Means in spaces of tree-like shapes

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Abstract
The mean is often the most important statistic of a dataset as it provides a single point that summarizes the entire set. While the mean is readily defined and computed in Euclidean spaces, no commonly accepted solutions are currently available in more complicated spaces, such as spaces of tree-structured data. In this paper we study the notion of means, both generally in Gromov's CAT(0)-spaces (metric spaces of non-positive curvature), but also specifically in the space of tree-like shapes. We prove local existence and uniqueness of means in such spaces and discuss three different algorithms for computing means. We make an experimental evaluation of the three algorithms through experiments on three different sets of data with tree-like structure: a synthetic dataset, a leaf morphology dataset from images, and a set of human airway subtrees from medical CT scans. This experimental study provides great insight into the behavior of the different methods and how they relate to each other. More importantly, it also provides mathematically well-founded, tractable and robust “average trees”. This statistic is of utmost importance due to the ever-presence of tree-like structures in human anatomy, e.g., airways and vascularization systems.

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