  A ubiquitous problem in pattern recognition is that of matching an observed
time-evolving pattern (or signal) to a gold standard in order to recognize or
characterize the meaning of a dynamic phenomenon. Examples include matching
sequences of images in two videos, matching audio signals in speech
recognition, or matching framed trajectories in robot action recognition. This
paper shows that all of these problems can be aided by reparameterizing the
temporal dependence of each signal individually to a universal standard
timescale that allows pointwise comparison at each instance of time. Given two
sequences, each with $N$ timesteps, the complexity of the algorithm has a cost
of $O(N)$, which is an improvement on the most common method for matching two
signals, i.e., dynamic time warping. The core of the approach presented here is
that the universal standard timescale is found by solving a variational
calculus problem in which the cost functional reflects the amount of change
that takes place as measured in the original temporal variable, and then
produces a mapping to a new temporal variable in which the amount of change is
globally minimized. The result builds on known facts in differential geometry.
