Shape Analysis of Open Curves in R3 with Applications to Study of Fiber Tracts in DT-MRI Data

Motivated by the problem of analyzing shapes of fiber tracts in DT-MRI data, we present a geometric framework for studying shapes of open curves in R. We start with a space of unit-length curves and define the shape space to be its quotient space modulo rotation and reparametrization groups. Thus, the resulting shape analysis is invariant to parameterizations of curves. Furthermore, a Riemannian structure on this quotient shape space allows us to compute geodesic paths between given curves and helps develop algorithms for: (i) computing statistical summaries of a collection of curves using means and covariances, and (ii) clustering a given set of curves into clusters of similar shapes. Examples using fiber tracts, extracted as parameterized curves from DT-MRI images, are presented to demonstrate this framework.