电阻网络电阻抗成像中噪声影响的研究
Study of noise effects in electrical impedance tomography with resistor networks
摘要：
我们提出了一个研究的二维电阻抗层析成像问题的数值解，与噪声测量的狄利克雷诺依曼映射。反演使用最佳网格上的电导率参数化。网格是最佳的意义上说，有限体积离散他们给出谱精确近似的狄利克雷诺依曼映射。近似是特殊电阻网络的Dirichlet到Neumann映射，它是唯一可从测量中恢复的。近年来，在最优网格上进行反演的方法已被提出并进行了分析，但关于噪声对反演的影响的研究还没有开展。在本文中，我们提出了一个数值研究的线性和非线性反问题。我们采取三种不同的参数化未知的电导率，具有相同的自由度。我们得到的参数化引起的反演最佳网格是最有效的三个，因为它给出了最小的标准偏差的最大后验估计的电导率，均匀的域。对于非线性问题，我们计算的平均值和方差的最大后验估计的电导率，最佳的网格。对于小噪声，我们得到的估计是无偏的，他们的方差是非常接近的最佳之一，给出的Cramer-Rao界。对于较大的噪声，我们使用正则化并量化在解决方案中减少方差和引入偏差之间的权衡。考虑了全部和部分测量设置。
英文摘要：
We present a study of the numerical solution of the two dimensional electrical impedance tomography problem, with noisy measurements of the Dirichlet to Neumann map. The inversion uses parametrizations of the conductivity on optimal grids. The grids are optimal in the sense that finite volume discretizations on them give spectrally accurate approximations of the Dirichlet to Neumann map. The approximations are Dirichlet to Neumann maps of special resistor networks, that are uniquely recoverable from the measurements. Inversion on optimal grids has been proposed and analyzed recently, but the study of noise effects on the inversion has not been carried out. In this paper we present a numerical study of both the linearized and the nonlinear inverse problem. We take three different parametrizations of the unknown conductivity, with the same number of degrees of freedom. We obtain that the parametrization induced by the inversion on optimal grids is the most efficient of the three, because it gives the smallest standard deviation of the maximum a posteriori estimates of the conductivity, uniformly in the domain. For the nonlinear problem we compute the mean and variance of the maximum a posteriori estimates of the conductivity, on optimal grids. For small noise, we obtain that the estimates are unbiased and their variance is very close to the optimal one, given by the Cramer-Rao bound. For larger noise we use regularization and quantify the trade-off between reducing the variance and introducing bias in the solution. Both the full and partial measurement setups are considered.
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