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We present an algorithm for computing joint state, smoothed, density estimates for non-linear dynamical systems in a Bayesian setting. Many visual tracking problems can be formulated as probabilistic inference over time series, but we are not aware of mixture smoothers that would apply to weakly identifiable models, where multimodality is persistent rather than transient (e.g. monocular 3D human tracking). Such processes, in principle, exclude iterated Kalman smoothers, whereas flexible MCMC methods or sample based particle smoothers encounter computational difficulties: accurately locating an exponential number of probable joint state modes representing high-dimensional trajectories, rapidly mixing between those or resampling probable configurations missed during filtering. In this paper we present an alternative, layered, mixture density smoothing algorithm that exploits the accuracy of efficient optimization within a Bayesian approximation framework. The distribution is progressively refined by combining polynomial time search over the embedded network of temporal observation likelihood peaks, MAP continuous trajectory estimates, and Bayesian variational adjustment of the resulting joint mixture approximation. Our results demonstrate the effectiveness of the method on the problem of inferring multiple plausible 3D human motion trajectories from monocular video.

Variational mixture smoothing for non-linear dynamical systems