Reconstruction of tomographic images from boundary measurements requires
flexibility with respect to target domains. For instance, when the system
equations are modeled by partial differential equations the reconstruction is
usually done on finite element (FE) meshes, allowing for flexible geometries.
Thus, any processing of the obtained reconstructions should be ideally done on
the FE mesh as well. For this purpose, we extend the hugely successful U-Net
architecture that is limited to rectangular pixel or voxel domains to an
equivalent that works flexibly on FE meshes. To achieve this, the FE mesh is
converted into a graph and we formulate a graph U-Net with a new cluster
pooling and unpooling on the graph that mimics the classic neighborhood based
max-pooling. We demonstrate effectiveness and flexibility of the graph U-Net
for improving reconstructions from electrical impedance tomographic (EIT)
measurements, a nonlinear and highly ill-posed inverse problem. The performance
is evaluated for simulated data and from three measurement devices with
different measurement geometries and instrumentations. We successfully show
that such networks can be trained with a simple two-dimensional simulated
training set and generalize to very different domains, including measurements
from a three-dimensional device and subsequent 3D reconstructions.

本文提出一种基于图卷积神经网络的重建方法，通过对有限元网格进行图化处理构建图，使用新型的聚集池化和反池化方式设计了灵活的图 U-Net。我们将该方法应用于电阻抗层析成像重建，经过模拟数据和三种不同测量设备的实验验证，表明该模型能够取得较好的性能和灵活性，可实现从三维设备到不同领域的重建。