基于强度传输方程的相位恢复算法研究
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摘要
相位是波场的一种内在特性,研究表明四分之一左右的信息在振幅中,而大约四分之三的信息则在相位中。在实际中,采用现有的检测器设备,仅能直接测量波场的强度,而不能同时测量波场的相位。这些丢失的相位信息在表面重建、显微镜学、位置检测和深度测量中至关重要。利用己知的强度信息获得这些丢失的相位信息,这就是相位恢复技术,在不同的物理学分支学科以及工程应用中有极其重要的意义。
相位恢复技术目前主要包括GSF(Gerchberg-Saxton-Fienup)迭代算法和基于强度传输方程(Transport of Intensity Equation, TIE)的确定性求解算法。由于TIE方法只需要测量与光轴垂直的二到三个平面上的强度,就可通过求解二阶微分方程(亦即用非迭代的确定性算法)来重构波的空间相位,克服了迭代算法的迭代不确定性、抗噪性能差等缺点。因此,本文主要对在强度测量基础上的相位恢复进行研究。重点研究和分析了求解TIE的多重网格算法,该算法从强度信息中恢复出来的相位解是精确解。本文主要研究工作和贡献如下所示：
(1)探讨了基于TIE的确定性相位恢复算法。对求解TIE的四种经典的算法进行了介绍,如傅里叶法,Green函数法,Zernike多项式法以及多重网格法。并对多重网格的数学计算进行了描述。
(2)重点研究和分析了基于TIE的多重网格算法,提出了一种改进的多重网格算法。该算法的主要思想：从最粗层开始计算,给定初值,迭代出一个解,将此解作为最细层的初值,然后在最细层计算出一个近似解。接着计算其残差,并将残差限制到较粗网格层求解,直至最粗层,然后逐层修正细网格层的解。利用循环在粗细不同的网格层来消除不同频率的误差分量,得到相位的精确解。
对改进的多重网格算法进行了模拟实验,实验结果表明,该算法提高了相位恢复的精度,并具有一定的抗噪能力。另外,还搭建了图像数据采集实验平台,利用平台拍摄真实散焦图像,并用改进的多重网格算法进行了真实实验。实验结果表明,该算法能够较好的从强度图像恢复物体的真实相位。
(3)研究了利用多幅散焦图求解TIE恢复相位的算法。对求解TIE过程中的重要参数的计算进行了分析,即强度微分的计算,将之用于改进的多重网格算法。实验结果表明,该算法进一步提高了抗噪性能。并通过搭建的实验平台进行了真实实验,实验结果表明,该算法能较好的恢复物体的真实相位。
(4)介绍了基于TIE的相位恢复系统。该系统包括了求解TIE的四种经典算法,重点介绍了系统中多重网格算法的实验操作。该系统软件界面简洁,直观,操作简单方便,并申请了软件著作权。
Phase is an intrinsic property of the wave field, some researches show that about a quarter of the information in amplitude, and about three-quarters of the information is in the phase. In practice, with the existing detector equipment, only the intensity of the wave field can be measured directly, the phase of the wave field can not be. The lost phase information is critical in the surface renewal, microscopy, the position detection and the depth measurement. Obtained the lost phase information by using the intensity knowned information, this is phase retrieval technique, it has extremely important implication in different branches of physics and engineering applications.
At present, phase retrieval technology mainly includes the deterministic algorithm that based on the intensity transport equation(TIE) and Gerchberg-Saxton-Fienup iterative algorithm. As the TIE method can reconstruct the wave spatial phase by solving the second order differential equations(that is, use non-iterative deterministic algorithm) which only need to measure two or three planes intensities that perpendicular to the optical axis, overcomes the uncertainty and poor antinoise of the iterative algorithm. This thesis concentrate on the research of deterministic phase retrieval technology base on the intensity measurement. This thesis is focus on research and analysis of the multigrid algorithm, the phase solution recovered fromed the intensity information is the extact solution. The main research work and contributions as follows:
(1) This thesis discusses the deterministic phase retrieval algorithm based on TIE. Four kind of classic algorithm for solving the TIE is summarized, such as Fourier method, Green function method, Zernike polynomial method and Multigrid method. And mathematical calculation of multiple grid is described.
(2) The thesis concentrate on the research of deterministic phase retrieval technology of Multigrid algorithm base on the intensity measurement, and made some improvement. The main idea of this method is that:stared from the coarsest grid, given a initial value, then got a solution by calculating, regard the calculated solution as the finest initial value, got an approximate solution by calculating, then calculate its residual, restricted it to coarser grid and solving, until it to the coarsest grid, followed with the correction step by step of the solution of a fine grid layer. Finally, we eliminate different frequency error components by looping in different degree of the grid layer, so get the accurate solution.
We verify the algorithm using simulation experiments, and examine the algorithm performance of noise immunity. We find that the improved multi-grid algorithm improves the accuracy of phase retrieval. And we build the image data acquisition platform, use it shoot real defocus image and make real experiment with the algorithm. The simulation and real experiments show that the multigrid algorithm can better take true phase objects from the intensity image.
(3) We research multigrid algorithm by using multiple defocus images to solve the TIE. We discuss important parameters calculate in solving the TIE process, that is calculation of the intensity differential, and use it in improved multigrid algorithm. And use simulation experiment verify the method, the results shows that it improve the algorithm performance of noise immunity further. The author make real experiment by using the platform, which can better retrieval the actual phase of the object from the intensity image.
(4) We make detail introduce for the developed phase retrieval system, the system contains four classical algorithms used in solve TIE, and we introduce how to handle Multigrid algrorithms manily. The system software interface is simple, intuitive, easy and simple to handle. The author has applied software copyright.
相位恢复技术目前主要包括GSF(Gerchberg-Saxton-Fienup)迭代算法和基于强度传输方程(Transport of Intensity Equation, TIE)的确定性求解算法。由于TIE方法只需要测量与光轴垂直的二到三个平面上的强度,就可通过求解二阶微分方程(亦即用非迭代的确定性算法)来重构波的空间相位,克服了迭代算法的迭代不确定性、抗噪性能差等缺点。因此,本文主要对在强度测量基础上的相位恢复进行研究。重点研究和分析了求解TIE的多重网格算法,该算法从强度信息中恢复出来的相位解是精确解。本文主要研究工作和贡献如下所示：
(1)探讨了基于TIE的确定性相位恢复算法。对求解TIE的四种经典的算法进行了介绍,如傅里叶法,Green函数法,Zernike多项式法以及多重网格法。并对多重网格的数学计算进行了描述。
(2)重点研究和分析了基于TIE的多重网格算法,提出了一种改进的多重网格算法。该算法的主要思想：从最粗层开始计算,给定初值,迭代出一个解,将此解作为最细层的初值,然后在最细层计算出一个近似解。接着计算其残差,并将残差限制到较粗网格层求解,直至最粗层,然后逐层修正细网格层的解。利用循环在粗细不同的网格层来消除不同频率的误差分量,得到相位的精确解。
对改进的多重网格算法进行了模拟实验,实验结果表明,该算法提高了相位恢复的精度,并具有一定的抗噪能力。另外,还搭建了图像数据采集实验平台,利用平台拍摄真实散焦图像,并用改进的多重网格算法进行了真实实验。实验结果表明,该算法能够较好的从强度图像恢复物体的真实相位。
(3)研究了利用多幅散焦图求解TIE恢复相位的算法。对求解TIE过程中的重要参数的计算进行了分析,即强度微分的计算,将之用于改进的多重网格算法。实验结果表明,该算法进一步提高了抗噪性能。并通过搭建的实验平台进行了真实实验,实验结果表明,该算法能较好的恢复物体的真实相位。
(4)介绍了基于TIE的相位恢复系统。该系统包括了求解TIE的四种经典算法,重点介绍了系统中多重网格算法的实验操作。该系统软件界面简洁,直观,操作简单方便,并申请了软件著作权。
Phase is an intrinsic property of the wave field, some researches show that about a quarter of the information in amplitude, and about three-quarters of the information is in the phase. In practice, with the existing detector equipment, only the intensity of the wave field can be measured directly, the phase of the wave field can not be. The lost phase information is critical in the surface renewal, microscopy, the position detection and the depth measurement. Obtained the lost phase information by using the intensity knowned information, this is phase retrieval technique, it has extremely important implication in different branches of physics and engineering applications.
At present, phase retrieval technology mainly includes the deterministic algorithm that based on the intensity transport equation(TIE) and Gerchberg-Saxton-Fienup iterative algorithm. As the TIE method can reconstruct the wave spatial phase by solving the second order differential equations(that is, use non-iterative deterministic algorithm) which only need to measure two or three planes intensities that perpendicular to the optical axis, overcomes the uncertainty and poor antinoise of the iterative algorithm. This thesis concentrate on the research of deterministic phase retrieval technology base on the intensity measurement. This thesis is focus on research and analysis of the multigrid algorithm, the phase solution recovered fromed the intensity information is the extact solution. The main research work and contributions as follows:
(1) This thesis discusses the deterministic phase retrieval algorithm based on TIE. Four kind of classic algorithm for solving the TIE is summarized, such as Fourier method, Green function method, Zernike polynomial method and Multigrid method. And mathematical calculation of multiple grid is described.
(2) The thesis concentrate on the research of deterministic phase retrieval technology of Multigrid algorithm base on the intensity measurement, and made some improvement. The main idea of this method is that:stared from the coarsest grid, given a initial value, then got a solution by calculating, regard the calculated solution as the finest initial value, got an approximate solution by calculating, then calculate its residual, restricted it to coarser grid and solving, until it to the coarsest grid, followed with the correction step by step of the solution of a fine grid layer. Finally, we eliminate different frequency error components by looping in different degree of the grid layer, so get the accurate solution.
We verify the algorithm using simulation experiments, and examine the algorithm performance of noise immunity. We find that the improved multi-grid algorithm improves the accuracy of phase retrieval. And we build the image data acquisition platform, use it shoot real defocus image and make real experiment with the algorithm. The simulation and real experiments show that the multigrid algorithm can better take true phase objects from the intensity image.
(3) We research multigrid algorithm by using multiple defocus images to solve the TIE. We discuss important parameters calculate in solving the TIE process, that is calculation of the intensity differential, and use it in improved multigrid algorithm. And use simulation experiment verify the method, the results shows that it improve the algorithm performance of noise immunity further. The author make real experiment by using the platform, which can better retrieval the actual phase of the object from the intensity image.
(4) We make detail introduce for the developed phase retrieval system, the system contains four classical algorithms used in solve TIE, and we introduce how to handle Multigrid algrorithms manily. The system software interface is simple, intuitive, easy and simple to handle. The author has applied software copyright.
引文
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[3]A. Tonomura. Direct observation of vortex motion in high-TC superconductors by Lorentz microscopy[J]. Physica B,2000,280(1-4):227-228.
[4]CHEN Xiu-shan, LI En-pu, ZHAO Jian-lin, etal. Measurement and reconstruction of 3D acoustic standing wave filed using digital holographic interferometry[J]. Acta Photonica Sinica,2010,39(1):95-99.
[5]P. Benzie, J. Watson, etal. A Survey of 3DTV Displays:Techniques and Technologies[C]. IEEE Transaction on circuits and systems for video technology, 2007,17(11):1647-1658.
[6]Yiying Pu, Jianwen Dong, Bingchu Chen, Yuanzhi Liu, Hezhou Wang. Three-dimensional imaging with monocular cues using holographic stereogrophay[J]. Opt. Lett.,2010,35(19):3279-3281.
[7]CHEN Daqing, GU Jihua, TAO Zhi. Audio watermarking based on phase retrieval algorithm and digital holography[J]. Acta Photonica Sinica,2009,38(12): 3333-3338.
[8]DING Lingyan, WU Yulie, LI Shengyi. Surface measurement for long focal length mirror with phase retrieval[J]. Acta Photonica Sinica,2010, 39(8):1431-1437.
[9]R. W. Gerchberg, W. O. Saxton. A practical algorithm for the determiniation of phase from image and diffraction plane pictures[J]. Optik,1972,35:237-246.
[10]J. R. Fienup. Phase retrieval algorithms:a comparison[J]. Appl. Opt.1982, 21:2758-2769.
[11]G. Z. Yang, B.Y. Gu and B. Z. Dong. Theory of the amplitude-phase retrieval in an any linear transform system and its applications[J], Modern Phys,1993, B7:3153-3224.
[12]黄利新,姚新,蔡冬梅等.一种快速高精度的相位恢复迭代算法[J].中国激光,2010,37(5)：1218-1221.
[13]于斌,彭翔,田劲东,牛憨笨等.硬x射线同轴相称成像相位恢复的模拟[J].中国科学G辑物理学力学天文学,2005,35(3)：233-40.
[14]彭金锰,杜少军,蒋鹏志.基于GS加权改进算法的相位恢复[J].强激光与粒子束,2013,25(2)：315-318.
[15]C. Dorrer, J. D. Zuegel. Optical testing using the transport-of-intensity equation[J]. OPTICS EXPRESS,2007,15(12):7165-7175.
[16]J. Frank, S. Altmeyer, G. Wernicke. Non-interferometric, non-iterative phase retrieval by Green's functions[J]. Journal Optical Society of America A,2010, 27(10):2244-2251.
[17]A. V. Matrin, L. J. Allen. Measuring the phase of a Bose-Einstein condensatefJ]. Physical Review A,2007,76(5):053606.
[18]XUE Bin-dang, ZHENG Shi-ling. Phase retrieval using the transport of intensity equation solved by the FMG-CG method[J]. Optik-International Light and Electron Optics,2011,122(23):2101-2106.
[19]薛斌党,郑世玲,姜志国.完全多重网格法求解强度传输方程的相位恢复方法[J].光学学报,2009,29(6)：1514-1518.
[20]王潇,毛珩,赵达尊.基于光强传播方程的相位恢复[J].光学学报,2007,27(12)：2117-2122.
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