  We introduce a novel geometric framework for separating the phase and the
amplitude variability in functional data of the type frequently studied in
growth curve analysis. This framework uses the Fisher-Rao Riemannian metric to
derive a proper distance on the quotient space of functions modulo the
time-warping group. A convenient square-root velocity function (SRVF)
representation transforms the Fisher-Rao metric into the standard $\ltwo$
metric, simplifying the computations. This distance is then used to define a
Karcher mean template and warp the individual functions to align them with the
Karcher mean template. The strength of this framework is demonstrated by
deriving a consistent estimator of a signal observed under random warping,
scaling, and vertical translation. These ideas are demonstrated using both
simulated and real data from different application domains: the Berkeley growth
study, handwritten signature curves, neuroscience spike trains, and gene
expression signals. The proposed method is empirically shown to be be superior
in performance to several recently published methods for functional alignment.
