Electrical impedance tomography (EIT) is an imaging modality where a patient or object is probed using harmless electric currents. The currents are fed through electrodes placed on the surface of the target, and the data consists of voltages measured at the electrodes resulting from a linearly independent set of current injection patterns. EIT aims to recover the internal distribution of electrical conductivity inside the target. The inverse problem underlying the EIT image formation task is nonlinear and severely ill-posed, and hence sensitive to modeling errors and measurement noise. Therefore, the inversion process needs to be regularized. However, traditional variational regularization methods, based on optimization, often suffer from local minima because of nonlinearity. This is what makes regularized direct (non-iterative) methods attractive for EIT. The most developed direct EIT algorithm is the D-bar method, based on Complex Geometric Optics solutions and a nonlinear Fourier transform. Variants and recent developments of D-bar methods are reviewed, and their practical numerical implementation is explained.

中文翻译：

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电阻抗断层扫描的D-bar法---揭秘

电阻抗断层扫描 (EIT) 是一种使用无害电流探测患者或物体的成像方式。电流通过放置在目标表面上的电极馈送，数据包括在电极处测得的电压，这些电压由一组线性无关的电流注入模式产生。EIT 旨在恢复目标内部电导率的内部分布。EIT 图像形成任务背后的逆问题是非线性和严重不适定的，因此对建模误差和测量噪声敏感。因此，需要对反演过程进行正则化。然而，传统的基于优化的变分正则化方法往往由于非线性而受到局部最小值的影响。这就是正则化直接（非迭代）方法对 EIT 有吸引力的原因。最发达的直接 EIT 算法是 D-bar 方法，它基于复杂几何光学解决方案和非线性傅立叶变换。回顾了 D-bar 方法的变体和最新发展，并解释了它们的实际数值实现。