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Bayesian filtering methods are widely used in many applications, one
such example being the problem of tracking moving shapes in a digital
video. These methods have been typically employed in finite
dimensional spaces, but it is not always desirable to work in finite
dimension. Recent attempts have been made to implement Bayesian
filtering techniques in infinite dimensional shape spaces as well.
However, to our knowledge, there is no rigorous formulation of
Bayesian filtering in such spaces and few examples of probability
measures on them are known. We are especially interested in a space of
curves which can be endowed with the differential structure of a
Stiefel manifold, and hence it can be embedded in a Hilbert space.

We study generalizations to the infinite dimensional case of methods
typically used to construct probability measures on finite dimensional
manifolds. We focus in particular on the Hausdorff measure, the
projection of measures defined on the ambient space and the push
forward of measures under the exponential map. Gaussian measures in
Hilbert spaces are an essential tool for these generalizations.

We provide an introduction to Gaussian measures in Hilbert spaces,
Bayesian filtering techniques and Stiefel manifolds.