基于非光滑多参数正则化的EIT图像重构算法研究

EIT图像重构问题数学上可归结为求解一类具有严重不适定性的二阶椭圆型偏微分方程参数识别反问题。随着稀疏约束和全变差约束等非光滑约束的引入，多参数正则化及其涉及的迭代法研究得以蓬勃开展。本项目将改进和发展近来流行的非光滑正则化，以多参数非光滑约束为出发点，探索反演参数的有效自适应稀疏表示，结合同伦摄动技术和加速技巧，开展多参数约束下的EIT图像重构的数值研究。主要内容包括：1)构造基于非凸多参数约束的Tikhonov型正则化，借助光滑化函数的凸化逼近思想，设计快速有效的迭代算法，并给出整体优化算法的收敛性分析；2)构造基于非光滑多参数约束的加速同伦摄动型迭代法，考虑到反演问题的规模，通过对测量数据进行分组处理，结合Kaczmarz思想研究快速算法；3)探讨应用启发式参数选取准则确定权重因子，实现不依赖噪音水平的多参数选取。本项目成果将进一步丰富和发展EIT图像重构的非光滑正则化算法研究。

The image reconstruction of electrical impedance tomography (EIT) mathematically can be regarded as solving a class of the parameter identification problem inverse problems for the second order elliptic partial differential equations with serious ill-posedness. With the introduction of nonsmooth constraints such as sparsity constraint and total variation constraint, the studies on the multi-parameters regularization methods and the iterative regularization methods have been developed vigorously. This project will improve and develop the recently popular nonsmooth regularization method. Starting from the multi-parameter non-smooth constraint, this project first explores the effective adaptive sparse representation for inversion parameters, and then combines the homotopy perturbation technique and acceleration technique, to carry out the numerical studies of EIT image reconstruction under multi-parameter constraint. This project mainly contains the following aspects: 1）We will construct Tikhonov-type regularization with the nonconvex and nonsmooth multi-parameter constraints. With the aid of smoothing function to realize the convexification approximation, the fast and effective iterative algorithms will be established, and the convergence analysis for the whole optimization algorithm will be given. 2) We will develop the accelerated homotopy perturbation iteration regularization method based on the nonsmooth multi-parameter constraints. Considering the scale of the inversion problem, Kaczmarz method by grouping the measurement data will be introduced for the research of the fast approaches. 3) Aiming at the problem of multiple regularization parameter selection caused by multi-parameter constraints, the heuristic parameter choice rules is applied and discussed to determine the weight factor to realize the choice of multi-parameters with unknown noise level. The results of this project will further enrich and develop the research on nonsmooth regularization algorithm of EIT image reconstruction, and have a certain role in promoting the application of EIT.