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

本文提出了一种直接求解二维电导率反问题的数值重建算法。二维电导率反问题是根据Dirichlet-to-Neumann映射的知识确定有界区域上未知的电导率分布。从物理上讲，知道狄里克莱特到诺依曼映射就等于知道施加在边界上的任何给定电压分布所产生的区域边界上的电流密度分布。以电导率为未知参数，用广义拉普拉斯方程对问题进行了数学建模。1995年，A.Nachman证明了Dirichlet-to-Neumann映射的知识唯一地决定了二维光滑有界区域内部的导电性。Nachman证明的一个重要特点是它提供了一种无需迭代即可直接求解电导率的方法。证明是基于逆散射技术和d-bar方法，d-bar方法是一种求解散射问题的方法，而不是一种数值技术。该方案的主要目标是使用d-bar方法数值求解电导率逆问题，开发一种实用的医学应用重建算法，并测试对物理相关电导率分布的实现。二维电导率逆问题在地球物理、无损检测和被称为电阻抗断层成像(EIT)的医学成像技术中都有应用。EIT的一个应用是心肺功能的实时成像。在这种应用中，将电极放置在患者躯干周围，在电极上施加电流，并测量产生的电压。由此产生的二维电导率逆问题随后被数值求解，以重建电流如何通过内部并形成患者胸部的横断面图像。其他应用包括检测乳腺癌、监测内出血和诊断肺血栓(肺内的一种血块)。该方法代表了一类新的EIT图像重建算法。到目前为止的工作已经表明，该算法产生的图像比现有的快速算法更准确，因为它解决了整个方程组，而不是问题的更简化版本。这在医学应用中尤其重要，例如乳腺癌检测，在这些应用中，测量值区分肿瘤的存在或良性囊肿的存在。该算法将在真实数据上进行测试，并在精度和效率方面与现有算法进行比较。

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.