Department of Mathematics Calendar
Chris Tralie, Duke University, From Musical Rhythms To Vibrating Vocal Folds: Geometric (Quasi)Periodicity Quantification in Multimedia Time Series
A large variety of multimedia data inference problems require analysis of repeated structures. In audio, for instance, the rhythm, or “pulse” of the music, occurs in a periodic pattern, and understanding this pattern is an important preprocessing step in music information retrieval. In medical video analysis, there is interest in determining heart pulse rate in ordinary webcam videos, analyzing stereotypical repetitive motor motions in videos of autistic children, and analyzing voice pathologist from high speed videos of vocal folds. In this work, we provide a unified sliding window embedding framework for quantifying repetitive patterns geometrically in the aforementioned applications using tools from topological data analysis. Periodic patterns show up as persistent loops, whose prominence can be measured with persistent H1, and quasi-periodic patterns (non-commensurate periodic activity) show up as flat tori, which can be measured with both persistent H1 and H2. Somewhat surprisingly, we also show that some periodic processes with harmonic structures lie on loops which bound twisted spaces such as the Mobius strip, and quantifying these structures is applicable both to rhythm hierarchy analysis and detection of “biphonation” due to mucous in vibrating vocal folds. Finally, in addition to detecting these classes of periodicity, we can parametrize data with maps to the circle using spectral graph theory, which is useful for figuring out the “tempo” of the periodic process in each application.