  We propose an approach for capturing the signal variability in hyperspectral
imagery using the framework of the Grassmann manifold. Labeled points from each
class are sampled and used to form abstract points on the Grassmannian. The
resulting points on the Grassmannian have representations as orthonormal
matrices and as such do not reside in Euclidean space in the usual sense. There
are a variety of metrics which allow us to determine a distance matrices that
can be used to realize the Grassmannian as an embedding in Euclidean space. We
illustrate that we can achieve an approximately isometric embedding of the
Grassmann manifold using the chordal metric while this is not the case with
geodesic distances. However, non-isometric embeddings generated by using a
pseudometric on the Grassmannian lead to the best classification results. We
observe that as the dimension of the Grassmannian grows, the accuracy of the
classification grows to 100% on two illustrative examples. We also observe a
decrease in classification rates if the dimension of the points on the
Grassmannian is too large for the dimension of the Euclidean space. We use
sparse support vector machines to perform additional model reduction. The
resulting classifier selects a subset of dimensions of the embedding without
loss in classification performance.
