Toward a theory of statistical tree-shape analysis

Aasa Feragen, Pechin Chien Pau Lo, Marleen de Bruijne, Mads Nielsen, Francois Bernard Lauze

Research output: Contribution to journalJournal articleResearchpeer-review

41 Citations (Scopus)

Abstract

In order to develop statistical methods for shapes with a tree-structure, we construct a shape space framework for tree-shapes and study metrics on the shape space. This shape space has singularities, which correspond 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 are needed for statistical analysis: geodesics always exist, and are generically locally unique. Following this we can also show existence and generic local uniqueness of average trees for QED. TED, while having some algorithmic advantages, does not share these advantages. Along with the theoretical framework we provide experimental proof-of-concept results on synthetic data trees as well as small airway trees from pulmonary CT scans. This way, we illustrate that our framework has promising theoretical and qualitative properties necessary to build a theory of statistical tree-shape analysis.
Original languageEnglish
JournalI E E E Transactions on Pattern Analysis and Machine Intelligence
Volume35
Issue number8
Pages (from-to)2008-2021
Number of pages14
ISSN0162-8828
DOIs
Publication statusPublished - 2013

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