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We introduce a novel geometric framework for separating the phase and theamplitude variability in functional data of the type frequently studied ingrowth curve analysis. This framework uses the Fisher-Rao Riemannian metric toderive a proper distance on the quotient space of functions modulo thetime-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 aKarcher mean template and warp the individual functions to align them with theKarcher mean template. The strength of this framework is demonstrated byderiving a consistent estimator of a signal observed under random warping,scaling, and vertical translation. These ideas are demonstrated using bothsimulated and real data from different application domains: the Berkeley growthstudy, handwritten signature curves, neuroscience spike trains, and geneexpression signals. The proposed method is empirically shown to be be superiorin performance to several recently published methods for functional alignment.

Registration of Functional Data Using Fisher-Rao Metric