Mixtures of Gaussians, factor analyzers (probabilistic PCA) and hidden Markov models are staples of static and dynamic data modeling and image and video modeling in particular. We show how topographic trans-formations in the input, such as translation and shearing in images, can be accounted for in these models by including a discrete transformation variable. The resulting mod-els perform clustering, dimensionality reduc-tion and time-series analysis in a way that is invariant to transformations in the input. Us-ing the EM algorithm, these transformation-invariant models can be fit to static data and time series. We give results on filtering mi-croscopy images, face and facial pose cluster-ing, handwritten digit modeling and recog-nition, video clustering, object tracking, and removal of distractions from video sequences.
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