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Tree-Space Statistics and Approximations for Large-Scale Analysis of Anatomical Trees

  • Aasa Feragen
  • Megan Owen
  • Jens Petersen
  • Mathilde M. W. Wille
  • Laura H. Thomsen
  • Asger Dirksen
  • Marleen de Bruijne
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7917)

Abstract

Statistical analysis of anatomical trees is hard to perform due to differences in the topological structure of the trees. In this paper we define statistical properties of leaf-labeled anatomical trees with geometric edge attributes by considering the anatomical trees as points in the geometric space of leaf-labeled trees. This tree-space is a geodesic metric space where any two trees are connected by a unique shortest path, which corresponds to a tree deformation. However, tree-space is not a manifold, and the usual strategy of performing statistical analysis in a tangent space and projecting onto tree-space is not available. Using tree-space and its shortest paths, a variety of statistical properties, such as mean, principal component, hypothesis testing and linear discriminant analysis can be defined. For some of these properties it is still an open problem how to compute them; others (like the mean) can be computed, but efficient alternatives are helpful in speeding up algorithms that use means iteratively, like hypothesis testing. In this paper, we take advantage of a very large dataset (N = 8016) to obtain computable approximations, under the assumption that the data trees parametrize the relevant parts of tree-space well. Using the developed approximate statistics, we illustrate how the structure and geometry of airway trees vary across a population and show that airway trees with Chronic Obstructive Pulmonary Disease come from a different distribution in tree-space than healthy ones. Software is available from http://image.diku.dk/aasa/software.php .

Keywords

Chronic Obstructive Pulmonary Disease Chronic Obstructive Pulmonary Disease Patient Linear Discriminant Analysis Geodesic Segment Computable Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Aasa Feragen
    • 1
    • 2
  • Megan Owen
    • 3
  • Jens Petersen
    • 1
  • Mathilde M. W. Wille
    • 4
  • Laura H. Thomsen
    • 4
  • Asger Dirksen
    • 4
  • Marleen de Bruijne
    • 1
    • 5
  1. 1.Department of Computer ScienceUniversity of CopenhagenDenmark
  2. 2.Max Planck Institute for Intelligent Systems and Max Planck Institute for Developmental BiologyTübingenGermany
  3. 3.David R. Cheriton School of Computer ScienceUniversity of WaterlooCanada
  4. 4.Lungemedicinsk Afdeling, Gentofte HospitalDenmark
  5. 5.Erasmus MC - University Medical Center RotterdamThe Netherlands

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