  Graph refinement, or the task of obtaining subgraphs of interest from
over-complete graphs, can have many varied applications. In this work, we
extract trees or collection of sub-trees from image data by, first deriving a
graph-based representation of the volumetric data and then, posing the tree
extraction as a graph refinement task. We present two methods to perform graph
refinement. First, we use mean-field approximation (MFA) to approximate the
posterior density over the subgraphs from which the optimal subgraph of
interest can be estimated. Mean field networks (MFNs) are used for inference
based on the interpretation that iterations of MFA can be seen as feed-forward
operations in a neural network. This allows us to learn the model parameters
using gradient descent. Second, we present a supervised learning approach using
graph neural networks (GNNs) which can be seen as generalisations of MFNs.
Subgraphs are obtained by training a GNN-based graph refinement model to
directly predict edge probabilities. We discuss connections between the two
classes of methods and compare them for the task of extracting airways from 3D,
low-dose, chest CT data. We show that both the MFN and GNN models show
significant improvement when compared to one baseline method, that is similar
to a top performing method in the EXACT'09 Challenge, and a 3D U-Net based
airway segmentation model, in detecting more branches with fewer false
positives.
