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.
Statistics on tree-shaped objects receive growing interest in the statistical community. Wang and Marron @cite_18 study metric spaces of trees and define a notion of average tree called the median-mean as well as a version of PCA, which finds modes of variation in terms of , encoding the maximum amount of structural and attributal variation. @cite_6 extend this work by finding efficient algorithms for PCA. This is applied to analysis of brain blood vessels. The metric defined by Wang and Marron does not give a natural geodesic structure on the space of trees, as it places a large emphasis on the tree-topological structure of the trees. The metric has discontinuities in the sense that a sequence of trees with a shrinking branch will not converge to a tree that does not have that branch. Such a metric is not suitable for studying trees with continuous topological variations and noise, such as anatomical tree-structures extracted from medical images, since the emphasis on topology makes the metric sensitive to structural noise.
@cite_18 @cite_6
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In order to develop statistical methods for shapes with a tree-structure, we construct a shape space framework for treelike shapes and study metrics on the shape space. The shape space has singularities, which correspond to topological transitions in the represented trees. We study two closely related metrics, TED and QED. The QED is a quotient euclidean distance arising from the new shape space formulation, while TED is essentially the classical tree edit distance. Using Gromov's metric geometry we gain new insight into the geometries defined by TED and QED. In particular, 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, has algorithmic advantages, while it does not share the geometric strongpoints of QED. We provide a theoretical framework as well as computational results such as matching of airway trees from pulmonary CT scans and geodesics between synthetic data trees illustrating the dynamic and geometric properties of the QED metric. The mean is often the most important statistic of a dataset as it provides a single point that summarizes the entire set. While the mean is readily defined and computed in Euclidean spaces, no commonly accepted solutions are currently available in more complicated spaces, such as spaces of tree-structured data. In this paper we study the notion of means, both generally in Gromov's CAT(0)-spaces (metric spaces of non-positive curvature), but also specifically in the space of tree-like shapes. We prove local existence and uniqueness of means in such spaces and discuss three different algorithms for computing means. We make an experimental evaluation of the three algorithms through experiments on three different sets of data with tree-like structure: a synthetic dataset, a leaf morphology dataset from images, and a set of human airway subtrees from medical CT scans. This experimental study provides great insight into the behavior of the different methods and how they relate to each other. More importantly, it also provides mathematically well-founded, tractable and robust average trees. This statistic is of utmost importance due to the ever-presence of tree-like structures in human anatomy, e.g., airways and vascularization systems.