IPMI 2013 - LNCS 7917

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

Aasa Feragen^1,2, Megan Owen³, Jens Petersen¹, Mathilde M.W. Wille⁴, Laura H. Thomsen⁴, Asger Dirksen⁴, and Marleen de Bruijne^1,5

¹Department of Computer Science, University of Copenhagen, Denmark
aasa@diku.dk
phup@diku.dk
marleen@diku.dk
http://www.image.diku.dk/aasa

²Max Planck Institute for Intelligent Systems and Max Planck Institute for Developmental Biology, Tübingen, Germany

³David R. Cheriton School of Computer Science, University of Waterloo, Canada

⁴Lungemedicinsk Afdeling, Gentofte Hospital, Denmark

⁵Erasmus MC - University Medical Center Rotterdam, The Netherlands

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.

LNCS 7917, p. 74 ff.

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