Towards a theory of statistical tree-shape analysis
This work addresses the need for robust statistical analysis in domains like medical imaging, though it is incremental as it builds on existing tree shape and metric concepts.
The paper tackled the problem of developing statistical methods for tree-like shapes by constructing a shape space framework with metrics like QED and TED, showing that QED has better geometric properties for statistical analysis while TED has algorithmic advantages, with experimental validation on synthetic and airway tree data.
In order to develop statistical methods for shapes with a tree-structure, we construct a shape space framework for tree-like shapes and study metrics on the shape space. This shape space has singularities, corresponding to topological transitions in the represented trees. We study two closely related metrics on the shape space, TED and QED. QED is a quotient Euclidean distance arising naturally from the shape space formulation, while TED is the classical tree edit distance. Using Gromov's metric geometry we gain new insight into the geometries defined by TED and QED. We show that the new metric QED has nice geometric properties which facilitate statistical analysis, such as existence and local uniqueness of geodesics and averages. TED, on the other hand, does not share the geometric advantages of QED, but has nice algorithmic properties. We provide a theoretical framework and experimental results on synthetic data trees as well as airway trees from pulmonary CT scans. This way, we effectively illustrate that our framework has both the theoretical and qualitative properties necessary to build a theory of statistical tree-shape analysis.