  Images encode both the state of the world and its content. The former is
useful for tasks such as planning and control, and the latter for
classification. The automatic extraction of this information is challenging
because of the high-dimensionality and entangled encoding inherent to the image
representation. This article introduces two theoretical approaches aimed at the
resolution of these challenges. The approaches allow for the interpolation and
extrapolation of images from an image sequence by joint estimation of the image
representation and the generators of the sequence dynamics. In the first
approach, the image representations are learned using probabilistic PCA
\cite{tipping1999probabilistic}. The linear-Gaussian conditional distributions
allow for a closed form analytical description of the latent distributions but
assumes the underlying image manifold is a linear subspace. In the second
approach, the image representations are learned using probabilistic nonlinear
PCA which relieves the linear manifold assumption at the cost of requiring a
variational approximation of the latent distributions. In both approaches, the
underlying dynamics of the image sequence are modelled explicitly to
disentangle them from the image representations. The dynamics themselves are
modelled with Lie group structure which enforces the desirable properties of
smoothness and composability of inter-image transformations.
