We present the application of a low dimensional manifold model (LDMM) on hyperspectral
image (HSI) reconstruction. An important property of hyperspectral images is that the
patch manifold, which is sampled by the three-dimensional blocks in the data cube, is generally of
a low dimensional nature. This is a generalization of low-rank models in that hyperspectral images
with nonlinear mixing terms can also fit in this framework. The point integral method (PIM) is used
to solve a Laplace-Beltrami equation over a point cloud sampling the patch manifold in LDMM.
Both numerical simulations and theoretical analysis show that the sample points constraint is correctly
enforced by PIM. The framework is demonstrated by experiments on the reconstruction of
both linear and nonlinear mixed hyperspectral images with a significant number of missing voxels
and several entirely missing spectral bands.
We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, aspherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple identity component, then the local isometry orbits in M are roughly ¯bers of a¯ber bundle. A corollary is that if M has an open, dense, locally homogeneous subset, then M is locally homogeneous.