Geometries on spaces of treelike shapes

TY - GEN

T1 - Geometries on spaces of treelike shapes

AU - Feragen, Aasa

AU - Lauze, Francois Bernard

AU - Lo, Pechin Chien Pau

AU - de Bruijne, Marleen

AU - Nielsen, Mads

N1 - Conference code: 10

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

U2 - 10.1007/978-3-642-19309-5_13

DO - 10.1007/978-3-642-19309-5_13

M3 - Article in proceedings

SN - 978-3-642-19308-8

T3 - Lecture notes in computer science

SP - 160

EP - 173

BT - Computer Vision – ACCV 2010

A2 - Kimmel, Ron

A2 - Klette, Reinhard

A2 - Sugimoto, Akihiro

PB - Springer

T2 - 10th Asian Conference on Computer Vision

Y2 - 8 November 2010 through 12 November 2010

ER -