  This work introduces a novel framework for quantifying the presence and
strength of recurrent dynamics in video data. Specifically, we provide
continuous measures of periodicity (perfect repetition) and quasiperiodicity
(superposition of periodic modes with non-commensurate periods), in a way which
does not require segmentation, training, object tracking or 1-dimensional
surrogate signals. Our methodology operates directly on video data. The
approach combines ideas from nonlinear time series analysis (delay embeddings)
and computational topology (persistent homology), by translating the problem of
finding recurrent dynamics in video data, into the problem of determining the
circularity or toroidality of an associated geometric space. Through extensive
testing, we show the robustness of our scores with respect to several noise
models/levels, we show that our periodicity score is superior to other methods
when compared to human-generated periodicity rankings, and furthermore, we show
that our quasiperiodicity score clearly indicates the presence of biphonation
in videos of vibrating vocal folds, which has never before been accomplished
end to end quantitatively.
