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

本文提出了一种二维反电导率问题的直接数值重建算法。二维逆电导率问题是根据狄利克雷-诺伊曼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，了解Dirichlet-to-Neumann映射等于了解由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。将电导率作为一个未知参数，用广义拉普拉斯方程对该问题进行数学建模。在1995年，选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电导率。Nachman证明的一个重要特征是它给出了一种不用迭代求解电导率的直接方法。证明是基于逆散射技术和d-杆法，这是一种求解散射问题的方法，而不是一种数值技术。本提案的主要目标是使用d-bar方法在数值上解决反电导率问题，开发用于医疗应用的实用重建算法，并测试在物理相关电导率分布上的实现。二维逆电导率问题在地球物理学、无损检测和医学成像技术——电阻抗断层扫描（EIT）中都有应用。EIT的一个应用是实时的心肺功能成像。在这种应用中，电极被放置在病人躯干的周围，电流被施加在电极上并测量产生的电压。由此产生的二维反电导率问题，然后用数值方法解决，以重建电流如何通过内部，并形成患者胸部的横截面图像。其他应用包括检测乳腺癌、监测内出血和诊断肺栓塞（肺中的血凝块）。该方法为EIT问题提供了一种新的图像重建算法。迄今为止的工作表明，该算法产生的图像比现有的快速算法更准确，因为它解决了整套方程，而不是一个更简化的问题。这在医学应用中尤其重要，如乳腺癌检测，其中测量值可以区分肿瘤或良性囊肿的存在。该算法将在实际数据上进行测试，并与现有算法在准确性和效率方面进行比较。

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.