摘要
当一束光波照射到一个物体上时,在忽略光与物体媒质发生偏振作用的情况下,物体对波的三个特性产生影响：振幅(亮度)、波长(颜色)和相位(一个波长内相位等同于深度)。相位是任何波场的一种内在的特性,统计表明大于25%的信息被编码在振幅项中而75%的信息在相位项中。但是,光场的振荡接近1015Hz,目前没有一个现有的光传感装置能直接并同时地记录光场的振幅和相位。干涉采样技术可以同时获取光波的振幅和相位,但是要求光源的空间和时间相干,因此,此技术难以走出实验室。换句话说,CCD摄像机、胶卷等检测器只能直接测量衍射光场的强度；为了得到深度信息,通常需要对数据进行复杂的转换和处理。
目前,基于强度测量的相位检索技术包括迭代相位检索算法、散焦求深度、基于强度的确定性相位检索以及基于二次约束的最优化相位检索等。其中,迭代相位检索使用物而和像而的强度数据来恢复输入和输出平而上的光场相位分布,实时性差；散焦求深度利用图像的模糊信息获得三维景物的深度信息,该技术需要通过调整摄像机参数以达到点与点的对应；基于强度测量的确定性相位检索方法通过光学与计算的结合通过解强度传输方程来恢复相位信息,理论和实验证明是获取相位信息的一种可行的途径。基于二次约束的最优化相位检索方法是新近由矩阵完备理论发展的一个新的思想,目前主要还处于理论研究阶段。
本论文围绕散焦求深度和基于强度测量的确定性相位检索两种技术展开研究,重点探讨了利用强度传输方程从强度图像中恢复相位信息的基于强度的确定性相位检索技术。
论文的主要工作和创新点如下：
(1)改进了传统的散焦恢复深度中的信息散度算法。原算法采用等焦而假设并忽略了在不同相机参数下图像大小会发生变化的事实。改进的算法利用单应矩阵对获取的图像进行矫正以获得相同尺寸的散焦图像对。同时,利用空间变化窗口克服等焦而假设的局限性,避免平滑非连续域的深度值并提高估计的精度。
(2)在总结文献中求解强度传输方程的各种算法,例如Green函数法、Zernike多项式法法、多重网格法和傅里叶变换法的基础上，论文将整体变分技术引入强度传输方程的求解中,并与傅里叶变换法进行比较。实验结果表明该算法恢复相位的同时可保持图像良好的边缘信息。
(3)给出了Neumann和Dirichlet边界条件下新的Green函数法推导方法。该方法中,Green函数本身是四维矩阵,随着图像分辨率的提高,直接求解所占的内存空间比较大。论文提出了新的Green函数数位解方法,可以降低计算量,提高运算速度。
(4)在自然光条件下成像通常需要应用透镜聚光,其传递函数比在光学显微镜、电子显微镜等微观领域广泛使用的物理环境下的传递函数更复杂。论文详细地阐述了在透镜聚焦区域附近的菲涅尔传播物理和相位重构的模型,提出了一种在此模型下的相位检索方法。
(5)论文设计了一台单CCD图像数据采集光学系统,该系统配有焦距移动的机械装代,以便给出聚焦平而和散焦平而的图像精确度量。该系统结构简单,易于携带,并申请了专利。同时,论文还搭建了3-CCD图像数据采集光学平台系统,在稳定的光学平台上可以得到更精确的实验结果。
When a wave, such a light, interacts with a body, details of the body are imprinted on the wave's three properties:amplitude (brightness),wavelength(colour) and phase(in a wavelength, the phase is equivalent to the depth). Phase is an inherent characteristic of any wave field. Statistics show that greater than25%of the information is encoded in the amplitude term and75%of the information is in the phase term. However, the oscillations of light waves about1015Hz, there are not existing light-sensing devices can record the amplitude and phase of diffraction field directly and simultaneously. In other words, the intensity of the light field can be directly measured through the CCD camera, film and other detector. Therefore, complex conversion and processing are needed to get depth information.
At present, the phase retrieval based on the intensity measurement includes four typical techniques:the iterative phase retrieval algorithm, depth from defocus, deterministic phase retrieval based on the intensity measurement and optimization phase retrieval based on quadratic constraints. Among them, intensity data of the object and image plane is used to recover the phase distribution of the input and output of the light field in iterative phase retrieval. There are many shortcomings such as poor real-time. Blur information is used to obtain depth information of the three-dimensional scene in depth from defocus. The technique requires adjusting the camera parameters in order to achieve correspondence between point and point. The transport of intensity equation is solved to recover the phase information through the combination of optics and calculation in the method of phase retrieval based on the intensity measurement. This method is proved to be a feasible way to obtain phase information theoretically and experimentally. Optimization phase retrieval based on quadratic constraints is a new idea by the development of the matrix complete theory recently. So far, this method is merely in the theoretical exploration stage.
Both technologies of depth from defocus and deterministic phase retrieval based on the intensity measurement are studied. Deterministic phase retrieval based on the intensity measurement which recovers phase information from the transport of intensity equation using the intensity images is focused in this thesis.
The main research works and contributions of this thesis are outlined as follows:
(1) Information-divergence algorithm in depth from defocus is improved. The original algorithm is yet based on equifocal assumption and ignores the fact that the image sizes of same object are changed in different parameters of the camera. In the improved algorithm, the homography between the original defocus images is calculated. Then original images are rectified and new defocus images of the same size are obtained. Space-variant window scheme is studied to overcome the limitation of equifocal assumption. Simulation and real experimental results prove the proposed algorithm can avoid smoothing depth in discontinuities and improve the precision.
(2) On the basis of summing up various algorithms to solve the transport of intensity equation, such as the Green's function method, Zernike polynomial method, the multi-grid method and the Fourier transform, total variation is introduced to solve the transport of intensity equation in this theies. The experiments show that information in edge is remained at the same time of phase retrieval compared with the Fourier transform method.
(3) A new derivation of Green function method in Neumann and Dirichlet boundary conditions is deduced. Because the Green's function itself is a four-dimensional matrix, large memory space will be occupied with the improvement of image resolution by solving the Green's function directly. The numerical solutions of Green's function method to solve the problems of the large-scale calculation and slow speed are presented in this thesis.
(4) Imaging usually needs to apply the lens condenser in natural light conditions. The transfer function is more complex than that in the physical environment which is widely used in optical microscopy, electron microscopy and other microscopic fields. Fresnel propagation physics and phase reconstruction model are described detailedly. A phase retrieval method in this model is proposed.
(5) A single CCD image data acquisition optical system is designed in this thesis. The system is equipped with a mobile focal mechanical device in order to give the accurately measurement of image in focal and defocal planes. The system is simple and portable. A patent for this system has been granted. At same time, a3-CCD image optical data collection platform is built. More accurate experimental results are achieved in this stable optical platform.
引文
[1]C. Iemmi, A. Moreno, J. Campos. Digital holography with a point diffraction interferometer[J]. Opt. Express,2005,13(6):1885-1891.
[2]O. Matoba, T. J. Naughton, Y. Frauel, N. Bertaux, B. Javidi. Real-time three-dimensional object reconstruction by use of a phase-encoded digital hologram[J]. Appl. Opt.,2002,41(29): 6187-6192.
[3]Yiying Pu, Jianwen Dong, Bingchu Chen, Yuanzhi, Liu, Hezhou Wang. Three-dimensional imaging with monocular cues using holographic stereography[J]. Opt. Lett.,2010,35(19): 3279-3281.
[4]S. Chaudhuri, A.N. Rajagopalan. Depth from defocus:A Real Aperture Imaging Approach[M]. New York:Springer-Verlag Press,1999:10-100.
[5]C.T.Yong, Wavefront senors in adaptive optics[D]. University of Canterbury,2008.
[6]B. E. Allman. Limits Document, CTD Contract Reference 4500445823,2006.
[7]B. E. Allman. Shape imaging in land operations[C]. Land Warfare Conference,2008.
[8]K.A. Nugent, D.Paganin, T.E. Gureyev. A phase odyssey[J]. Physics Today,2001,54(8): 27-32.
[9]B. E. Allman, K. A. Nugent. Shape imaging in defence operations[C]. Land Warfare Conference,2006.
[10]B.E. Allman, G.P. Gregson, B.H.B Powell. Imaging by shape [C]. Land Warfare Conference, 2004,6206:620628.
[11]M. R. Teague. Deterministic phase retrieval:A Green's function solution[J]. J. Opt. Soc. Am., 1983,73(11):1434-1441.
[12]C. Dorrer, J. D. Zuege. Optical testing using the transport-of-intensity equation [J]. Opt. Express,2007,15(12):7165-7175.
[13]S.McVitie, D.T.Ngo. Quantitative measurements of phase using the transport of intensity equation[J]. Journal of Physics:Conference Series,2008,126:012041.
[14]T. E. Gureyev, A. Pogany, D. M. Paganin, S.W.Wilkins. Linear algorithms for phase retrieval in the Fresnel region [J]. Opt. Commun.,2004,231(1-6):53-70.
[15]曾发,谭峭峰,魏晓峰,向勇,严瑛白,金国藩.一种可对复杂光场进行相位检索的算法[J].中国激光.2006,33(3)：339-342.
[16]S. Marchesini. Phase retrieval and saddle-point optimization[J]. J. Opt. Soc. Am. A,2007, 24(10):3289-3296.
[17]S. Marchesini. A unified evaluation of iterative projection algorithms for phase retrieval [J]. Rev. Sci. Inst.,2007,78(1):011301.
[18]A. B.Tal, A.S.Nemirovski. Lectures on modern convex optimization:analysis, algorithms, and engineering applications[M],Society for Industrial Mathematics,2001.
[19]E.J.Candes, Y. Eldar, T. Strohmer, V. Voroninski. Phase retrieval via matrix completion. http://arxiv.org/abs/1109.0573.
[20]R. W. Gerchberg, W. O. Saxton. A practical algorithm for the determination of phase from image and diffraction plane pictures[J]. Optik,1972,35:237-246.
[21]J. R. Fienup, Phase retrieval algorithms:A comparison [J]. Appl. Opt.,1982,21(15): 2758-2769.
[22]A. P. Pentland. A new sense for depth of fleld[J]. Transactions on Pattern Analysis and Machine Intelligence,1987,9(4):523-531.
[23]M. Subbarao. Parallel depth recovery by changing camera parameters[C]. Proceedings of the Second International Conference on Computer Vision,1988:149-155.
[24]A. Horii. Depth from defocusing[M]. Royal Institute of Technology,1992.
[25]V. P. Namboodiri, S. Chaudhuri. On defocus, diffusion and depth estimation[J]. Pattern Recognition Letters,2007,28(3):311-319.
[26]章权兵,张爱明.基于相对模糊和边缘强化的散焦求深度方法[J].华南理工大学学报(自然科学版),2010,38(4)：136-140.
[27]程鸿,宫炎焱,章权兵,张伟.基于空间变化窗口的信息散度算法[J].华南理工大学学报,2011,39(10)：50-54.
[28]P. Favaro, S.Soatto.3-D shape estimation and image restoration[M]. London:Springer-Verlag Press,2006:40-90.
[29]C. Y. Zhou, S. K. Nayar. What are Good Apertures for Defocus Deblurring[C]. International Conference on Computational Photography,2009:1-8.
[30]S. C. Woods, A. H. Greenaway. Wave-frontsensing by use of a Green's function solutionto the intensity transport equation:reply to comment[J]. J. OPT. SOC. AM. A,2003:24(8):2482-2484.
[31]J.Frank, S.Altmeyer, G.Wernicke. Non-interferometric, non-iterative phase retrieval by Green's functions [J]. J. OPT. SOC. AM. A,2010,27(10):2244-2251.
[32]J.Desachy, S.Tajbakhsh. Robust algorithms for phase unwrapping in SAR interferometry[C]. Proc. SPIE,1997,3217:176.
[33]T. E. Gureyev, A. Roberts, K. A. Nugent. Phase retrieval with the transport of intensityequation:matrix solution with use of Zernike polynomials[J]. J. OPT. SOC. AM. A,1995, 12(9):1932-1942.
[34]M. Born, E. Wolf. Principles of Optics (Sixth Edition)[M]. Oxford:Pergamon Press. 1980:51-70.
[35]D. Stab. Reconstruction of optical fields from Phase Diversity.2005. http://scholar.google.c om.hk/scholar ?q=Reconstruction+of+optical+fields+from+Phase+Diversity&hl=zh-CN&as_sdt= 0&as_vis=1&oi=scholart&sa=X&ei=AyZ8T5mMIaWfiAeYodClCQ&ved=0CBoQgQMwAA
[36]T. E. Gureyev, K. A. Nugent. Phase retrieval with the transport-of-intensity equation. Ⅱ. Orthogonal series solution for nonuniform illumination[J], J. OPT. SOC. AM. A,1996,13(8): 1670-1682.
[37]A. Brandt. Multi-level adaptive solutions to boundary-value problems[J]. Math. Comp.,1977, 31(38):333-390.
[38]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.
[39]薛斌党,郑世玲,姜志国.完全多重网格法求解光强度传播方程的相位检索方法[J].光学学报,2009,29(6)：1514-1518.
[40]Bindang Xue, ShilingZheng. Phase retrieval using the transport of intensity equation solved by the FMG-CG method[J], Optik,2011,122(23):2101-2106.
[41]李永华.X射线类同轴法及光栅法相衬成像的相位复原算法研究[D].清华大学,2010.
[42]T. E. Gureyev, K. A. Nugent. Rapid quantitative phase imaging using the transport of intensity equation[J]. Opt. Commun.,1997,133(1-6):339-346.
[43]于斌,彭翔,田劲东,牛憨笨,刁麓弘,李华.硬X射线同轴相衬成像相位恢复的模拟[J].中国科学G辑,2005,35(3)：233-240.
[44]D. Paganin, K.. A. Nugent. Noninterferometric phase imaging with partially coherent light[J]. Phys. Rev. Lett.,1998,80(12):2586-2589.
[45]L. J.Allen, M. P. Oxley. Phase retrieval from series of images obtained by defocus variation[J]. Opt. Commun.,2001,199(1-4):65-75.
[46]程鸿,章权兵,韦穗.基于整体变分的相位检索[J].中国图象图形学报，2010,15(10)：1425-1429.
[47]谢文国,陈福荣.穿透式电子显微镜相位恢复技术及其应用[J].物理双月刊,2004,26(3)：503-511.
[48]M. X. Goemans, D. P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming[J]. J. ACM,1995,42:1115-1145.
[49]R. Balan, B. Bodmann, P.G. Casazza, D. Edidin. Painless reconstruction from magnitudes of frame coefficients[J]. J. Four. Anal. Appl.,2009,15:488-501.
[50]A. Chai, M. Moscoso, G. Papanicolaou. Array imaging using intensity-only measurements[J]. Inverse Problems,2011,27(1):015005.
[1]M. Born, E.Wolf. Principles of optics(7th Ed)[M]. Cambridge:Cambridge University Press, 1999.
[2]陈家璧,苏显渝.光信息技术原理及应用[M].北京：高等教育出版社,2001,16-40.
[3]A. D. Polyanin.Handbook of linear partial differential equations for engineers and scientists[M]. Chapman & Hall,2002.
[4]P. H. Lehmann, W. Osten, K. Gastinger. Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique[C]. Proc. SPIE,2011,8082:80820N.
[5]G. R. Hadley. Transparent boundary condition for the beam propagation[J]. Optics Letters, 1991,16(9):624-626.
[6]G. R. Hadley, Wide-Angle beam propagation using pade approximant operators [J]. Optics Letters,1992,17(20):1426-1428.
[7]J.W. Goodman傅里叶光学导论[M],秦克诚，刘培森,陈家璧,曹其智.3版,北京：电子工业出版社,2006.
[8]C.T.Yong. Wavefront senors in adaptive optics[D]. University of Canterbury,2008.
[9]S. D. Mellin, G. P. Nordin. Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design[J]. Opt. Express,2001, 8(13):705-722.
[10]J. H. Poynting. On the transfer of energy in the electromagnetic field:communicated by rayleigh[J]. The Royal Society,1885,175:343-361.
[11]A. Sommerfeld. Optics:volume IV of lectures on theoretical physics[M]. New York: Academic Press,1954.
[12]S. Mezouari, A. R. Harvey. Validity of Fresnel and Fraunhofer approximations in scalar diffraction[J]. Journal of Optics A:Pure and Appled Optics,2003,5(4):86-91.
[13]W. H. Southwell, Validity of the Fresnel approximation in the near field[J]. J. Opt. Soc. Am., 1981,71(1):7-14.
[1]程鸿,章权兵,宫炎焱.一种新的基于散焦图像的深度恢复算法[J].计算机应用与软件,2010,27(2)： 271-273.
[2]A. P. Witkin. Scale-space filtering[C]. Proceedings of the Fourth International Joint Conference on Artificial Intelligence,1983,1019-1022.
[3]J. J. Koenderink, A. J. van Doom. Dynamic shape[J]. Biological Cybernetics,1986,53(6): 383-396.
[4]V. P. Namboodiri, S. Chaudhuri. Use of linear diffusion in depth estimation based on defocus cue[C]. Proceedings of the Fourth Indian Conference on Computer Vision,2004:133-138.
[5]P. Favaro, S.Soatto.3-D shape estimation and image restoration[M]. London:Springer-Verlag Press,2006:40-90.
[6]H. Jin, P. Favaro. A variational approach to shape from defocus[C]. Proc. of the European Conference on Computer Vision,2002, Part Ⅱ,18-30.
[7]程鸿,宫炎焱,章权兵,张伟.基于空间变化窗口的信息散度算法[J].华南理工大学学报,2011,39(10)：50-54.
[8]C. Dorrer, J. D. Zuege. Optical testing using the transport-of-intensity equation [J]. Opt. Express,2007,15(12):7165-7175.
[9]V. P. Namboodiri, S. Chaudhuri. On defocus, diffusion and depth estimation[J]. Pattern Recognition Letters,2007,28(3):311-319.
[10]B. E. Allman, K. A. Nugent. Shape imaging in defence operations[C], Land Warfare Conference,2006.
[11]B. E. Allman, K. A. Nugent. Shape imaging in land operations[C]. Land Warfare Conference, 2008.
[12]B. E. Allman, G. P. Gregson, B. H. B Powell. Imaging by shape[C]. Land Warfare Conference, 2004.
[13]M. R. Teague. Deterministic phase retrieval:A Green's function solution[J]. J. Opt. Soc. Am., 1983,73(11):1434-1441.
[14]J.Frank, S.Altmeyer, G.Wernicke. Non-interferometric, non-iterative phase retrieval by Green's functions [J]. J. OPT. SOC. AM. A,2010,27(10):2244-2251.
[15]K. Ishizuka, B. Allman. Phase measurement in electron microscopy using the transport of intensity equation[J]. Microscopy Today,2005:22-24.
[16]T. E. Gureyev, K. A. Nugent. Phase retrieval with the transport-of-intensity equation. Ⅱ. Orthogonal series solution for nonuniform illumination[J]. J. Opt. Soc. Am. A,1996,13(8): 1670-1682.
[17]T. E. Gureyev, K. A. Nugent. Rapid quantitative phase imaging using the transport of intensty equation [J]. Opt. Commun.,1997,133(1-6):339-346.
[18]L. J. Allen, M. P. Oxley. Phase retrieval from series of images obtained by defocus variation[J]. Opt. Commun.,2001,199(1-4):65-75.
[19]程鸿,章权兵,韦穗.基于整体变分的相位恢复[J].中国图象图形学报,2010,15(10)1425-1429.
[20]A. N. Tikhonov, V. Y. Arsenin. Solutions of ill-posed problems[M].Winston,1977.
[21]L. Rudin, S. J. Osher, E. Fatemi. Nonlinear total variation based noise removal algorithms[J]. Physica D.,1992,60:259-268.
[22]T. F. Chan, S. J. Hong. Mathematical models for local non-texture inpaintings[J]. Siam Journal on Applied Mathematics,2001,62(3):1019-1043.
[23]张航,罗大庸.图像盲复原算法研究现状及其展望[J].中国图象图形学报,2004,10(9)：1145-1152.
[24]程鸿,章权兵,韦穗,沈川.基于强度传输方程的相位检索.光子学报,2011,40(10)：1566-1570.
[1]程建春.数学物理方程及其近似方法[M].北京：科学出版社,2004.
[2]王登营,李富,胡永明,周旭华.基于Dirichlet边界条件的Green函数节块法[J].清华大学学报(自然科学版),2010,50(6)：905-908.
[3]南京工学院数学教研组.数学物理方法与特殊函数(第二版)fM].北京：高等教育出版社,1982.
[4]汪德新.数学物理方法(第三版)[M].北京：科学出版社,2007.
[5]A. D. Polyanin. Handbook of Linear Partial Differential Equations for Engineers and Scientists[M]. Chapman & Hall,2002.
[6]白正阳,关于用格林函数法求解电场问题的讨论,科技传播,2011,16：110-111.
[7]M. R. Teague. Deterministic phase retrieval:a Green's function solution[J]. JOSA,1983,73(11): 1434-1441.
[8]G. Fornaro, G. Franceschetti, R. Lanari. Interferometric SAR phase unwrapping using Green's formulation[J]. Trans. Geosci. Remote Sens.,1996,34(3):720-727.
[9]I. Lyuboshenko, H. Maitre. Phase unwrapping for interferometric synthetic aperture radar by use of Helmholtz equation eigenfunctions and the first Green's identity[J]. J. Opt. Soc. Am. A,1999, 16(2):378-395.
[10]E. G. Abramochkin, V. G. Volostnikov. Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone[J]. Opt. Commun.,1989,74(3-4):144-148.
[11]J. Desachy, S. Tajbakhsh. Robust algorithms for phase unwrapping in SAR interferometry[C]. Proc. SPIE,1997,176:3217.
[12]P. H. Lehmann, W. Osten. K. Gastinger. Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique[C]. Proc. SPIE,2011,8082:80820N.
[13]J. Frank, S. Altmeyer, G. Wernicke. Non-interferometric, non-iterative phase retrieval by Green's functions[J]. J. Opt. Soc. Am.,2010,27(10):2244-2251.
[14]宋福.用本征函数求静电场格林函数[J].山西师范大学学报(自然科学版),2000,14(1)： 40-44.
[15]M. Beleggia, M. A. Schofield, V. V. Volkov, Y. Zhu. On the transport of intensity technique for phase retrieval[J]. Ultramicroscopy,2004,102(1):37-49.
[16]C. Dorrer, J. D. Zuegel. Optical testing using the transport-of-intensity equation [J]. Opt. Express,2007,15(2):7165-7175.
[17]J. Frank, J. Matrisch, J. Horstmann, S. Altmeyer. Refractive index determination of transparent samples by noniterative phase retrieval[J]. Appl. Opt.,2011,50(4):427-433.
[18]D. Paganin, K. A. Nugent. Noninterferometric phase imaging with partially coherent light[J]. Phys. Rev. Lett.,1998,80(12):2586-2589.
[19]B. E. Allman, K. A. Nugent. Shape imaging in defence operations[C], Proceedings Land Warfare Conference,2006.
[20]L. J. Allen, M. P. Oxley. Phase retrieval from series of images obtained by defocus variation[J]. Opt. Commun.,2001,199(1-4):65-75.
[21]V. V. Volkov, Y. Zhu, M. De Graef. A new symmetrized solution for phase retrieval using the transport of intensity equation[J]. Micron,2002,33(5):411-416.
[22]A. V. Martin, F.-R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, L. J. Allen. Spatial incoherence in phase retrieval based on focus variation[J]. Ultramicroscopy,2006,106(10):914-924.
[23]D. Paganin, A. Barty, P. J. McMohan, K. A. Nugent. Quantitative phase- amplitude microscopy Ⅲ:the effects of noise[J]. J. Microsc,2004,214(1):51-61.
[1]J. A. Schmalz, T. E. Gureyev, D. M. Paganin, etc. Phase retrieval using radiation and matter-wave fields:Validity of Teague's method for solution of the transport-of-intensity equation[J]. Phy. Rev. A,2011,84(2):023808.
[2]J. A. Schmalz, G. Schmalz, T. E. Gureyev, etc. On the derivation of the Green's function for the Helmholtz equation using generalized functions[J]. Am. J. Phys.,78 (2):181-186.
[3]Bindang Xue, ShilingZheng. Phase retrieval using the transport of intensity equation solved by the FMG-CG method[J], Optik,2011,122(23):2101-2106.
[4]J. Goodman. Introduction to Fourier Optics[M]. NewYork, McGraw-Hill Press,1996.
[5]J. Frank, J. Matrisch, J. Horstmann, S. Altmeyer. Refractive index determination of transparent samples by noniterative phase retrieval[J]. Applied Optics,2011,50 (4):427-433.
[6]K. Ishizuka, B. Allman. Phase measurement in electron microscopy using the transport of intensity equation[J]. Microscopy Today,2005,13:22-24.
[7]T. E. Gureyev. Composite techniques for phase retrieval in the Fresnel region[J]. Optics Communications,2003,220(1-3):49-58.
[8]L. Mandel, E. Wolf, Optical Coherence and Quantum Optics[M]. Cambridge, Cambridge University Press,1995, Sec.5.6.
[9]M. R. Teague. Deterministic phase retrieval:A Green's function solution[J]. J. Opt. Soc. Am., 1983,73(11):1434-1441.