Jesper Holst Pedersen

Raghavendra Selvan, Thomas Kipf, Max Welling et al.

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.

Raghavendra Selvan, Jens Petersen, Jesper H Pedersen et al.

In this work, we adapt a method based on multiple hypothesis tracking (MHT) that has been shown to give state-of-the-art vessel segmentation results in interactive settings, for the purpose of extracting trees. Regularly spaced tubular templates are fit to image data forming local hypotheses. These local hypotheses are used to construct the MHT tree, which is then traversed to make segmentation decisions. However, some critical parameters in this method are scale-dependent and have an adverse effect when tracking structures of varying dimensions. We propose to use statistical ranking of local hypotheses in constructing the MHT tree, which yields a probabilistic interpretation of scores across scales and helps alleviate the scale-dependence of MHT parameters. This enables our method to track trees starting from a single seed point. Our method is evaluated on chest CT data to extract airway trees and coronary arteries. In both cases, we show that our method performs significantly better than the original MHT method.

Raghavendra Selvan, Max Welling, Jesper H. Pedersen et al.

We present tree extraction in 3D images as a graph refinement task, of obtaining a subgraph from an over-complete input graph. To this end, we formulate an approximate Bayesian inference framework on undirected graphs using mean field approximation (MFA). Mean field networks 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 from training data using back-propagation algorithm. We demonstrate usefulness of the model to extract airway trees from 3D chest CT data. We first obtain probability images using a voxel classifier that distinguishes airways from background and use Bayesian smoothing to model individual airway branches. This yields us joint Gaussian density estimates of position, orientation and scale as node features of the input graph. Performance of the method is compared with two methods: the first uses probability images from a trained voxel classifier with region growing, which is similar to one of the best performing methods at EXACT'09 airway challenge, and the second method is based on Bayesian smoothing on these probability images. Using centerline distance as error measure the presented method shows significant improvement compared to these two methods.

Raghavendra Selvan, Jens Petersen, Jesper H. Pedersen et al.

Segmenting tree structures is common in several image processing applications. In medical image analysis, reliable segmentations of airways, vessels, neurons and other tree structures can enable important clinical applications. We present a framework for tracking tree structures comprising of elongated branches using probabilistic state-space models and Bayesian smoothing. Unlike most existing methods that proceed with sequential tracking of branches, we present an exploratory method, that is less sensitive to local anomalies in the data due to acquisition noise and/or interfering structures. The evolution of individual branches is modelled using a process model and the observed data is incorporated into the update step of the Bayesian smoother using a measurement model that is based on a multi-scale blob detector. Bayesian smoothing is performed using the RTS (Rauch-Tung-Striebel) smoother, which provides Gaussian density estimates of branch states at each tracking step. We select likely branch seed points automatically based on the response of the blob detection and track from all such seed points using the RTS smoother. We use covariance of the marginal posterior density estimated for each branch to discriminate false positive and true positive branches. The method is evaluated on 3D chest CT scans to track airways. We show that the presented method results in additional branches compared to a baseline method based on region growing on probability images.

Aasa Feragen, Jens Petersen, Dominik Grimm et al.

Methodological contributions: This paper introduces a family of kernels for analyzing (anatomical) trees endowed with vector valued measurements made along the tree. While state-of-the-art graph and tree kernels use combinatorial tree/graph structure with discrete node and edge labels, the kernels presented in this paper can include geometric information such as branch shape, branch radius or other vector valued properties. In addition to being flexible in their ability to model different types of attributes, the presented kernels are computationally efficient and some of them can easily be computed for large datasets (N of the order 10.000) of trees with 30-600 branches. Combining the kernels with standard machine learning tools enables us to analyze the relation between disease and anatomical tree structure and geometry. Experimental results: The kernels are used to compare airway trees segmented from low-dose CT, endowed with branch shape descriptors and airway wall area percentage measurements made along the tree. Using kernelized hypothesis testing we show that the geometric airway trees are significantly differently distributed in patients with Chronic Obstructive Pulmonary Disease (COPD) than in healthy individuals. The geometric tree kernels also give a significant increase in the classification accuracy of COPD from geometric tree structure endowed with airway wall thickness measurements in comparison with state-of-the-art methods, giving further insight into the relationship between airway wall thickness and COPD. Software: Software for computing kernels and statistical tests is available at http://image.diku.dk/aasa/software.php.