基于质量图的相位展开方法研究
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摘要
相位展开是光学干涉面形测量、干涉合成孔径雷达、医学磁共振成像等研究领域数据处理的一个关键步骤,在一定程度上决定了测量结果的精确性和可靠性。多年来,人们对相位展开技术做了大量研究,但仍有许多问题值得深入地探讨和研究。本文在参照国内外相位展开方法的基础上,对相位展开方法进行了深入地研究,主要工作如下:
本文首先介绍了相位展开在光学三维面形测量技术中的应用,讨论了两种主要的三维传感技术,重点介绍了傅里叶变换轮廓术的原理。阐述相位展开的原理,残差点和质量图等基本概念,及相位展开算法的数学框架。二维相位展开问题的范数优化框架根据p值的不同可以分为路径积分法和最小范数法等。介绍传统路径积分法的原理,模拟用其展开时出现的问题。Lp
然后在传统路径积分法的基础上通过一个表征包裹相位质量的质量图来导引相位展开。通过识别质量图数据,从相位图中质量值最高像素点开始,按相位质量高低的顺序对相位图进行路径积分,最后展开质量值较低的不可靠区域,这样可以把不可避免的误差局限在最小范围内。论文在研究相结合的传统质量图基础上,对最大相位梯度质量图进行改进,并提出一种结合边缘检测信息的改进质量图。并针对几种典型的相位不连续问题,如噪声、阴影和剪切图,用这几种不同的质量图进行了相位展开的数值模拟,获得较为理想的相位展开结果,由此证明新提出的这几种质量图具有很强的可靠性和抗噪声能力。
最后,相位展开的最小二乘算法可看作在纽曼边界条件下求解泊松方程,深入研究了不加权的的最小范数法——基于离散余弦变换求解泊松方程的相位展开方法,以及以它的结果作为初始迭代条件的加权的最小范数法——预处理共轭梯度法。利用质量图、像素掩膜等作为权重,如前面提出的新质量图及二值掩膜,它们可以有效区分残差点和非残差点,从而得到较好的相位展开结果。模拟了这两种方法针对不同包裹相位图的展开结果,可以看出基于离散余弦变换的方法速度快,对某些情况展开结果不够理想;预处理共轭梯度法由于涉及大型矩阵运算速度很慢,但是利用质量图、像素掩膜等作为权重,通过迭代可以获得较理想的结果。最后补充了Hunt关于相位展开最小范数法的矩阵理论的相关推导,作为前面理论及算法的基础。
Phase unwrapping is one of the key steps of data processing in the field of the shape measurement of optical interference, interferometric synthetic aperture radar (INSAR) and medical magnetic resonance imaging, which determines the accuracy and reliability of measurement results to some extent. Over the years, people made a lot of research on phase unwrapping techniques, but the phase unwrapping problem has many issues worthy of in-depth study and discussion. Based on the literatures of phase unwrapping from home and abroad, the in-depth researches on the phase unwrapping method are carried out in this thesis. The main work of the thesis are as follows:
First of all, we introduce the application of phase unwrapping in the optical shape measurement technology, discuss two main three-dimensional sensor technology, and highlight the principle of Fourier transform profilometry. Then we expound the principle of phase unwrapping, the basic concept of residues and quality map, and the mathematical framework for phase unwrapping algorithm. Two dimensional phase unwrapping problem is essentially a constrained optimization problem. Its norm optimization framework can be divided into path integral method and the minimum-norm method. We introduce the principle of path integral method, and simulate the disadvantage of the method. Lp
A quality map which characterizes the quality of the phase can guide phase unwrapping based on the traditional method of phase wrapping. By identifying datas of quality map , the phase path integral begins from the highest quality value, in order of the phase quality value. The unreliable regions which have low quality value are unwrapped at last so that the inevitable errors can be limited to minimum. We improve the maximum phase gradient quality map and propose a new quality map combined edge detection in the study of traditional quality map. For several typical phase discontinuities, such as noises, shadow and sheared plane, the simulations show the relatively optimal phase unwrapping results with several different quality maps. Therefore the quality maps we proposed have strong reliability and noise resistance.
The least squares phase unwrapping algorithm can be seen as solving the Poisson equation with Neumann boundary conditions. The non-weighted minimum norm method is based on discrete cosine transform, and with its results as the initial conditions the weighted iterative minimum norm method is represented by Preconditioned Conjugate Gradient Method. Quality map and pixel mask can be the weight, such as previously proposed new quality map and the binary mask. They can distinguish non-residues and residues, and thus get a better phase unwrapping results. The simulations of the two methods for different wrapping phase show discrete cosine transform method is faster, but sometimes the results are not ideal, and preconditioned conjugate gradient method is slow as it involves a large number of matrix computation, but with quality map and pixel mask as the weight, can get a better result through iteration. At last the matrix theory of the minimum norm method about phase unwrapping from Hunt is derived and added as the basis of previous theory and algorithm.
本文首先介绍了相位展开在光学三维面形测量技术中的应用,讨论了两种主要的三维传感技术,重点介绍了傅里叶变换轮廓术的原理。阐述相位展开的原理,残差点和质量图等基本概念,及相位展开算法的数学框架。二维相位展开问题的范数优化框架根据p值的不同可以分为路径积分法和最小范数法等。介绍传统路径积分法的原理,模拟用其展开时出现的问题。Lp
然后在传统路径积分法的基础上通过一个表征包裹相位质量的质量图来导引相位展开。通过识别质量图数据,从相位图中质量值最高像素点开始,按相位质量高低的顺序对相位图进行路径积分,最后展开质量值较低的不可靠区域,这样可以把不可避免的误差局限在最小范围内。论文在研究相结合的传统质量图基础上,对最大相位梯度质量图进行改进,并提出一种结合边缘检测信息的改进质量图。并针对几种典型的相位不连续问题,如噪声、阴影和剪切图,用这几种不同的质量图进行了相位展开的数值模拟,获得较为理想的相位展开结果,由此证明新提出的这几种质量图具有很强的可靠性和抗噪声能力。
最后,相位展开的最小二乘算法可看作在纽曼边界条件下求解泊松方程,深入研究了不加权的的最小范数法——基于离散余弦变换求解泊松方程的相位展开方法,以及以它的结果作为初始迭代条件的加权的最小范数法——预处理共轭梯度法。利用质量图、像素掩膜等作为权重,如前面提出的新质量图及二值掩膜,它们可以有效区分残差点和非残差点,从而得到较好的相位展开结果。模拟了这两种方法针对不同包裹相位图的展开结果,可以看出基于离散余弦变换的方法速度快,对某些情况展开结果不够理想;预处理共轭梯度法由于涉及大型矩阵运算速度很慢,但是利用质量图、像素掩膜等作为权重,通过迭代可以获得较理想的结果。最后补充了Hunt关于相位展开最小范数法的矩阵理论的相关推导,作为前面理论及算法的基础。
Phase unwrapping is one of the key steps of data processing in the field of the shape measurement of optical interference, interferometric synthetic aperture radar (INSAR) and medical magnetic resonance imaging, which determines the accuracy and reliability of measurement results to some extent. Over the years, people made a lot of research on phase unwrapping techniques, but the phase unwrapping problem has many issues worthy of in-depth study and discussion. Based on the literatures of phase unwrapping from home and abroad, the in-depth researches on the phase unwrapping method are carried out in this thesis. The main work of the thesis are as follows:
First of all, we introduce the application of phase unwrapping in the optical shape measurement technology, discuss two main three-dimensional sensor technology, and highlight the principle of Fourier transform profilometry. Then we expound the principle of phase unwrapping, the basic concept of residues and quality map, and the mathematical framework for phase unwrapping algorithm. Two dimensional phase unwrapping problem is essentially a constrained optimization problem. Its norm optimization framework can be divided into path integral method and the minimum-norm method. We introduce the principle of path integral method, and simulate the disadvantage of the method. Lp
A quality map which characterizes the quality of the phase can guide phase unwrapping based on the traditional method of phase wrapping. By identifying datas of quality map , the phase path integral begins from the highest quality value, in order of the phase quality value. The unreliable regions which have low quality value are unwrapped at last so that the inevitable errors can be limited to minimum. We improve the maximum phase gradient quality map and propose a new quality map combined edge detection in the study of traditional quality map. For several typical phase discontinuities, such as noises, shadow and sheared plane, the simulations show the relatively optimal phase unwrapping results with several different quality maps. Therefore the quality maps we proposed have strong reliability and noise resistance.
The least squares phase unwrapping algorithm can be seen as solving the Poisson equation with Neumann boundary conditions. The non-weighted minimum norm method is based on discrete cosine transform, and with its results as the initial conditions the weighted iterative minimum norm method is represented by Preconditioned Conjugate Gradient Method. Quality map and pixel mask can be the weight, such as previously proposed new quality map and the binary mask. They can distinguish non-residues and residues, and thus get a better phase unwrapping results. The simulations of the two methods for different wrapping phase show discrete cosine transform method is faster, but sometimes the results are not ideal, and preconditioned conjugate gradient method is slow as it involves a large number of matrix computation, but with quality map and pixel mask as the weight, can get a better result through iteration. At last the matrix theory of the minimum norm method about phase unwrapping from Hunt is derived and added as the basis of previous theory and algorithm.
引文
1. F. Chen. Overview of Three-Dimensional Shape Measurement Using Optical Methods. Opt. Eng. 2000, 39 (1): 10-22
2.苏显渝.三维面形测量技术的新进展.物理. 1996, 25(10): 614-620
3. M. K. Kalms, W. Osten, W. Juptner. Active Industrial Surface Inspection with the Inverse Projected-Fringe-Technique. Proc.SPIE. 2001, 4596: 37-47
4. R. Kowarschik, J. Gerber, G. Notni. Adaptive Optical 3D-Measurement with Structured Light. Proc. SPIE. 1999, 3824: 169-178
5. M. Tkeda, K. Motoh. Fourier Transform Profilometry for the Automatic Measurement of 3-D Object Shapes. Appl. Opt.. 1983, 22(24): 3977-3982.
6. X. Su, W. Chen, Q. Zhang. Dynamic 3-D Shape Measurement Method Based on FTP. Optics and Lasers in Engineering. 2001, 36: 49-64
7. J. Esteve-Taboada, J. Garcia, C. Ferreira. Application of Fourier Transform Profilometry to the Recognition of 3-D Objects. Proc. SPIE. 2003, 5227: 59-66
8. F. Berryman, P. Pynsent, J. Cubillo. The Effect of Windowing in Fourier Transform Porfilometry Applied to Noisy Images. Optics and Lasers in Engineering. 2004, 41 (6): 815-825.
9.陈锋.动态三维面形测量中相位展开问题的研究.四川大学. 2005: 5-6.
10. R. Zhu, J. Chen. Accuracy of Phase Shifting Interferometry—Phase Shifter Calibration and Error Compensation. Proc. SPIE. 1992, 1553: 702-710
11. J. Schmit, K. Creath. Window Function Influence on Phase Error in Phase-Shifting Algorithm. Appl. Opt.. 1996, 35(28): 5642-5465
12. D. C. Ghiglia, M. D. Pritt. Two-Dimensional Phase Unwrapping Theory, Algorithms and Software. John Wiley & Sons. 1998: 68-73
13. V. V. Volkov, Y. Zhu. Deterministic Phase Unwrapping in the Presence of Noise. Optics Letters. 2003, 28(22): 2156-2158
14. D. Kerr, G. H. Kaufmann, G.E. Galizzi. Unwrapping of Interferometric Phase-Fringe Maps by the Discrete Cosine Transform. Appl. Opt.. 1996, 35(5): 810-816
15. D. J. Bone. Fourier Fringe Analysis: the Two-Dimensional Phase Un-wrapping Problem. Applied Optics. 1991, 30(25): 3627-3632.
16. K. Itoh. Analysis of the Phase Unwrapping Algorithm. Applied Optics. 1982, 21(14): 2470
17. T. J. Flynn. Two-Dimensional Phase Unwrapping with Minimum Weighted Discontinuity. J. Opt. Soc. Am. A.. 1997, 14(10): 2692-2701
18. L. Su, X. Su, W. Li. Application of Modulation Measurement Profilometry(MMP) to Objects with Holes on Surface. Appl. Opt.. 1999, 38(7): 1153-1158
19. D. C. Ghigla, L. A. Romero. Minimun Lp-norm Two-Dimnesional Phase Unwrapping. J. Opt. Soc.Am.A.. 1996, 13(10): 1999-2013
20. Y. G. Lu, X. Z. Wang, X. P. Zhang. Weighted Least-Squares Phase Unwrapping Algorithm based on Derivative Variance Correlation Map. Optik. 2007, 118(2): 62-66.
21. M. D. Pritt, J. S. Shipman. Least-Squares Two-Dimensional Phase Unwrapping Using FFTs. IEEE Transactions on Geoscience and Remote Sensing. 1994, 32(3): 706-708
22.彭海良,云日升.干涉合成孔径雷达二维相位展开问题及其算法.测试技术. 2003, 17(2): 99-104
23. D . Just, R. Bamler. Phase Statistics of Interferograms with Applications to Synthetic Aperture Radar. Applied Optics. 1994, 33(20): 4361- 4368.
24. D. C. Ghiglia, G. A. Mastin, L. A. Romero. Cellular Automata Method for Phase Unwrapping. J. Opt. Soc. Am.. 1987, 4(1): 267-280.
25. R. M. Goldstein, H. A. Zebker, L. Werner. Satellite Radar Interferometry: Two-Dimensional Phase Unwrapping. Radio Science. 1988, 23(4): 713- 720.
26. M. D. Pritt. Phase Unwrapping by Means of Multigrid Techniques for Inter-ferometric SAR. IEEE Transactions on Geoscience and Remote Sensing. 1996 , 34 (16): 728-738.
27. X. Su, G. V. Bally, D. Vukicevic. Phase-Stepping Grating Profilometry: Utilization of Intensity Modulation Analysis in Complex Object Evaluation. Opt. Comm.. 1993, 98(1-3): 141-150
28. X. Su, W. Chen. Reliability-Guided Phase Unwrapping Algorithm: a Review. Opt.& Lasers Eng.. 2004, 42(3): 245-261
29. C. Quan, C. J. Tay, L. Chen. Spatial-Fringe-Modulation-Based Quality Map for Phase Unwrapping. Appl. Opt. 2003, 42(35): 7060-7065
30.刘景峰,李艳秋.一种新的质量图导引路径积分相位展开算法.光电工程. 2007, 34(12): 104-107
31. H. Guo, P. Huang. Quality-Guided Phase Unwrapping for the Modified Fourier Transform Method. Proc. SPIE.. 2009, 7389: 73890B
32. S. Zhang, X. Li, S. T. Yau, Multilevel Quality-Guided Phase Unwrapping Algorithm for Real-Time Three Dimensional Shape Reconstruction. Appl. Opt.. 2007: 46(1), 50-57.
33.赵万成,路元刚,张婷.用于二维相位展开的相位梯度相关质量图.光学学报. 2009, 29(2): 149-154
34. T. J. Flynn. Consistent 2-D Phase Unwrapping Guided by a Quality Map. Proc. IGARSS96, Lincoln, NE. 1996: 2057–2059.
35. X. Su, and L. Xue. Phase Unwrapping Algorithm Based on Fringe Frequency Analysis in Fourier-Transform Profilometry. Opt. Eng. 2001, 40(4): 637–643
36. C. Yu, Q. Peng. A Correlation-Based Phase Unwrapping Method for Fourier-Transform Profilometry. Optics and Lasers in Engineering. 2007, 45: 730-736
37. T. R. Judge, P. J. Bryanston-Cross. A Review of Phase Unwrapping Techniques in Fringe Analysis. Optics and laser in Engineering. 1994, 21(4): 199-293
38. D. C. Ghiglia, L. A. Romero. Robust Two-Dimensional Weighted and Unweighted Phase Unwrapping That Uses Fast Transforms and Iterative Methods. Opt. Soc. Am. 1994, 11(l): 107-118
39. B. R. Hunt. Matrix Formulation of the Reconstruction of Phase Values from Phase Differences. J. Opt. Soc. Am. 1979, 69(3): 393-399
40.惠梅,王东生.基于离散泊松解的相位展开方法.光学学报. 2003, 23(10): 1245-1248
41.刘景峰,李艳秋,刘克.含噪声包裹相位图的加权最小二乘相位展开算法研究.光学技术. 2008, 34(5): 643-650
42. J. L. Marroquin, M. Rivera. Quadratic Regularization Functionals for Phase Unwrapping. J. Opt. Soc. Am. A. 1995, 12(11): 2393-2400.
43.林成森.数值计算方法.北京,科学出版社. 1998.
2.苏显渝.三维面形测量技术的新进展.物理. 1996, 25(10): 614-620
3. M. K. Kalms, W. Osten, W. Juptner. Active Industrial Surface Inspection with the Inverse Projected-Fringe-Technique. Proc.SPIE. 2001, 4596: 37-47
4. R. Kowarschik, J. Gerber, G. Notni. Adaptive Optical 3D-Measurement with Structured Light. Proc. SPIE. 1999, 3824: 169-178
5. M. Tkeda, K. Motoh. Fourier Transform Profilometry for the Automatic Measurement of 3-D Object Shapes. Appl. Opt.. 1983, 22(24): 3977-3982.
6. X. Su, W. Chen, Q. Zhang. Dynamic 3-D Shape Measurement Method Based on FTP. Optics and Lasers in Engineering. 2001, 36: 49-64
7. J. Esteve-Taboada, J. Garcia, C. Ferreira. Application of Fourier Transform Profilometry to the Recognition of 3-D Objects. Proc. SPIE. 2003, 5227: 59-66
8. F. Berryman, P. Pynsent, J. Cubillo. The Effect of Windowing in Fourier Transform Porfilometry Applied to Noisy Images. Optics and Lasers in Engineering. 2004, 41 (6): 815-825.
9.陈锋.动态三维面形测量中相位展开问题的研究.四川大学. 2005: 5-6.
10. R. Zhu, J. Chen. Accuracy of Phase Shifting Interferometry—Phase Shifter Calibration and Error Compensation. Proc. SPIE. 1992, 1553: 702-710
11. J. Schmit, K. Creath. Window Function Influence on Phase Error in Phase-Shifting Algorithm. Appl. Opt.. 1996, 35(28): 5642-5465
12. D. C. Ghiglia, M. D. Pritt. Two-Dimensional Phase Unwrapping Theory, Algorithms and Software. John Wiley & Sons. 1998: 68-73
13. V. V. Volkov, Y. Zhu. Deterministic Phase Unwrapping in the Presence of Noise. Optics Letters. 2003, 28(22): 2156-2158
14. D. Kerr, G. H. Kaufmann, G.E. Galizzi. Unwrapping of Interferometric Phase-Fringe Maps by the Discrete Cosine Transform. Appl. Opt.. 1996, 35(5): 810-816
15. D. J. Bone. Fourier Fringe Analysis: the Two-Dimensional Phase Un-wrapping Problem. Applied Optics. 1991, 30(25): 3627-3632.
16. K. Itoh. Analysis of the Phase Unwrapping Algorithm. Applied Optics. 1982, 21(14): 2470
17. T. J. Flynn. Two-Dimensional Phase Unwrapping with Minimum Weighted Discontinuity. J. Opt. Soc. Am. A.. 1997, 14(10): 2692-2701
18. L. Su, X. Su, W. Li. Application of Modulation Measurement Profilometry(MMP) to Objects with Holes on Surface. Appl. Opt.. 1999, 38(7): 1153-1158
19. D. C. Ghigla, L. A. Romero. Minimun Lp-norm Two-Dimnesional Phase Unwrapping. J. Opt. Soc.Am.A.. 1996, 13(10): 1999-2013
20. Y. G. Lu, X. Z. Wang, X. P. Zhang. Weighted Least-Squares Phase Unwrapping Algorithm based on Derivative Variance Correlation Map. Optik. 2007, 118(2): 62-66.
21. M. D. Pritt, J. S. Shipman. Least-Squares Two-Dimensional Phase Unwrapping Using FFTs. IEEE Transactions on Geoscience and Remote Sensing. 1994, 32(3): 706-708
22.彭海良,云日升.干涉合成孔径雷达二维相位展开问题及其算法.测试技术. 2003, 17(2): 99-104
23. D . Just, R. Bamler. Phase Statistics of Interferograms with Applications to Synthetic Aperture Radar. Applied Optics. 1994, 33(20): 4361- 4368.
24. D. C. Ghiglia, G. A. Mastin, L. A. Romero. Cellular Automata Method for Phase Unwrapping. J. Opt. Soc. Am.. 1987, 4(1): 267-280.
25. R. M. Goldstein, H. A. Zebker, L. Werner. Satellite Radar Interferometry: Two-Dimensional Phase Unwrapping. Radio Science. 1988, 23(4): 713- 720.
26. M. D. Pritt. Phase Unwrapping by Means of Multigrid Techniques for Inter-ferometric SAR. IEEE Transactions on Geoscience and Remote Sensing. 1996 , 34 (16): 728-738.
27. X. Su, G. V. Bally, D. Vukicevic. Phase-Stepping Grating Profilometry: Utilization of Intensity Modulation Analysis in Complex Object Evaluation. Opt. Comm.. 1993, 98(1-3): 141-150
28. X. Su, W. Chen. Reliability-Guided Phase Unwrapping Algorithm: a Review. Opt.& Lasers Eng.. 2004, 42(3): 245-261
29. C. Quan, C. J. Tay, L. Chen. Spatial-Fringe-Modulation-Based Quality Map for Phase Unwrapping. Appl. Opt. 2003, 42(35): 7060-7065
30.刘景峰,李艳秋.一种新的质量图导引路径积分相位展开算法.光电工程. 2007, 34(12): 104-107
31. H. Guo, P. Huang. Quality-Guided Phase Unwrapping for the Modified Fourier Transform Method. Proc. SPIE.. 2009, 7389: 73890B
32. S. Zhang, X. Li, S. T. Yau, Multilevel Quality-Guided Phase Unwrapping Algorithm for Real-Time Three Dimensional Shape Reconstruction. Appl. Opt.. 2007: 46(1), 50-57.
33.赵万成,路元刚,张婷.用于二维相位展开的相位梯度相关质量图.光学学报. 2009, 29(2): 149-154
34. T. J. Flynn. Consistent 2-D Phase Unwrapping Guided by a Quality Map. Proc. IGARSS96, Lincoln, NE. 1996: 2057–2059.
35. X. Su, and L. Xue. Phase Unwrapping Algorithm Based on Fringe Frequency Analysis in Fourier-Transform Profilometry. Opt. Eng. 2001, 40(4): 637–643
36. C. Yu, Q. Peng. A Correlation-Based Phase Unwrapping Method for Fourier-Transform Profilometry. Optics and Lasers in Engineering. 2007, 45: 730-736
37. T. R. Judge, P. J. Bryanston-Cross. A Review of Phase Unwrapping Techniques in Fringe Analysis. Optics and laser in Engineering. 1994, 21(4): 199-293
38. D. C. Ghiglia, L. A. Romero. Robust Two-Dimensional Weighted and Unweighted Phase Unwrapping That Uses Fast Transforms and Iterative Methods. Opt. Soc. Am. 1994, 11(l): 107-118
39. B. R. Hunt. Matrix Formulation of the Reconstruction of Phase Values from Phase Differences. J. Opt. Soc. Am. 1979, 69(3): 393-399
40.惠梅,王东生.基于离散泊松解的相位展开方法.光学学报. 2003, 23(10): 1245-1248
41.刘景峰,李艳秋,刘克.含噪声包裹相位图的加权最小二乘相位展开算法研究.光学技术. 2008, 34(5): 643-650
42. J. L. Marroquin, M. Rivera. Quadratic Regularization Functionals for Phase Unwrapping. J. Opt. Soc. Am. A. 1995, 12(11): 2393-2400.
43.林成森.数值计算方法.北京,科学出版社. 1998.