一种电容层析成像图像重建优化算法

提出一种电学层析成像(ECT)图像重建优化算法。通过将传统正则化算法转化为最小二乘问题进行求解,结合lp范数逼近正则化最小化问题,利用重新加权的方法进行迭代计算。以油-气两相流模型进行仿真及静态实验,将所提出的优化算法与常用的LBP、Landweber迭代及Tikhonov正则化算法进行对比。结果表明,与常用算法相比,采用该优化算法对管道中心物体及多物体分布流型进行图像重建,其图像相对误差均为最低,且重建图像的形状保真度明显提高。     

Implementation of edge-preserving regularization for frequency-domain diffuse optical tomography

In this study, we first propose the use of edge-preserving regularization in optimizing an ill-conditioned problem in the reconstruction procedure for diffuse optical tomography to prevent unwanted edge smoothing, which usually degrades the attributes of images for distinguishing tumors from background tissues when using Tikhonov regularization. In the edge-preserving regularization method presented here, a potential function with edge-preserving properties is introduced as a regularized term in an objective function. With the minimization of this proposed objective function, an iterative method to solve this optimization problem is presented in which half-quadratic regularization is introduced to simplify the minimization task. Both numerical and experimental data are employed to justify the proposed technique. The reconstruction results indicate that edge-preserving regularization provides a superior performance over Tikhonov regularization.     

改进的共轭梯度算法的图像重建算法

针对电容层析成像图像重建问题的病态性,在Tikhonov正则化的基础上,以正则化解决方案的规模和给定数据的质量为理论依据,引入一个数学变换,提出了一种新的正则化方法,该方法克服了常规正则化方法扰乱原系统的缺陷。同时,将ECT物理模型进行规范化,并对共轭梯度算法进行改进。仿真实验表明,改进的共轭梯度算法的成像质量高于LBP算法、 Tikhonov正则化算法和共轭梯度算法,并利用相关系数进行了验证。     

基于改进SR3模型算法的ECT图像重建研究

针对电容层析成像技术应用于气固两相流检测时,图像重建过程中存在的不适定性问题,提出一种稀疏松弛正则化回归模型(SR3)应用于ECT图像重建。采用软阈值迭代法和梯度下降法为SR3模型求解器,向SR3模型中加入L1、L2惩戒项,并设计滤值环节优化解向量。实验结果表明,改进SR3模型算法相比Tikhonov正则化算法、L1正则化算法及原SR3模型算法,重建图像精度明显提高,图像相对误差显著降低,有较好的成像效果。     

改进的Tikhonov正则化图像重建算法

Tikhonov正则化法可以解决电容层析成像中图像重建的病态问题,同时能够平衡解的稳定性与精确性,但其有效性和成像质量受到测量数据粗差的影响。改进的Tikhonov正则化法将2范数和M-估计结合,用一个缓慢增长的Cauchy函数代替最小二乘法的平方和函数,提高了估计稳健性和适应性。利用COMSOL和MATLAB软件对方法的有效性进行验证,重建结果表明,改进的Tikhonov正则化法能够有效减少粗差影响,提高重建图像精确度及分辨率。     

Inverse heat transfer analysis for design and control of a micro-heater array

A non-linear inverse heat source identification problem is described and solved. The inverse problem analysis is used in the design of an embedded micro-heater array and to estimate the required control settings, which are the input currents to each heating element, to generate as close as possible to a prescribed temperature profile on the surface of a thin copper film. The purpose of the micro-heater array is to control the local copper microstructure through control of the local temperature field. A finite element model of the micro-heater system is used to define a discrete set of non-linear equations used as a basis for the inverse problem solution. Two methods are explored to solve the inverse problem, a direct minimization method with Tikhonov regularization and a passivity-based feedback control algorithm. A uniform and a linear temperature distribution could be attained in the central region above the micro-heater array, but the temperatures near the edges of the domain could not be controlled due to heat loss at the edges. Thus, to control the temperature field over the full width of the domain, the heater array must extend beyond the domain of interest. Both methods to solve the inverse problem are found to perform well. The regularization method allows for a smoother solution, while the feedback control method is simpler as the coefficient matrix for which the update remains unchanged for each iteration.     

Numerical boundary identification for Helmholtz-type equations

We study the identification of an unknown portion of the boundary of a two-dimensional domain occupied by a material satisfying Helmholtz-type equations from additional Cauchy data on the remaining known portion of the boundary. This inverse geometric problem is approached using the boundary element method (BEM) in conjunction with the Tikhonov first-order regularization procedure, whilst the choice of the regularization parameter is based on the L-curve criterion. The numerical results obtained show that the proposed method produces a convergent and stable solution     

Active constrained truncated Newton method for simple-bound optical tomography

In the past, nonlinear unconstrained optimization of the optical imaging problem has focused on Newton-Raphson techniques. Besides requiring expensive computation of the Jacobian, the unconstrained minimization with Tikhonov regularization can pose significant storage problems for large-scale reconstructions, involving a large number of unknowns necessary for realization of optical imaging. We formulate the inverse optical imaging problem as both simple-bound constrained and unconstrained minimization problems in order to illustrate the reduction in computational time and storage associated with constrained image reconstructions. The forward simulator of excitation and generated fluorescence, consisting of the Galerkin finite-element formulation, is used in an inverse algorithm to find the spatial distribution of absorption and lifetime that minimizes the difference between predicted and synthetic frequency-domain measurements. The inverse approach employs the truncated Newton method with trust region and a modification of automatic reverse differentiation to speed the computation of the optimization problem. The reconstruction results confirm that the physically based, constrained minimization with efficient optimization schemes may offer a more logical approach to the large-scale optical imaging problem than unconstrained minimization with regularization.     

Image reconstruction algorithm based on the extended regularised total least squares method for electrical capacitance tomography

Electrical capacitance tomography (ECT) is a non-invasive imaging technology that aims at the visualisation of the cross-sectional permittivity distribution of a dielectric object based on the measured capacitance data. Successful applications of ECT depend greatly on the precision and speed of the image reconstruction algorithms. ECT image reconstruction is a typical ill-posed problem, and its solution is unstable, that is, the solution is sensitive to noises in the input data. Methods that ensure the stability of a solution while enhancing the quality of the reconstructed images should be used to obtain a meaningful reconstruction result. An image reconstruction algorithm based on the regularised total least squares (TLS) method that considers the errors in both the sensitivity field matrix and the capacitance data for ECT is presented. The regularised TLS method is extended using a combination robust estimation technique and an extended stabilising functional according to the ill-posed characteristics of ECT, which transforms the image reconstruction problem into an optimisation problem. In addition, the Newton algorithm is employed to solve the objective functional. Numerical simulations indicate that the algorithm is feasible and overcomes the numerical instability of ECT image reconstruction; for the cases of the reconstructed objects considered here, the spatial resolution of the reconstructed images obtained using the algorithm is enhanced; as a result, an efficient method for ECT image reconstruction is introduced.     

On the inverse problem of soil profile reconstruction: a comparison of time-domain approaches

We discuss a one-dimensional inverse material profile reconstruction problem that arises in layered media underlain by a rigid bottom, when total wavefield surficial measurements are used to guide the reconstruction. To tackle the problem, we adopt the systematic framework of PDE-constrained optimization and construct an augmented misfit functional that is further endowed by a regularization scheme. We report on a comparison of spatial regularization schemes such as Tikhonov and total variation against a temporal scheme that treats the model parameters as time-dependent. We study numerically the effects of inexact initial estimates, data noise, and regularization parameter choices for all three schemes, and report inverted profiles for the modulus, and for simultaneous inversion of both the modulus and viscous damping. Our numerical experiments demonstrate comparable or superior performance of the time-dependent regularization over the Tikhonov and total variation schemes for both smooth and sharp target profiles, albeit at increased computational cost. Support for this work was provided by the US National Science Foundation under grant awards ATM-0325125 and CMS-0348484.     

基于lp范数的ECT图像重建算法研究

针对在ECT图像重建过程中,基于lp-范数的非凸压缩感知算法常存在计算量较大以及现有的近端映射算法受一些特定的p值限制而导致成像分辨率较低的问题,利用改进的插值函数替换lp-范数xpp,通过调整参数使得改进的函数无限逼近lp-范数xpp,同时引入阈值表示理论,并在此基础上提出新的自适应阈值迭代算法对新模型进行求解。实验结果表明,改进后的自适应lp-范数重构算法相对于Landwebr算法、迭代重加权最小二乘法具有更强的适应性,更高的图像分辨率,更快的成像速度。     

Numerical identification for impedance coefficient by a MFS-based optimization method

In this paper, we propose a new numerical method to solve an inverse impedance problem for Laplace's equation. The Robin coefficient in the impedance boundary condition is recovered from Cauchy data on a part of boundary. A crucial step is to transform the problem into an optimization problem based on the MFS and Tikhonov regularization. Then the popular conjugate gradient method is used to solve the minimization problem. We compare several stopping rules in the iteration procedure and try to find an accurate and stable approximation. Numerical results for four examples in 2D and 3D cases will show the effectiveness of the proposed method.     

基于梯度投影稀疏重建算法的电容层析成像图像重建

为解决两相流中存在中心物体、物体比较小或存在多个物体且相距较近时电容层析成像(ECT)重建图像精度较差的问题,基于稀疏分布的流型其介电常数分布满足稀疏性的先验条件,采用梯度投影稀疏重建(GPSR-BB)算法进行ECT图像重建。仿真及实验测试结果表明:GPSR-BB算法对于流体中小目标以及复杂流型的图像重建质量较好,重建图像的形状保真度高。     

Dual imaging modality of granular flow based on ECT sensors

In this study, a previously developed dual modality imaging system is applied to image the flow of granular matter with different electrical properties in cylindrical vessels. The imaging system is based on both capacitance and power measurements acquired by an electrical capacitance tomography (ECT) sensor located around the vessel. The measurement data are then used to reconstruct cross-sectional images of both permittivity and conductivity distributions. A neural network multi-criterion optimization reconstruction technique (NN-MOIRT) is used for the inverse (reconstruction) problem. The contribution of this technology to the field of granular matters is explored through review of research articles that can be a direct application of this development. We discuss the capabilities of this dual-modality acquisition system using synthetic data for granular matter with different electrical properties.     

A meshless method for the stable solution of singular inverse problems for two-dimensional Helmholtz-type equations

We investigate a meshless method for the stable and accurate solution of inverse problems associated with two-dimensional Helmholtz-type equations in the presence of boundary singularities. The governing equation and boundary conditions are discretized by the method of fundamental solutions (MFS). The existence of boundary singularities affects adversely the accuracy and convergence of standard numerical methods. Solutions to such problems and/or their corresponding derivatives may have unbounded values in the vicinity of the singularity. Moreover, when dealing with inverse problems, the stability of solutions is a key issue and this is usually taken into account by employing a regularization method. These difficulties are overcome by combining the Tikhonov regularization method (TRM) with the subtraction from the original MFS solution of the corresponding singular solutions, without an appreciable increase in the computational effort and at the same time keeping the same MFS discretization. Three examples for both the Helmholtz and the modified Helmholtz equations are carefully investigated.     

基于Tikhonov正则化及奇异值分解的载荷识别方法

针对矩阵求逆法应用中存在的病态逆问题,用Tikhonov正则化及奇异值分解法解决。通过对平板模型仿真分析,利用频响函数法矩阵条件数评价系统的病态性,系统病态性不同时用奇异值分解法与基于不同正则化参数选择的Tikhonov方法对载荷进行识别。研究表明,条件数大于1000时,Tikhonov正则化方法识别误差较小;反之,奇异值分解法较优。提出综合使用Tikhonov正则化与奇异值分解的载荷识别方法,给出方法流程。仿真与实验结果表明该方法可提高结构载荷识别精度,具有一定工程应用价值。     

A Multimodal Tomography System Based on ECT Sensors

A new noninvasive system for multimodal electrical tomography based on electrical capacitance tomography (ECT) sensor hardware is proposed. Quasistatic electromagnetic fields are produced by ECT sensors and used for interrogating the sensing domain. The new system is noninvasive and based on capacitance measurements for permittivity and power balance measurements for conductivity (impedance) imaging. A dual sensitivity map of perturbations in permittivity and conductivity is constructed. The measured data along with the sensitivity matrix are used for the actual image reconstruction. The new multimodal tomography system has the advantage of using already established reconstruction techniques, and the potential for combination with new reconstruction techniques by taking advantage of combined conductivity/permittivity data. Moreover, it does not require direct contact between the sensor and the region of interest. The system performance has been tested on representative data, producing good results     

Use of non-negative constraint in Tikhonov regularization for particle sizing based on forward light scattering

Abstract

The set of linear equations in the inversion of particle size distribution (PSD) based on forward light scattering is an ill-posed problem. In order to solve the inverse problem of this kind, a number of inversion algorithms have been proposed. The regularization algorithm can reconstruct the PSD, but in usual case, the solution may contain negative values and is strongly oscillating. Owing to the natural reason, the solution should be non-negative and smooth. In this paper, a simple non-negative constraint (NNC) is used with a combination of the Tikhonov regularization. Simulations and experiments show that the regularization with NNC can achieve more reasonable results.     

Desingularized meshless method for solving Laplace equation with over-specified boundary conditions using regularization techniques

The desingularized meshless method (DMM) has been successfully used to solve boundary-value problems with specified boundary conditions (a direct problem) numerically. In this paper, the DMM is applied to deal with the problems with over-specified boundary conditions. The accompanied ill-posed problem in the inverse problem is remedied by using the Tikhonov regularization method and the truncated singular value decomposition method. The numerical evidences are given to verify the accuracy of the solutions after comparing with the results of analytical solutions through several numerical examples. The comparisons of results using Tikhonov method and truncated singular value decomposition method are also discussed in the examples.     

Simultaneous determination for a space-dependent heat source and the initial data by the MFS

In this paper we propose a numerical algorithm based on the method of fundamental solutions for recovering a space-dependent heat source and the initial data simultaneously in an inverse heat conduction problem. The problem is transformed into a homogeneous backward-type inverse heat conduction problem and a Dirichlet boundary value problem for Poisson's equation. We use an improved method of fundamental solutions to solve the backward-type inverse heat conduction problem and apply the finite element method for solving the well-posed direct problem. The Tikhonov regularization method combined with the generalized cross validation rule for selecting a suitable regularization parameter is applied to obtain a stable regularized solution for the backward-type inverse heat conduction problem. Numerical experiments for four examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed algorithm.