A Direct Reconstruction Algorithm for the 2-D Inverse Conductivity Problem

二维电导率反问题的直接重构算法

• 批准号: 0104861
• 负责人: Jennifer Mueller
• 金额: $ 8.1万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-01 至 2005-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104861&HistoricalAwards=false
• 关键词: Direct; Reconstruction; Algorithm; Inverse; Conductivity

This proposal addresses a direct numerical reconstruction algorithm for the 2-D inverse conductivity problem. The 2-D inverse conductivity problem is to determine an unknown conductivity distribution on a bounded region from knowledge of the Dirichlet-to-Neumann map. Physically, knowledge of the Dirichlet-to-Neumann map is tantamount to knowing the current density distribution on the boundary of the region resulting from any given voltage distribution applied on the boundary. The problem is modeled mathematically by the generalized Laplace's equation with the conductivity as an unknown parameter. In 1995 A. Nachman proved that knowledge of the Dirichlet-to-Neumann map uniquely determines the conductivity in the interior of a smooth bounded region in 2-D. An important feature of Nachman's proof is that it outlines a direct method for solving for the conductivity without iteration. The proof is based on techniques of inverse scattering and the d-bar method, which is a method of solution for scattering problems, not a numerical technique. The primary goals of this proposal are to solve the inverse conductivity problem numerically using the d-bar method, develop a practical reconstruction algorithm for medical applications, and to test the implementation on physically relevant conductivity distributions. The 2-D inverse conductivity problem has applications in geophysics, nondestructive testing, and a medical imaging technique known as electrical impedance tomography (EIT). One application of EIT is the imaging of heart and lung function in real time. In this application, electrodes are placed around the circumference of the patient's torso, current is applied on the electrodes and the resulting voltage is measured. The resulting 2-D inverse conductivity problem is then solved numerically to reconstruct how the electricity passes through the interior and to form a cross-sectional image of the patient's chest. Other applications include the detection of breast cancer, monitoring for internal bleeding, and the diagnosis of pulmonary embolis (a blood clot in the lung). The proposed approach represents a new class of image reconstruction algorithm for the EIT problem. Work thus far has indicated that the algorithm yields more accurate images than the existing fast algorithms, since it solves the full set of equations rather than a a more simplified version of the problem. This is particularly important in medical applications such as breast cancer detection, where the measured values distinguish between the presence of a tumor or a benign cyst. The algorithm will be tested on real data and compared to existing algorithnms in terms of accuracy and efficiency.

该提案提出了二维逆电导率问题的直接数值重建算法。 二维反电导率问题是根据狄利克雷到诺依曼图的知识确定有界区域上的未知电导率分布。 从物理上讲，了解狄利克雷到诺依曼图就等于了解由施加在边界上的任何给定电压分布产生的区域边界上的电流密度分布。 该问题通过广义拉普拉斯方程进行数学建模，其中电导率作为未知参数。 1995 年，A. Nachman 证明，狄利克雷到诺伊曼图的知识唯一地确定了二维平滑有界区域内部的电导率。 纳赫曼证明的一个重要特点是它概述了一种无需迭代即可求解电导率的直接方法。 该证明基于逆散射技术和 d-bar 方法，这是一种解决散射问题的方法，而不是数值技术。 该提案的主要目标是使用 d-bar 方法以数值方式解决电导率反演问题，开发适用于医学应用的实用重建算法，并测试物理相关电导率分布的实现。二维电导率反演问题在地球物理学、无损检测和称为电阻抗断层扫描 (EIT) 的医学成像技术中具有应用。 EIT 的应用之一是心肺功能的实时成像。 在此应用中，将电极放置在患者躯干的周围，在电极上施加电流并测量产生的电压。 然后对由此产生的二维逆电导率问题进行数值求解，以重建电流如何通过内部并形成患者胸部的横截面图像。 其他应用包括乳腺癌检测、内出血监测和肺栓塞（肺部血栓）的诊断。 所提出的方法代表了一类针对 EIT 问题的新型图像重建算法。 迄今为止的工作表明，该算法比现有的快速算法产生更准确的图像，因为它解决了全套方程组而不是问题的更简化版本。 这在乳腺癌检测等医疗应用中尤其重要，其中测量值区分肿瘤或良性囊肿的存在。 该算法将在真实数据上进行测试，并在准确性和效率方面与现有算法进行比较。