Left and right preconditioning for electrical impedance tomography with structural information
Left and right preconditioning for electrical impedance tomography with structural information
复制标题
具有结构信息的电阻抗断层扫描的左右预处理
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
E. Somersalo
中科院分区：
文献类型：
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作者：
D. Calvetti;Debra F. McGivney;E. Somersalo
在计算反问题中，一个常见的问题是寻找一种有效的方法来求解线性或非线性最小二乘问题。对于大规模问题，迭代求解器是求解相关线性系统的首选方法，而对于非线性问题，则需要另外一种有效的局部线性化方法。本文基于反问题的贝叶斯分析，讨论了Krylov子空间方法的一种有效的预条件方案。我们应用这种方法的模型问题是电阻抗断层成像(EIT)，它增加了来自互补模式的先验信息，例如X射线成像。这里考虑的特殊几何形状模拟了用于乳房成像的x射线引导的EIT。同时应用EIT和X光乳腺成像的兴趣源于实验观察，某些类型的恶性和良性组织的阻抗谱彼此显著不同，从而提供了一种在不需要更多侵入性组织采样的情况下进行诊断的可能性。在贝叶斯框架下建立EIT反问题后，我们给出了计算最大后验估计的内、外迭代格式。先验协方差提供了右预条件子，建模误差协方差为迭代方法提供了左预条件子，迭代方法用于在优化问题的每次外迭代中求解线性最小二乘问题。此外，内迭代的停止准则与外迭代解的进展相结合。除了预处理方案外，计算效率还依赖于一种非常有效的计算雅可比矩阵的方法，该方法是通过精心组织正演计算而获得的。算例表明了该算法的稳健性和计算效率。
A common problem in computational inverse problems is to find an efficient way of solving linear or nonlinear least-squares problems. For large-scale problems, iterative solvers are the method of choice for solving the associated linear systems, and for nonlinear problems, an additional effective local linearization method is required. In this paper, we discuss an efficient preconditioning scheme for Krylov subspace methods, based on the Bayesian analysis of the inverse problem. The model problem to which we apply this methodology is electrical impedance tomography (EIT) augmented with prior information coming from a complementary modality, such as x-ray imaging. The particular geometry considered here models the x-ray-guided EIT for breast imaging. The interest in applying EIT concurrently with x-ray breast imaging arises from the experimental observation that the impedivity spectra of certain types of malignant and benign tissues differ significantly from each other, thus offering a possibility of diagnosis without more invasive tissue sampling. After setting up the EIT inverse problem within a Bayesian framework, we present an inner and outer iteration scheme for computing a maximum a posteriori estimate. The prior covariance provides a right preconditioner and the modeling error covariance provides a left preconditioner for the iterative method used to solve the linear least-squares problem at each outer iteration of the optimization problem. Moreover, the stopping criterion for the inner iterations is coupled with the progress of the solution of the outer iteration. Besides the preconditioning scheme, the computational efficiency relies on a very efficient method to compute the Jacobian, obtained by carefully organizing the forward computation. Computed examples illustrate the robustness and computational efficiency of the proposed algorithm.