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Temporal alignment is an inherent task in most applications dealing withvideos: action recognition, motion transfer, virtual trainers, rehabilitation,etc. In this paper we dive into the understanding of this task from a geometricpoint of view: in particular, we show that the basic properties that areexpected from a temporal alignment procedure imply that the set of alignedmotions to a template form a slice to a principal fiber bundle for the group oftemporal reparameterizations. A temporal alignment procedure provides areparameterization invariant projection onto this particular slice. Thisgeometric presentation allows to elaborate a consistency check for testing theaccuracy of any temporal alignment procedure. We give examples of alignmentprocedures from the literature applied to motions of tennis players. Most ofthem use dynamic programming to compute the best correspondence between twomotions relative to a given cost function. This step is computationallyexpensive (of complexity $O(NM)$ where $N$ and $M$ are the numbers of frames).Moreover most methods use features that are invariant by translations androtations in $\mathbb{R}^3$, whereas most actions are only invariant bytranslation along and rotation around the vertical axis, where the verticalaxis is aligned with the gravitational field. The discarded informationcontained in the vertical direction is crucial for accurate synchronization ofmotions. We propose to incorporate keyframe correspondences into the dynamicprogramming algorithm based on coarse information extracted from the verticalvariations, in our case from the elevation of the arm holding the racket. Thetemporal alignment procedures produced are not only more accurate, but alsocomputationally more efficient.

Temporal Alignment of Human Motion Data: A Geometric Point of View