We present tree extraction in 3D images as a graph refinement task, of\nobtaining a subgraph from an over-complete input graph. To this end, we\nformulate an approximate Bayesian inference framework on undirected graphs\nusing mean field approximation (MFA). Mean field networks are used for\ninference based on the interpretation that iterations of MFA can be seen as\nfeed-forward operations in a neural network. This allows us to learn the model\nparameters from training data using back-propagation algorithm. We demonstrate\nusefulness of the model to extract airway trees from 3D chest CT data. We first\nobtain probability images using a voxel classifier that distinguishes airways\nfrom background and use Bayesian smoothing to model individual airway branches.\nThis yields us joint Gaussian density estimates of position, orientation and\nscale as node features of the input graph. Performance of the method is\ncompared with two methods: the first uses probability images from a trained\nvoxel classifier with region growing, which is similar to one of the best\nperforming methods at EXACT'09 airway challenge, and the second method is based\non Bayesian smoothing on these probability images. Using centerline distance as\nerror measure the presented method shows significant improvement compared to\nthese two methods.\n