基于混合约束正则化的电阻抗成像反演研究

电阻抗成像技术（Electrical Impedance Tomography，EIT）实际上可看作二阶椭圆型偏微分方程参数识别逆问题。由于EIT逆问题存在非线性性、欠定性和不适定性等难点，成像的精度和稳定性通常难以保证。当前现有的反演方法往往分辨率不高、对比度差和速度较慢，给EIT技术的实际应用带来困难。针对此问题，本项目首先探索EIT问题反演参数的稀疏性表示方法，同时鉴于Thilknov正则化、全变差(TV)正则化和稀疏约束正则化各自具有的特性，将Thilknov正则化、TV正则化和稀疏约束正则化联合使用，设计混合约束罚项正则化的联合反演策略，并给出权重的选择方法。然后，针对混合约束反演模型，基于分裂Bregman技术构造并分析快速的迭代优化算法。目的是实现对目标体的光滑部分、分段常值的边缘轮廓以及可稀疏表示的尖角边界的同步反演，达到优势互补，从而改进成像质量和对数据噪声的鲁棒性。

Electrical impedance tomography (EIT) can be by nature regarded as the inverse problem of parameter identification for two order elliptic partial differential equation. It is often hard to guarantee the quality and stability for the imaging because of the existence of nonlinearity, under-determinedness and ill-posedness for EIT inverse problem. The current inversion methods usually suffer from the low resolution, low contrast and slow imaging speed, which bring the difficulty for the practical application of EIT. According to the defects, the project will first explore the sparsity representation method for the inversion parameters. Then In view of the respective characteristics for Tikhonov regularization, total variation (TV) regularization and sparsity regularization, the project designs the joint inversion strategies of mixed constraint penalty regularizations, and proposes one selection method for weights. Meanwhile, one fast numerical scheme based on the split Bregman iteration technique is constructed and analyzed to solve the proposed joint inversion models efficiently. Our aim is to realize the simultaneous inversion for the smooth part, piecewise constant edge and sharp boundary of the object, so as to improve the imaging quality and the robustness to noisy data.