A Categorical Approach to Contour, Split and Join Trees with Application to Airway Segmentation

Contour, split and join trees can be defined as functors acting on the category of scalar fields, whose morphisms are value-preserving functions. The categorical definition provides a natural way to efficiently compute a variety of topological properties of all contours, sublevel or superlevel components in a scalar field. The result is a labeling of the contour, split or join tree and can be used to find all contours, sublevel or superlevel sets with desired properties. We describe an algorithm for airway segmentation from Computed Tomography (CT) scans based on this paradigm. It computes all sublevel components in thick slices of the input image that have simple topology and branching structure. The output is a connected component of the union of all such sublevel components. This procedure can be viewed as a local thresholding approach, where the local thresholds are determined based on topological analysis of sublevel sets.