(Quasi)Periodicity Quantification in Video Data, Using Topology (1704.08382v2)

Abstract: 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.

• The paper introduces a framework quantifying video periodicity and quasiperiodicity using delay embeddings and persistent homology, avoiding conventional preprocessing.
• Experimental validation demonstrates the framework's robustness against various noise types and its alignment with human perception of video periodicity.
• Applying the method to vocal fold videos shows its potential for detecting speech pathologies and highlights practical applications in biomechanics and surveillance.

Topological Quantification of (Quasi)Periodicity in Video Data

The paper by Christopher J. Tralie and Jose A. Perea presents a novel framework for quantifying periodic and quasiperiodic dynamics in video data, employing methodologies grounded in nonlinear time series analysis and computational topology. By leveraging delay embeddings combined with persistent homology, the authors introduce a robust approach to analyze video sequences without conventional preprocessing steps like segmentation, training, or object tracking.

Core Methodological Contributions

The primary methodological contribution lies in the use of delay embeddings to reconstruct the dynamics present within video data directly. This technique, commonly used in one-dimensional time series, is generalized to video by incorporating the spatial dimension into the embedding process. By mapping each video frame to a higher-dimensional space through these embeddings, the framework captures the intrinsic recurrent dynamics.

Persistent homology, an advanced tool from computational topology, is utilized to assess the geometric structure of these embeddings. The birth and death of various topological features across different scales enable a quantifiable measure of periodicity and quasiperiodicity. The periodicity score is derived by calculating the largest persistence of one-dimensional loops, while the quasiperiodicity score considers the combination of one- and two-dimensional topological features, hinting at the data's toroidal nature.

Experimental Validation

Extensive validation is conducted to demonstrate the robustness of the proposed measures against various noise models, including camera blur, Gaussian noise, and MPEG bit corruption. The results indicate high resilience, showcasing the reliability of persistent homology in preserving relevant topological features despite significant noise interference.

Furthermore, the framework's ability to align with human rankings of periodicity in videos is affirmed through a systematic comparison with existing methodologies, involving human judgment in assessing video periodicity. This alignment underscores the effectiveness of the tool in capturing perceptually important temporal dynamics.

Application to Vocal Fold Videos

An illuminating application of this approach is its examination of high-speed vocal fold videos, aimed at detecting speech pathologies characterized by biphonation. By segmenting vocal fold vibrations into distinct periodic and quasiperiodic classes and validating these classifications quantitatively, the framework offers promising capabilities for automatic diagnosis and monitoring in clinical settings.

Implications and Future Directions

The implications of this work are multifaceted. Practically, the proposed framework creates avenues for enhanced video analysis in numerous fields, such as biomechanics, surveillance, and animal behavior studies, where identifying recurrent patterns is essential. Theoretically, it paves the way for further exploration of topological techniques in understanding complex dynamical systems.

Looking ahead, the research invites extensions in various directions. Expanding the computational efficiency of persistent homology algorithms and adapting the approach to utilize color channels and high-definition video could enhance its applicability. Furthermore, discerning quasiperiodic phenomena from harmonic modes solely through geometric analysis promises breakthrough advances in acoustic and biomechanical diagnostics.

This paper significantly contributes to the intersection of topology and video analysis, offering a sophisticated framework that combines computational efficiency with analytical depth.