  We consider the problem of taking a video that is comprised of multiple
periods of repetitive motion, and reordering the frames of the video into a
single period, producing a detailed, single cycle video of motion. This problem
is challenging, as such videos often contain noise, drift due to camera motion
and from cycle to cycle, and irrelevant background motion/occlusions, and these
factors can confound the relevant periodic motion we seek in the video. To
address these issues in a simple and efficient manner, we introduce a tracking
free Eulerian approach for synthesizing a single cycle of motion. Our approach
is geometric: we treat each frame as a point in high-dimensional Euclidean
space, and analyze the sliding window embedding formed by this sequence of
points, which yields samples along a topological loop regardless of the type of
periodic motion. We combine tools from topological data analysis and spectral
geometric analysis to estimate the phase of each window, and we exploit the
sliding window structure to robustly reorder frames. We show quantitative
results that highlight the robustness of our technique to camera shake, noise,
and occlusions, and qualitative results of single-cycle motion synthesis across
a variety of scenarios.
