相位场的莫尔层析重建理论与算法研究
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摘要
光学计算层析技术是计算机层析技术的一个分支，是一种不干扰待测场分布的测量诊断技术，在流动显示、热物理量测试等众多领域显示出了强大的优越性。莫尔层析技术是以偏折角为投影数据的一种光学层析技术，具有装置简单，抗干扰能力强，适合于在有强震动的恶劣环境下的测试的特点，对于复杂流场的定性显示和定量测试都具有很高的实用价值。本文针对目前对莫尔层析的理论体系和重建算法还缺少系统研究的情况，重点研究了复杂流场中的莫尔层析重建问题，主要工作内容和研究成果简述如下：
1)莫尔层析的理论研究
研究了偏折层析投影与传统层析投影间的数学关系，从不同的角度给出两者之间明确的关系表达式。从而在为偏折层析在整个层析理论体系中进行明确定位的同时，又为将传统层析中的重建算法应用到偏折层析中提供的理论依据。
2)偏折层析的变换类算法研究
由两类层析投影间的关系出发，推导出了基于偏折层析的滤波反投影算法，并对其数值实现进行了探讨，给出了相应的在空域用卷积反投影重建的算法。重点分析了噪声对重建的影响，并提出了相应的改进算法。
3)偏折层析中的级数展开类算法研究
提出了直接将折射率梯度离散化建立线性方程组的方法，并在此基础上，研究了三种不同的线性方程组求解算法，数值模拟的结果表明该算法有较强的抗系统噪声的能力。同时又提出了以局部基函数为基的偏折层析的迭代重建算法，该算法在保证了系数矩阵稀疏性的同时，能以一种“自然”的方式对偏折层析问题进行离散化。数值模拟表明，其既能保证较高的重建精度，又能有效的抑制系统噪声的影响。
4)偏折层析的非完全数据重建方面的研究。
在有限角问题中，提出基于贝叶斯准则的迭代算法用于对有限角的偏折层析重建中。在内问题的研究中，借鉴了Farrokh等人提出的基于小波理论的局部层析重建算法对偏折层析中的局部重建进行了研究。
5)对真实流场的莫尔重建的研究
使用偏折层析滤波反投影算法和相应改进算法对火箭燃气射流密度场进行了重建，重建结果表明改进的算法确实能够有效的抑制场的倾斜现象。然后用基于汉宁窗函数展开的迭代算法对高超音速风洞内包含激波的密度场进行重建，并与只适用于轴对称场的逆Abel变换法的重建结果进行了对比，两者的重建结果是基本吻合的，但新算法可以用于任意分布的场的重建，具有更好的普适性。对这两个真实流场的重建有效验证了前文中的算法，为莫尔层析的使用打下了基础。
Optical Computerized Tomography (OCT) is a branch of ComputerizedTomography. It is a non-contact measurement technique and shows superioritiesin many domains, such as flow visualization and measurement of thermo physicalparameters. The moire tomography belongs to the Optical ComputerizedTomography and its projection data is deflection angles. It has many meritssuch as simple optical configuration, superior ability of anti-error andadaptation to tough environment. The moire tomography has great practicalvalue to the qualitative visualizations and quantitative measurements ofcomplex flow fields. But up to now, the theory system and reconstructionalgorithms of the moire tomography have not been studied systematically. Soemphasis of this dissertation is put on the reconstruction algorithms of moiretomography for the complex flow fields. The main work is described as follows:
1) Theory research of moire tomography
The relation between the deflection projection and the traditionalprojection is studied and is described from many different views. By thisrelation, the deflection tomography can have a explicit position in the wholetomographic theoretical frame and theoretical law can be found to thesemethods which apply the traditional tomographic reconstruction algorithms tothe reconstruction of deflection tomography.
2) Research work to the transform algorithms of deflection tomography
According to the relation between these two tomographic problems, thedeflection filtered back-projection algorithm is achieved and its numericalimplementation is studied. The corresponding convolution back-projectionalgorithm is also given. Then an error analysis is made to the DFBP algorithmand a modified algorithm is presented.
3) Research work to the series expansion algorithms of deflectiontomography
A model which directly numeralize the gradient of refractive index ispresented to get the discrete linear equations of deflection tomography.According to the model, three reconstruction algorithms are studied to solvethese linear equations. Numerical experiments show that these algorithms have strong abilities of noise immunity. Another deflection iterative algorithmbased on the expansion of local basis is presented. This algorithm can notonly reserve the sparse property Of the coefficient array, but also discretethe deflection tomography in a natural way. Good quality of anti-noise andreconstruction accuracy of this algorithm have been found by numericalexperiments.
4) Research work to the reconstructions of deflection tomography withincomplete data
In the limited angle problem, a deflection tomographic iterative algorithmwhich is based on the Bayesian Law is presented. In the research of the interiorproblem, the wavelet based local tomographic reconstruction algorithm whichis present by Farrokh is introduced to the local reconstruction of deflectiontomography.
5) Research work to the real flow field of moire tomography
The deflection filtered back-projection algorithm and its correspondmodified algorithm are applied to reconstruct the density field caused by therocket exhausted plumes. Results show that the modified algorithm can depressthe slope phenomena efficiently. The iterative algorithm which is based onthe expansion of harming function is applied to reconstruct thedensity fieldin a hypersonic wind tunnel. The reconstruction results are compared with theinverse Abel transform method which can only be applied with axial symmetricalfield. Results coincident well with each other. But it is obviously that thenovel algorithm have handle arbitrary flied and more universality.Reconstructions to these real flow fields show those algorithms are valid.So the application foundation of moire tomography has been provided in thisdissertation.
1)莫尔层析的理论研究
研究了偏折层析投影与传统层析投影间的数学关系，从不同的角度给出两者之间明确的关系表达式。从而在为偏折层析在整个层析理论体系中进行明确定位的同时，又为将传统层析中的重建算法应用到偏折层析中提供的理论依据。
2)偏折层析的变换类算法研究
由两类层析投影间的关系出发，推导出了基于偏折层析的滤波反投影算法，并对其数值实现进行了探讨，给出了相应的在空域用卷积反投影重建的算法。重点分析了噪声对重建的影响，并提出了相应的改进算法。
3)偏折层析中的级数展开类算法研究
提出了直接将折射率梯度离散化建立线性方程组的方法，并在此基础上，研究了三种不同的线性方程组求解算法，数值模拟的结果表明该算法有较强的抗系统噪声的能力。同时又提出了以局部基函数为基的偏折层析的迭代重建算法，该算法在保证了系数矩阵稀疏性的同时，能以一种“自然”的方式对偏折层析问题进行离散化。数值模拟表明，其既能保证较高的重建精度，又能有效的抑制系统噪声的影响。
4)偏折层析的非完全数据重建方面的研究。
在有限角问题中，提出基于贝叶斯准则的迭代算法用于对有限角的偏折层析重建中。在内问题的研究中，借鉴了Farrokh等人提出的基于小波理论的局部层析重建算法对偏折层析中的局部重建进行了研究。
5)对真实流场的莫尔重建的研究
使用偏折层析滤波反投影算法和相应改进算法对火箭燃气射流密度场进行了重建，重建结果表明改进的算法确实能够有效的抑制场的倾斜现象。然后用基于汉宁窗函数展开的迭代算法对高超音速风洞内包含激波的密度场进行重建，并与只适用于轴对称场的逆Abel变换法的重建结果进行了对比，两者的重建结果是基本吻合的，但新算法可以用于任意分布的场的重建，具有更好的普适性。对这两个真实流场的重建有效验证了前文中的算法，为莫尔层析的使用打下了基础。
Optical Computerized Tomography (OCT) is a branch of ComputerizedTomography. It is a non-contact measurement technique and shows superioritiesin many domains, such as flow visualization and measurement of thermo physicalparameters. The moire tomography belongs to the Optical ComputerizedTomography and its projection data is deflection angles. It has many meritssuch as simple optical configuration, superior ability of anti-error andadaptation to tough environment. The moire tomography has great practicalvalue to the qualitative visualizations and quantitative measurements ofcomplex flow fields. But up to now, the theory system and reconstructionalgorithms of the moire tomography have not been studied systematically. Soemphasis of this dissertation is put on the reconstruction algorithms of moiretomography for the complex flow fields. The main work is described as follows:
1) Theory research of moire tomography
The relation between the deflection projection and the traditionalprojection is studied and is described from many different views. By thisrelation, the deflection tomography can have a explicit position in the wholetomographic theoretical frame and theoretical law can be found to thesemethods which apply the traditional tomographic reconstruction algorithms tothe reconstruction of deflection tomography.
2) Research work to the transform algorithms of deflection tomography
According to the relation between these two tomographic problems, thedeflection filtered back-projection algorithm is achieved and its numericalimplementation is studied. The corresponding convolution back-projectionalgorithm is also given. Then an error analysis is made to the DFBP algorithmand a modified algorithm is presented.
3) Research work to the series expansion algorithms of deflectiontomography
A model which directly numeralize the gradient of refractive index ispresented to get the discrete linear equations of deflection tomography.According to the model, three reconstruction algorithms are studied to solvethese linear equations. Numerical experiments show that these algorithms have strong abilities of noise immunity. Another deflection iterative algorithmbased on the expansion of local basis is presented. This algorithm can notonly reserve the sparse property Of the coefficient array, but also discretethe deflection tomography in a natural way. Good quality of anti-noise andreconstruction accuracy of this algorithm have been found by numericalexperiments.
4) Research work to the reconstructions of deflection tomography withincomplete data
In the limited angle problem, a deflection tomographic iterative algorithmwhich is based on the Bayesian Law is presented. In the research of the interiorproblem, the wavelet based local tomographic reconstruction algorithm whichis present by Farrokh is introduced to the local reconstruction of deflectiontomography.
5) Research work to the real flow field of moire tomography
The deflection filtered back-projection algorithm and its correspondmodified algorithm are applied to reconstruct the density field caused by therocket exhausted plumes. Results show that the modified algorithm can depressthe slope phenomena efficiently. The iterative algorithm which is based onthe expansion of harming function is applied to reconstruct thedensity fieldin a hypersonic wind tunnel. The reconstruction results are compared with theinverse Abel transform method which can only be applied with axial symmetricalfield. Results coincident well with each other. But it is obviously that thenovel algorithm have handle arbitrary flied and more universality.Reconstructions to these real flow fields show those algorithms are valid.So the application foundation of moire tomography has been provided in thisdissertation.
引文
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