Texture Map and Video Compression Using Beltrami Representation

Abstract

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.

Landmark-Matching Transformation with Large Deformation Via n-dimensional Quasi-conformal Maps

“…In fact, using the same argument, we can show that the optimization problem (38) is convex in any one of the octant regions. Hence, the optimization is a global minimizer in each octant region.…”

“…Hence, the optimization is a global minimizer in each octant region. Now, we will explain the convergence of Algorithm 4 to the minimizer of the optimization problem (38). 1 2 , which is a minimizer of (38).…”

“…The sign of y n i is chosen as to minimize the energy functional (38), which is given by the following:…”

“…(44) with a rate 1 2 . In other words, algorithm 4 converges to a solution of the Euler-Lagrange equation of (38). Depending on the sign of det(U V * ), our algorithm search for a critical point at the first octant region if det(U V * ) > 0 and at the other octant (with x i < 0) if det(U V * ) < 0.…”

“…Finding an optimal transformation between corresponding data, such as images or geometric shapes, is an important task in various fields, such as computer vision [61], computer graphics [6,22,24], video processing [38,45,59,60] and medical imaging [1,25,28,29,32,49,55]. Such a process is called registration.…”