{\left| {\nabla u} \right|}^2}} \right)}^3}}}$

 ${\rm{div}}\frac{{\nabla \phi - W\left( {\nabla \psi } \right)}}{{{{\left( {1 + {{\left| {\nabla \phi - W\left( {\nabla \psi } \right)} \right|}^2}} \right)}^2}}} = 0$

 ${\rm{div}}\frac{{\nabla \phi - W\left( {\nabla \psi } \right)}}{{{{\left( {\alpha + {{\left| {\nabla \phi - W\left( {\nabla \psi } \right)} \right|}^2}} \right)}^2}}} = 0$ (7)

 图 2 不同尺度系数的代价函数 Fig. 2 Cost function of different scale coefficients

3 解缠方法的数值实现

 ${\rm{div}}\left( {{b_n}\nabla {\phi _{n + 1}}} \right) = {\rm{div}}\left( {{b_n}W\left( {\nabla \psi } \right)} \right)$
 ${b_n} = \frac{1}{{{{\left( {\alpha + {{\left| {\nabla {\phi _n} - W\left( {\nabla \psi } \right)} \right|}^2}} \right)}^2}}}$

 ${\rm{div}}\left( {{b_n}\nabla \delta {\phi _{n + 1}}} \right) = {\rm{div}}\left( {{b_n}W\left( {\nabla \psi } \right) - \nabla \delta {\phi _n}} \right)$
 $\delta {\phi _n} = {\phi _{n + 1}} - {\phi _n}$

 $\delta {\psi _n} = \psi - {\phi _n}$

 图 3 数据分块解缠 Fig. 3 Unwrapping with data partitioning

4 数据实验与分析

 图 4 合成缠绕相位、解缠结果及其对应的不连续边界 Fig. 4 Artificial wrapped phase, unwrapping results and its discontinuous boundary

 图 5 前4次和最后1次方程系数以及对应的相位残量 Fig. 5 First four and last coefficients and the corresponding phase residue

 图 6 使用2×2和3×3分块获得解缠结果和不连续边界分布 Fig. 6 Unwrapping results and discontinuous boundaries of 2×2 and 3×3 partitions

 图 7 InSAR干涉相位数据14×6分块解缠处理 Fig. 7 Unwrapping process of InSAR phase of 14×6 partitions

5 结语

 [1] Huang Q, Zhou H Q, Dong S C, et al. Parallel Branch-Cut Algorithm Based on Simulated Annealing for Large-Scale Phase Unwrapping[J]. IEEE Transactions on Geoscience and Remote Sensing, 2015, 53(7): 3833-3846 DOI:10.1109/TGRS.2014.2385482 (0) [2] Gao D P, Yin F L. Mask Cut Optimization in Two-Dimensional Phase Unwrapping[J]. IEEE Geoscience and Remote Sensing Letters, 2012, 9(3): 338-342 DOI:10.1109/LGRS.2011.2168940 (0) [3] Zhong H P, Tang J S, Zhang S, et al. A Quality-Guided and Local Minimum Discontinuity Based Phase Unwrapping Algorithm for InSAR/InSAS Interferograms[J]. IEEE Geoscience and Remote Sensing Letters, 2014, 11(1): 215-219 DOI:10.1109/LGRS.2013.2252880 (0) [4] 钟何平, 唐劲松, 张森. 一种基于质量引导和最小不连续合成的InSAR相位解缠算法[J]. 电子与信息学报, 2011, 33(2): 369-374 (Zhong Heping, Tang Jinsong, Zhang Sen. A Combined Phase Unwrapping Algorithm Based on Quality-Guided and Minimum Discontinuity for InSAR[J]. Journal of Electronics & Information Technology, 2011, 33(2): 369-374) (0) [5] Flynn T J. Phase Unwrapping Using Discontinuity Optimization[C]. Geoscience and Remote Sensing IEEE International Symposium, Seattle, 1998 https://www.researchgate.net/publication/3760930_Phase_unwrapping_using_discontinuity_optimization (0) [6] 陈强, 杨莹辉, 刘国祥, 等. 基于边界探测的InSAR最小二乘整周相位解缠方法[J]. 测绘学报, 2012, 41(3): 441-448 (Chen Qiang, Yang Yinghui, Liu Guoxiang, et al. InSAR Phase Unwrapping Using Least Squares Method with Integer Ambiguity Resolution and Edge Detection[J]. Acta Geodaeticaet Cartographica Sinica, 2012, 41(3): 441-448) (0) [7] 陆军, 李积江, 王成成, 等. 基于构造边的精确快速相位解缠算法[J]. 光电子·激光, 2015, 26(1): 122-129 (Lu Jun, Li Jijiang, Wang Chengcheng, et al. An Accurate and Fast Phase Unwrapping Algorithm Based on Constructed Edge[J]. Journal of Optoelectronics·Laser, 2015, 26(1): 122-129) (0) [8] Ghiglia D C, Romero L A. Minimum Lp-Norm Two-Dimensional Phase Unwrapping[J]. Journal of the Optical Society of America A, 1996, 13(10): 1999-2013 DOI:10.1364/JOSAA.13.001999 (0) [9] Ghiglia D C, Romero L A. Robust Two-Dimensional Weighted and Unweighted Phase Unwrapping That Uses Fast Transforms and Iterative Methods[J]. Journal of the Optical Society of America A, 1994, 11(1): 107-117 DOI:10.1364/JOSAA.11.000107 (0) [10] Liu H T, Xing M D, Bao Z. A Novel Mixed-Norm Multibaseline Phase-Unwrapping Algorithm Based on Linear Programming[J]. IEEE Geoscience and Remote Sensing Letters, 2015, 12(5): 1086-1090 DOI:10.1109/LGRS.2014.2381666 (0) [11] 刘国林, 郝华东, 陶秋香. 卡尔曼滤波相位解缠及其与其他方法的对比分析[J]. 武汉大学学报:信息科学版, 2010, 35(10): 1174-1178 (Liu Guolin, Hao Huadong, Tao Qiuxiang. Kalman Filter Phase Unwrapping Algorithm and Comparison and Analysis with Other Methods[J]. Geomatics and Information Science of Wuhan University, 2010, 35(10): 1174-1178) (0) [12] 魏志强, 金亚秋. 基于蚁群算法的InSAR相位解缠算法[J]. 电子与信息学报, 2008, 30(3): 518-523 (Wei Zhiqiang, Jin Yaqiu. InSAR Phase Unwrapping Algorithm Based on Ant Colony Algorithm[J]. Journal of Electronics & Information Technology, 2008, 30(3): 518-523) (0) [13] Aubert G, Kornprobst P. Mathematical Problems in Image Processing[M]. New York: Springer, 2006 (0) [14] Chen C W. Statistical-Cost Network-Flow Approaches to Two-Dimensional Phase Unwrapping for Radar Interferometry[D]. Palo Alto: Stanford University, 2001 http://citeseerx.ist.psu.edu/showciting?cid=488208 (0) [15] Aubert G, Kornprobst P. Mathematical Problems in Image Processing[M]. New York: Spinger, 2006 (0) [16] Kahan W M. Numerical Linear Algebra[M]. Lincoln: Mathematics & Computer Education, 2013 (0) [17] Demmel J W. Applied Numerical Linear Algebra[M]. USA: Society for Industrial and Applied Mathematics, 1997 (0)
An Effective Two-Dimensional Phase Unwrapping Method Using Cost Function Model of L0 Norm
SHI Juan1     GAO Jian2     ZHANG Chengbin3     FEI Xianyun1
1. School of Surveying and Oceanographic Information Engineering, Huaihai Institute of Technology, 59 Cangwu Road, Lianyungang 222005, China;
2. School of Geographic and Biologic Information, Nanjing University of Posts and Telecommunications, 9 Wenyuan Road, Nanjing 210023, China;
3. Shandong Provincal Institute of Land Surveying and Mapping, 2289 East-Jingshi Road, Ji'nan 250013, China
Abstract: In order to improve the cost model and convergence speed of the minimum norm unwrapping, we propose a two-dimensional global phase unwrapping method optimized with L0 norm. By analyzing the features of cost model in phase unwrapping with minimum norm, a cost function definition is employed in line with the L0 norm, which imposes a stronger constraint in the tangent direction of phase discontinuity boundary than that in normal direction. Meanwhile, to overcome slow convergence of low-frequency error compensation in linear equations, a data partitioning strategy is adopted in unwrapping processing, in view of the intrinsic independence of the minimum norm method, which only focuses on high-frequency information to improve efficiency in the iterative process, but processing of the low-frequency part is transferred to offsetting between data partitions. The experiment shows that reliability and efficiency of the proposed phase unwrapping method are enhanced.
Key words: phase unwrapping; minimum norm; cost function; partitioning processing