Pattern recognition in video is a challenging task because of the multitude of spatio-temporal variations that occur in different videos capturing the exact same event. While traditional pattern-theoretic approaches account for the spatial changes that occur due to lighting and pose, very little has been done to address the effect of temporal rate changes in the executions of an event. In this paper, we provide a systematic model-based approach to learn the nature of such temporal variations (time warps) while simultaneously allowing for the spatial variations in the descriptors. We illustrate our approach for the problem of action recognition and provide experimental justification for the importance of accounting for rate variations in action recognition. The model is composed of a nominal activity trajectory and a function space capturing the probability distribution of activity-specific time warping transformations. We use the square-root parameterization of time warps to derive geodesics, distance measures, and probability distributions on the space of time warping functions. We then design a Bayesian algorithm which treats the execution rate function as a nuisance variable and integrates it out using Monte Carlo sampling, to generate estimates of class posteriors. This approach allows us to learn the space of time warps for each activity while simultaneously capturing other intra- and interclass variations. Next, we discuss a special case of this approach which assumes a uniform distribution on the space of time warping functions and show how computationally efficient inference algorithms may be derived for this special case. We discuss the relative advantages and disadvantages of both approaches and show their efficacy using experiments on gait-based person identification and activity recognition.
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