Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems
This work addresses a bottleneck in medical imaging for nonlinear inverse problems like Electrical Impedance Tomography, enabling more flexible and generalizable reconstruction methods on nonuniform meshes, though it is incremental as it builds on existing graph convolutional and model-based learning approaches.
The authors tackled the challenge of applying model-based learned image reconstruction to nonuniform meshes in nonlinear inverse problems by developing a framework using graph convolutional neural networks, resulting in the Graph Convolutional Newton-type Method (GCNM) that demonstrated strong generalizability to different domain shapes and meshes, including out-of-distribution and experimental data, without requiring transfer training.
The majority of model-based learned image reconstruction methods in medical imaging have been limited to uniform domains, such as pixelated images. If the underlying model is solved on nonuniform meshes, arising from a finite element method typical for nonlinear inverse problems, interpolation and embeddings are needed. To overcome this, we present a flexible framework to extend model-based learning directly to nonuniform meshes, by interpreting the mesh as a graph and formulating our network architectures using graph convolutional neural networks. This gives rise to the proposed iterative Graph Convolutional Newton-type Method (GCNM), which includes the forward model in the solution of the inverse problem, while all updates are directly computed by the network on the problem specific mesh. We present results for Electrical Impedance Tomography, a severely ill-posed nonlinear inverse problem that is frequently solved via optimization-based methods, where the forward problem is solved by finite element methods. Results for absolute EIT imaging are compared to standard iterative methods as well as a graph residual network. We show that the GCNM has strong generalizability to different domain shapes and meshes, out of distribution data as well as experimental data, from purely simulated training data and without transfer training.