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Visual observations of dynamic phenomena, such as human actions, are often represented as sequences of smoothly-varying features. In cases where the feature spaces can be structured as Riemannian manifolds, the corresponding representations become trajectories on manifolds. Analysis of these trajectories is challenging due to non-linearity of underlying spaces and high-dimensionality of trajectories. In vision problems, given the nature of physical systems involved, these phenomena are better characterized on a low-dimensional manifold compared to the space of Riemannian trajectories. For instance, if one does not impose physical constraints of the human body, in data involving human action analysis, the resulting representation space will have highly redundant features. Learning an effective, low-dimensional embedding for action representations will have a huge impact in the areas of search and retrieval, visualization, learning, and recognition. Traditional manifold learning addresses this problem for static points in the euclidean space, but its extension to Riemannian trajectories is non-trivial and remains unexplored. The difficulty lies in inherent non-linearity of the domain and temporal variability of actions that can distort any traditional metric between trajectories. To overcome these issues, we use the framework based on transported square-root velocity fields (TSRVF); this framework has several desirable properties, including a rate-invariant metric and vector space representations. We propose to learn an embedding such that each action trajectory is mapped to a single point in a low-dimensional euclidean space, and the trajectories that differ only in temporal rates map to the same point. We utilize the TSRVF representation, and accompanying statistical summaries of Riemannian trajectories, to extend existing coding methods such as PCA, KSVD and Label Consistent KSVD to Riemannian trajectories or more generally to Riemannian functions. We show that such coding efficiently captures trajectories in applications such as action recognition, stroke rehabilitation, visual speech recognition, clustering and diverse sequence sampling. Using this framework, we obtain state-of-the-art recognition results, while reducing the dimensionality/complexity by a factor of $100-250\times$ . Since these mappings and codes are invertible, they can also be used to interactively visualize Riemannian trajectories and synthesize actions. [ABSTRACT FROM AUTHOR]

We develop a combinatorial approach to represent and infer semantic interpretations of video contents using tools from Grenander’s pattern theory. Semantic structures for video interpretation are formed using generators and bonds, the fundamental units of representation in pattern theory. Generators represent features and ontological items, such as actions and objects, whereas bonds are threads used to connect generators while respecting appropriate constraints. The resulting configurations of partially-connected generators are termed scene interpretations. Our goal is to parse a given video data set into high-probability configurations. The probabilistic models are imposed using energies that have contributions from both data (classification scores) and prior information (ontological constraints, co-occurrence frequencies, etc). The search for optimal configurations is based on an MCMC, simulated-annealing algorithm that uses simple moves to propose configuration changes and to accept/reject them according to the posterior energy. In contrast to current graphical methods, this framework does not preselect a neighborhood structure but tries to infer it from the data. The proposed framework is able to obtain 20% higher classification rates, compared to a purely machine learning-based baseline, despite artificial insertion of low-level processing errors. In an uncontrolled scenario, video interpretation performance rates are found to be double that of the baseline. [ABSTRACT FROM AUTHOR]