  Surface parameterizations and registrations are important in computer
graphics and imaging, where 1-1 correspondences between meshes are computed. In
practice, surface maps are usually represented and stored as 3D coordinates
each vertex is mapped to, which often requires lots of storage memory. This
causes inconvenience in data transmission and data storage. To tackle this
problem, we propose an effective algorithm for compressing surface
homeomorphisms using Fourier approximation of the Beltrami representation. The
Beltrami representation is a complex-valued function defined on triangular
faces of the surface mesh with supreme norm strictly less than 1. Under
suitable normalization, there is a 1-1 correspondence between the set of
surface homeomorphisms and the set of Beltrami representations. Hence, every
bijective surface map is associated with a unique Beltrami representation.
Conversely, given a Beltrami representation, the corresponding bijective
surface map can be exactly reconstructed using the Linear Beltrami Solver
introduced in this paper. Using the Beltrami representation, the surface
homeomorphism can be easily compressed by Fourier approximation, without
distorting the bijectivity of the map. The storage memory can be effectively
reduced, which is useful for many practical problems in computer graphics and
imaging. In this paper, we proposed to apply the algorithm to texture map
compression and video compression. With our proposed algorithm, the storage
requirement for the texture properties of a textured surface can be
significantly reduced. Our algorithm can further be applied to compressing
motion vector fields for video compression, which effectively improve the
compression ratio.
