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 three-dimensional coordinates each vertex is mapped to, which often requires lots of 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 requirement can be effectively reduced, which is useful for many practical problems in computer graphics and imaging. In this paper, we propose applying 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 improves the compression ratio.

In shape analysis, finding an optimal 1-1 correspondence between surfaces within a large class of admissible bijective mappings is of great importance. Such process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making exhaustive search for the best mapping challenging. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs), which are complex-valued functions defined on surfaces with supreme norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of BCs. Hence, every bijective surface map can be represented by a unique BC. Conversely, given a BC, we can reconstruct the unique surface map associated to it using the Beltrami Holomorphic flow (BHF) method. Using BCs to represent surface maps is advantageous because it is a much simpler functional space, which captures many essential features of a surface map. By adjusting BCs, we equivalently adjust surface diffeomorphisms to obtain the optimal map with desired properties. More specifically, BHF gives us the variation of the associated map under the variation of BC. Using this, a variational problem over the space of surface diffeomorphisms can be easily reformulated into a variational problem over the space of BCs. This makes the minimization procedure much easier. More importantly, the diffeomorphic property is always preserved. We test our method on synthetic examples and real medical applications. Experimental results demonstrate the effectiveness of our proposed algorithm for surface registration.

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The manipulation of surface homeomorphisms is an important aspect in three-dimensional modeling and surface processing. Every homeomorphic surface map can be considered as a quasi-conformal map, with its local nonconformal distortion given by its Beltrami differential. As a generalization of conformal maps, quasi-conformal maps are of great interest in mathematical study and real applications. Efficient and accurate computational construction of desirable quasi-conformal maps between general surfaces is crucial. However, in the literature we have reviewed, all existing computational works on construction of quasi-conformal maps to or from a compact domain require global parametrization onto the plane and are difficult to directly apply to maps between arbitrary surfaces. This work fills the gap by proposing to compute quasi-conformal homeomorphisms between arbitrary Riemann surfaces using discrete Beltrami flow, which is a vector field corresponding to the adjustment to the intrinsic Beltrami differential of the map. The vector field is defined by a partial differential equation in a local conformal coordinate. Based on this formulation and a composition formula, we can compute the Beltrami flow of any homeomorphism adjustment as a vector field on the target domain defined from the source domain, with appropriate boundary conditions and correspondences. Numerical tests show that our method provides a robust and efficient way of adjusting surface homeomorphisms. It is also insensitive to surface representation and has no limitation to the classes of surfaces that can be processed. Extensive numerical examples will be shown.

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