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

Abstract

Document Version Peer reviewed version Citation for published version (APA):Feragen, A., Owen, M., Petersen, J., Wille, M. M. W., Thomsen, L. H., Dirksen, A., & de Bruijne, M. (2013). Tree-space statistics and approximations for large-scale analysis of anatomical trees. In J. C. Gee, S. 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.

Principal Curves on Riemannian Manifolds

“…These non-linear models often only rely on local computations, which make them ideal for nonlinear representation spaces. We further expect the principal curves to be extendable to more general metric spaces [42], [43] as they do not depend on tangent space constructions.…”

Geometry and Statistics: Manifolds and Stratified Spaces

“…A stratified space is a union of smooth manifolds, meeting in a "controlled" way [16]. Stratified spaces lend themselves well to modeling data with variable topology, such as weighted trees [17][18][19][20][21][22][23] or graphs [24]. Approaches to stratified data spaces include the thesis of Bendich [25], who addressed the inverse problem of estimating stratified data spaces from data using persistent homology.…”

“…Some approaches model the first principal component as a geodesic optimizing a least squares cost function [19,22]. However, it is unclear both how well a geodesic can describe data in a stratified space, how such principal components might be computed, and how to pass to the second principal component.…”

Sample-Limited $$L_p$$ Barycentric Subspace Analysis on Constant Curvature Spaces

“…In neuroimaging studies, the individual image minimizing the sum of square deformation distance to other subject images is a good alternative to the mean template (a Fréchet mean in deformation and intensity space) because it conserves the original characteristics of a real subject image [7]. Beyond the Fréchet mean, [3] proposed to define the first principal component mode as the unexplained variance minimizing geodesic going through two of the data points. The method named set statistics was aiming to accelerate the computation of statistics on tree spaces.…”

“…The main drawback is the combinatorial explosion of the computational complexity with the dimension for the optimal order-k flag of affine spans, which is involving O(N k+1 ) operations, where N is the number of data points. In this paper we perform an exhaustive search, but approximate optima can be sought using a limited number of randomly sampled points [3].…”