Functional time warping

Functional time warping represents phase variation by composing a function with an increasing endpoint-preserving diffeomorphism of its domain. Registration seeks to separate this phase variation from amplitude variation in the observed functions.

For an absolutely continuous real function $f$, its square-root velocity function is $Q(f)=f^{\prime }/\sqrt{|f^{\prime }|}$ where $f^{\prime }\ne 0$, with value zero where $f^{\prime }=0$. Under an increasing reparameterization $\gamma$, it transforms as $Q(f\circ \gamma )=(Q(f)\circ \gamma )\sqrt{{\gamma }^{\prime }}$.