  Past approaches for statistical shape analysis of objects have focused mainly
on objects within the same topological classes, e.g., scalar functions,
Euclidean curves, or surfaces, etc. For objects that differ in more complex
ways, the current literature offers only topological methods. This paper
introduces a far-reaching geometric approach for analyzing shapes of graphical
objects, such as road networks, blood vessels, brain fiber tracts, etc. It
represents such objects, exhibiting differences in both geometries and
topologies, as graphs made of curves with arbitrary shapes (edges) and
connected at arbitrary junctions (nodes). To perform statistical analyses, one
needs mathematical representations, metrics and other geometrical tools, such
as geodesics, means, and covariances. This paper utilizes a quotient structure
to develop efficient algorithms for computing these quantities, leading to
useful statistical tools, including principal component analysis and analytical
statistical testing and modeling of graphical shapes. The efficacy of this
framework is demonstrated using various simulated as well as the real data from
neurons and brain arterial networks.
