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