Estimating the mean manifold of a deformable object from noisy observations

Assume we have a set of noisy observations (for example, images) of different objects, each undergoing a different geometric deformation, yet all the deformations belong to the same family. As a result of the action of these deformations, the set of different observations on each object is generally a manifold in the ambient space of observations. It has been shown, [1], that in the absence of noise, in those cases where the set of deformations admits a finite-dimensional representation, the universal manifold embedding (UME) provides a mapping from the space of observations to a low dimensional linear space. The manifold corresponding to each object is mapped to a distinct linear subspace of Euclidean space, and the dimension of the subspace is the same as that of the manifold. In the presence of noise, different observations are mapped to different subspaces. In this paper we derive a method for "averaging" the different subspaces, obtained from different observations made on the same object, in order to estimate the mean representation of the object manifold. The mean manifold representation is then employed to minimize the effects of noise in matched manifold detectors and to improve the separability of data sets in the context of object detection and classification.