多相流的电容层析成像图像重建研究
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摘要
电容层析成像(ECT)是一种可用于多相流浓度测量的可视化技术，该技术基于测量的电容数据重建被测物场的介质分布图。ECT因具有具有快速、安全、非侵入传感、廉价等优点而被认为是极具广阔发展前景的过程成像技术。ECT技术的成功应用在很大程度上取决于图像重建算法的精度与速度。本文主要探讨ECT图像重建算法，具体内容安排如下：
详细分析了电容层析成像技术的基本原理、逆问题的数学基础、三种典型的正则化方法(Landweber迭代法、Tikhonov正则法和截断奇异值分解法)、总结了七种广义逆的迭代算法，并进行了对比研究。
ECT图像重建问题常常被转化为一个最优化问题，本文详细地探讨了三类无约束最优化算法：最速下降法、共轭梯度法和牛顿法。特别地，由于共轭梯度算法具有简便、所需要存储量与最速下降法差别不大，而收敛速度比最速下降法要快、初值不敏感性、大范围收敛等特性，本文基于解的先验信息改进共轭梯度算法，并对四种典型的共轭梯度算法进行了比较评价。
Tikhonov正则法是一个求解病态问题的有效方法，一个Tikhonov正则化解是个平衡解的精确性与稳定性的结果。根据ECT图像重建的病态本质，本文推导了两种广义Tikhonov泛函：其一，利用组合估计技术改进标准Tikhonov泛函；其二，用Tikhonov正则化技术改进标准Minimax估计。数值结果表明，两种算法均表现出良好的数值性能，能够有效克服ECT图像重建的数值不稳定性。所重建图像的空间分辨率得到了显著提高，被重建图像的失真较小且伪影较少；同时，用噪声污染电容数据重建的结果也表明算法具有良好的稳健性。
传统ECT图像重建算法仅考虑测量电容数据的噪声，敏感矩阵的不精确性并未得到考虑。事实上，由于图像重建模型的线性化近似，使得敏感矩阵可能是不精确的。为此，在ECT图像重建过程中同时考虑电容数据和敏感矩阵的不精确性是合理的。基于正则化总体最小二乘估计，本文提出了一个同时考虑电容数据和敏感矩阵不精确性的图像重建算法，并利用稳健估计技术改进标准正则化总体最小二乘法。数值结果表明，该算法是可行的，能够有效克服ECT图像重建的数值不稳定性。所重建图像的空间分辨率得到了显著提高，被重建图像的失真较小且伪影较少。
基于小波多尺度分析技术和总体最小二乘估计方法，本文推导了一个广义多尺度ECT图像重建目标泛函。此外，本文用同伦算法求解所构造的目标泛函，其同伦方程由固定点同伦设计，并用固定点迭代算法求解。数值实验结果表明，该算法是可行的，能够有效克服ECT图像重建的数值不稳定性。所重建图像的空间分辨率得到了显著提高，被重建图像的失真较小且伪影较少。
基于小波多尺度分析方法，本文对ECT图像进行了相应的后处理，它包括ECT图像的多尺度增强、多尺度去噪和多尺度融合三个方面。数值结果表明，经过处理后的ECT图像能够更好地突出被重建对象的细节特征，增强了ECT图像的解释能力和可靠性，进而为后续的定量分析提供必要的条件。
传统ECT图像重建算法一般考虑线性化模型，当被重建对象的高介电常数与低介电常数的差别不大时是可行的，而且，也获得了许多成功的应用。然而，在实际的应用中，当被重建对象的高介电常数与低介电常数的差别大时，线性化近似可能引入较大的误差，此时，在ECT图像重建过程中考虑线性化近似所引入的误差是合理的。基于半参数方法，本文推导了一个同时考虑电容测量噪声和线性化近似所引入误差的ECT图像重建目标泛函，在此基础上用同伦算法求解所建立的目标泛函。数值结果表明，该算法是可行的，所重建图像的质量得到了显著的提高。
Electrical capacitance tomography (ECT) is a visualization technology that can be applied to concentration measurement for the multiphase flow. It attempts to image the permittivity distribution of an object by measuring the electrical capacitances between sets of electrodes placed around its periphery. ECT is considered as a promising tomography technology due to its advantages such as high speed, high safety, non-intrusive sensing and low cost. Successful applications of ECT technology depend on the speed and precision of the image reconstruction algorithms. A main motivation for this dissertation is to investigate image reconstruction algorithms for ECT, and the main contents are as follows:
The principles of ECT technique and the mathematics theory of the inverse problem are reviewed. Three typical regularization methods, such as the Landweber iteration algorithm, the Tikhonov regularization method and the truncated singular value decomposition method, are discussed in detail. Seven kinds of iterative algorithms of solving the generalized inverse are introduced and compared.
The image reconstruction problem for ECT is often transformed into an optimization problem. In this dissertation, three kinds of unconstrained optimization algorithms, such as the steepest descent algorithm, the conjugate gradient algorithm and the Newton algorithm, are analyzed in detail. Especially, owing to the advantages such as the simplicity, smaller storage and faster convergence when compared with the steepest descent method, larger convergence domain, the conjugate gradient algorithms are detailedly discussed and improved according to the prior information of a solution. At the same time, four kinds of typical conjugate gradient algorithms are compared and evaluated by numerical simulations.
The Tikhonov regularization method is an effective method to solve the inverse problems. A Tikhonov regularization solution is a result of balancing the precision and stabilization of a solution. Two generalized Tikhonov functionals are deduced according to the ill posed characteristics of ECT image reconstruction problem. In the first objective functional, the standard Tikhonov functional is improved using a combination robust estimation technique. In the second objective functional, the standard Minimax estimation is developed by the Tikhonov regularization technique. The numerical results indicate that the both algorithms are feasible and effectively overcome the numerical instability of ECT image reconstruction process. The spatial resolution of the reconstructed images is remarkably enhanced. The distortion of the reconstructed images is relative small, and the artifacts in the reconstructed images can be eliminated effectively. At the same time, the reconstructed results by the noise-contaminated capacitance data also show that the proposed algorithms hold a good robustness to the noises in the capacitance data.
Traditional image reconstruction algorithms for ECT only consider the noises in the measured capacitance data; however, the inaccurate characteristics in the sensitivity matrix are not to be considered. In fact, the sensitivity matrix may be inaccurate due to the linearization approximation of the image reconstruction model. Therefore, considering the inaccurate nature in the measured capacitance data and the sensitivity matrix in the process of image reconstruction is reasonable. Based on the regularized total least squares method that has been developed using the robust estimation technique, an image reconstruction algorithm that considers the inaccurate nature in the measured capacitance data and the sensitivity matrix is proposed. The numerical results indicate that the proposed algorithm is feasible and effectively overcomes the numerical instability of ECT image reconstruction process. The spatial resolution of the reconstructed images is remarkably enhanced. The distortion of the reconstructed images is relatively small, and the artifacts in the reconstructed images can be eliminated effectively.
A multiscale objective functional based on the wavelet multiscale analysis technique and the total least squares method is deduced. The homotopy algorithm is employed to solve the objective functional. The homotopy equation is designed by the fixed-point homotopy and is solved by the fixed-point iterative algorithm. The numerical results indicate that the proposed algorithm is feasible and effectively overcomes the numerical instability of ECT image reconstruction process. The spatial resolution of the reconstructed images is remarkably enhanced. The distortion of the reconstructed images is relatively small, and the artifacts in the reconstructed images can be eliminated effectively.
Post-processing techniques for ECT images, such as the multiscale image enhancing, the multiscale image de-noising and the multiscale image fusion, have been carried out using the wavelet-based multiscale method. Numerical results indicate that the processed images can better highlight the detailed characteristics of the reconstructed objects and the explanatory ability and reliability of ECT images are increased, which provide necessary elements for the sequent quantitative analysis.
Traditional ECT image reconstruction algorithms often consider the linearized model, which has obtained many successful applications when the difference between the high permittivity and low permittivity in the measured region is not large. However, the linearized model may bring large error when the difference between the high permittivity and low permittivity in the measured region is large. Therefore, considering the errors that linearization process brings is reasonable in the process of image reconstruction. A new objective functional based on the semiparametric method that considers the noises in the measured capacitance data and the linearization error is established. The homotopy algorithm is employed to solve this objective functional. Numerical results indicate that this algorithm is feasible, and quality of the reconstructed images is enhanced remarkably.
详细分析了电容层析成像技术的基本原理、逆问题的数学基础、三种典型的正则化方法(Landweber迭代法、Tikhonov正则法和截断奇异值分解法)、总结了七种广义逆的迭代算法，并进行了对比研究。
ECT图像重建问题常常被转化为一个最优化问题，本文详细地探讨了三类无约束最优化算法：最速下降法、共轭梯度法和牛顿法。特别地，由于共轭梯度算法具有简便、所需要存储量与最速下降法差别不大，而收敛速度比最速下降法要快、初值不敏感性、大范围收敛等特性，本文基于解的先验信息改进共轭梯度算法，并对四种典型的共轭梯度算法进行了比较评价。
Tikhonov正则法是一个求解病态问题的有效方法，一个Tikhonov正则化解是个平衡解的精确性与稳定性的结果。根据ECT图像重建的病态本质，本文推导了两种广义Tikhonov泛函：其一，利用组合估计技术改进标准Tikhonov泛函；其二，用Tikhonov正则化技术改进标准Minimax估计。数值结果表明，两种算法均表现出良好的数值性能，能够有效克服ECT图像重建的数值不稳定性。所重建图像的空间分辨率得到了显著提高，被重建图像的失真较小且伪影较少；同时，用噪声污染电容数据重建的结果也表明算法具有良好的稳健性。
传统ECT图像重建算法仅考虑测量电容数据的噪声，敏感矩阵的不精确性并未得到考虑。事实上，由于图像重建模型的线性化近似，使得敏感矩阵可能是不精确的。为此，在ECT图像重建过程中同时考虑电容数据和敏感矩阵的不精确性是合理的。基于正则化总体最小二乘估计，本文提出了一个同时考虑电容数据和敏感矩阵不精确性的图像重建算法，并利用稳健估计技术改进标准正则化总体最小二乘法。数值结果表明，该算法是可行的，能够有效克服ECT图像重建的数值不稳定性。所重建图像的空间分辨率得到了显著提高，被重建图像的失真较小且伪影较少。
基于小波多尺度分析技术和总体最小二乘估计方法，本文推导了一个广义多尺度ECT图像重建目标泛函。此外，本文用同伦算法求解所构造的目标泛函，其同伦方程由固定点同伦设计，并用固定点迭代算法求解。数值实验结果表明，该算法是可行的，能够有效克服ECT图像重建的数值不稳定性。所重建图像的空间分辨率得到了显著提高，被重建图像的失真较小且伪影较少。
基于小波多尺度分析方法，本文对ECT图像进行了相应的后处理，它包括ECT图像的多尺度增强、多尺度去噪和多尺度融合三个方面。数值结果表明，经过处理后的ECT图像能够更好地突出被重建对象的细节特征，增强了ECT图像的解释能力和可靠性，进而为后续的定量分析提供必要的条件。
传统ECT图像重建算法一般考虑线性化模型，当被重建对象的高介电常数与低介电常数的差别不大时是可行的，而且，也获得了许多成功的应用。然而，在实际的应用中，当被重建对象的高介电常数与低介电常数的差别大时，线性化近似可能引入较大的误差，此时，在ECT图像重建过程中考虑线性化近似所引入的误差是合理的。基于半参数方法，本文推导了一个同时考虑电容测量噪声和线性化近似所引入误差的ECT图像重建目标泛函，在此基础上用同伦算法求解所建立的目标泛函。数值结果表明，该算法是可行的，所重建图像的质量得到了显著的提高。
Electrical capacitance tomography (ECT) is a visualization technology that can be applied to concentration measurement for the multiphase flow. It attempts to image the permittivity distribution of an object by measuring the electrical capacitances between sets of electrodes placed around its periphery. ECT is considered as a promising tomography technology due to its advantages such as high speed, high safety, non-intrusive sensing and low cost. Successful applications of ECT technology depend on the speed and precision of the image reconstruction algorithms. A main motivation for this dissertation is to investigate image reconstruction algorithms for ECT, and the main contents are as follows:
The principles of ECT technique and the mathematics theory of the inverse problem are reviewed. Three typical regularization methods, such as the Landweber iteration algorithm, the Tikhonov regularization method and the truncated singular value decomposition method, are discussed in detail. Seven kinds of iterative algorithms of solving the generalized inverse are introduced and compared.
The image reconstruction problem for ECT is often transformed into an optimization problem. In this dissertation, three kinds of unconstrained optimization algorithms, such as the steepest descent algorithm, the conjugate gradient algorithm and the Newton algorithm, are analyzed in detail. Especially, owing to the advantages such as the simplicity, smaller storage and faster convergence when compared with the steepest descent method, larger convergence domain, the conjugate gradient algorithms are detailedly discussed and improved according to the prior information of a solution. At the same time, four kinds of typical conjugate gradient algorithms are compared and evaluated by numerical simulations.
The Tikhonov regularization method is an effective method to solve the inverse problems. A Tikhonov regularization solution is a result of balancing the precision and stabilization of a solution. Two generalized Tikhonov functionals are deduced according to the ill posed characteristics of ECT image reconstruction problem. In the first objective functional, the standard Tikhonov functional is improved using a combination robust estimation technique. In the second objective functional, the standard Minimax estimation is developed by the Tikhonov regularization technique. The numerical results indicate that the both algorithms are feasible and effectively overcome the numerical instability of ECT image reconstruction process. The spatial resolution of the reconstructed images is remarkably enhanced. The distortion of the reconstructed images is relative small, and the artifacts in the reconstructed images can be eliminated effectively. At the same time, the reconstructed results by the noise-contaminated capacitance data also show that the proposed algorithms hold a good robustness to the noises in the capacitance data.
Traditional image reconstruction algorithms for ECT only consider the noises in the measured capacitance data; however, the inaccurate characteristics in the sensitivity matrix are not to be considered. In fact, the sensitivity matrix may be inaccurate due to the linearization approximation of the image reconstruction model. Therefore, considering the inaccurate nature in the measured capacitance data and the sensitivity matrix in the process of image reconstruction is reasonable. Based on the regularized total least squares method that has been developed using the robust estimation technique, an image reconstruction algorithm that considers the inaccurate nature in the measured capacitance data and the sensitivity matrix is proposed. The numerical results indicate that the proposed algorithm is feasible and effectively overcomes the numerical instability of ECT image reconstruction process. The spatial resolution of the reconstructed images is remarkably enhanced. The distortion of the reconstructed images is relatively small, and the artifacts in the reconstructed images can be eliminated effectively.
A multiscale objective functional based on the wavelet multiscale analysis technique and the total least squares method is deduced. The homotopy algorithm is employed to solve the objective functional. The homotopy equation is designed by the fixed-point homotopy and is solved by the fixed-point iterative algorithm. The numerical results indicate that the proposed algorithm is feasible and effectively overcomes the numerical instability of ECT image reconstruction process. The spatial resolution of the reconstructed images is remarkably enhanced. The distortion of the reconstructed images is relatively small, and the artifacts in the reconstructed images can be eliminated effectively.
Post-processing techniques for ECT images, such as the multiscale image enhancing, the multiscale image de-noising and the multiscale image fusion, have been carried out using the wavelet-based multiscale method. Numerical results indicate that the processed images can better highlight the detailed characteristics of the reconstructed objects and the explanatory ability and reliability of ECT images are increased, which provide necessary elements for the sequent quantitative analysis.
Traditional ECT image reconstruction algorithms often consider the linearized model, which has obtained many successful applications when the difference between the high permittivity and low permittivity in the measured region is not large. However, the linearized model may bring large error when the difference between the high permittivity and low permittivity in the measured region is large. Therefore, considering the errors that linearization process brings is reasonable in the process of image reconstruction. A new objective functional based on the semiparametric method that considers the noises in the measured capacitance data and the linearization error is established. The homotopy algorithm is employed to solve this objective functional. Numerical results indicate that this algorithm is feasible, and quality of the reconstructed images is enhanced remarkably.
引文
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