Differential Structure in non-Linear Image Embedding Functions

Many natural image sets are samples of a low dimensional manifold in the space of all possible images. When the image data set is not a linear combination of a small number of basis images, then linear dimensionality reduction techniques such as PCA and ICA fail, and non-linear dimensionality reduction techniques are required to automatically determine the intrinsic structure of the image set. Recent techniques such as ISOMAP and LLE provide a mapping between the images and a low dimensional parameterization of the images. In this paper we consider how choosing different image distance metrics affects the low-dimensional parameterization. For image sets that arise from non-rigid and human motion analysis, and MRI applications, differential motions in some directions of the low-dimensional space correspond to common transformations in the image domain. Defining distance measures that are invariant to these transformations makes Isomap a powerful tool for automatic registration of large image or video data sets.