An activity can be seen as a resultant of coordinated movement of body joints and their respective interdependencies to achieve a goal-directed task. This idea is further supported by Johansson's demonstrations that visual perception of the entire human body motion can be represented by a few bright spots which holistically describe the motion of important joints. Traditional dynamical modeling approaches usually operate on the level of individual joints of the human body, lacking any information about the interdependencies between joints. We propose a novel approach for dynamical modeling by extending conventional ideas to quantify the interdependencies between body joints. Towards this end, we propose a new approach -approximate entropy-based feature representation to model the dynamics in human movement by quantifying dynamical regularity. In this paper, we utilize the algorithmic framework of [3] for estimating approximate entropy from time series data and extend it to model the dynamics in human activities for applications such as temporal segmentation and fine-grained quality assessment of actions.Approximate entropy is a statistical tool proposed by Pincus [3, 4] for quantification of regularity of time series data and system complexity, based on the log-likelihood of repetitions of patterns of length m being close within a defined tolerance window that will exhibit similar characteristics as patterns of length (m+1) [2,3]. It assigns a non-negative number to time series data, with lower values for predictable (ordered) signals and higher values for signals with increased irregularity (or randomness). It is defined using three parameters: embedding dimension (m), radius (r), and time delay (τ). Here, m represents the length of pattern (also called as embedding vector) in the data which is checked for repeatability, τ is selected so that the components of the embedding vector are sufficiently independent, and r is used for the estimation of local probabilities. Given N data samples {x 1 , x 2 , . . . , x N }, we can define embedding vector x(i) as,The frequency of repeatable patterns of the embedding vector within a tolerance r is given by C m i (r) asApproximate Entropy is given bywhere:C m i (r) represents the frequency of repeatable patterns in the embedding vector x(i), Θ(a) is the Heaviside step function, and Φ m (r) represents the conditional frequency estimates.Multivariate Cross Approximate Entropy (XAPEN): Recent theoretical and empirical findings have demonstrated that multivariate embedding of time series data by simple concatenation of individual univariate embedding vectors achieves good state space reconstruction as evaluated by the shape and dynamics distortion measures. In this work, we propose to use the multivariate embedding procedure as described by Cao et al.[1] per body joint and estimate the approximate entropy feature representation. In addition, natural human movement involves multiple body joints interacting with each other to together accomplish a particular action task. Hence, i...

Topological data analysis is becoming a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature, which is the number of d−dimensional holes in the dataset, also known as the Betti−d number. The persistence of the Betti numbers over various scales is encoded into a persistence diagram (PD), which indicates the birth and death times of these holes as scale varies. A common way to compare PDs is by a pointto-point matching, which is given by the n-Wasserstein metric. However, a big drawback of this approach is the need to solve correspondence between points before computing the distance; for n points, the complexity grows according to O(n 3 ). Instead, we propose to use an entirely new framework built on Riemannian geometry, that models PDs as 2D probability density functions that are represented in the square-root framework on a Hilbert Sphere. The resulting space is much more intuitive with closed form expressions for common operations. The distance metric is 1) correspondence-free and also 2) independent of the number of points in the dataset. The complexity of computing distance between PDs now grows according to O(K 2 ), for a K × K discretization of [0, 1] 2 . This also enables the use of existing machinery in differential geometry towards statistical analysis of PDs such as computing the mean, geodesics, classification etc. We report competitive results with the Wasserstein metric, at a much lower computational load, indicating the favorable properties of the proposed approach. arXiv:1605.08912v1 [math.AT]

In this paper, we propose a novel framework for dynamical analysis of human actions from 3D motion capture data using topological data analysis. We model human actions using the topological features of the attractor of the dynamical system. We reconstruct the phase-space of time series corresponding to actions using time-delay embedding, and compute the persistent homology of the phase-space reconstruction. In order to better represent the topological properties of the phase-space, we incorporate the temporal adjacency information when computing the homology groups. The persistence of these homology groups encoded using persistence diagrams are used as features for the actions. Our experiments with action recognition using these features demonstrate that the proposed approach outperforms other baseline methods.

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