|Abstract:|| We introduce a mathematical framework in which a functional data registration problem can be soundly and coherently set. In detail, we show
that the introduction of a metric/semi-metric and of a group of warping function respect to which the metric/semi-metric is invariant is the key to
a clear and not ambiguous denition of phase and amplitude variability.
Moreover, an amplitude-to-total variability index is proposed. This index turns to be useful in practical situations to measure to what extent amplitude variability aects the data and to compare the eectiveness of different