Since these parameterizations minimize the distortion of different intrinsic measures of the original mesh, we call them Intrinsic Parameterizations. Given a few desirable properties such as rotation and translation invariance, we show that the only admissible parameterizations form a two-dimensional set and each parameterization in this set can be computed using a sim- ple, sparse, linear system. In this paper, we present new theoretical and practical results on the parameterization of triangulated surface patches. As 3D data becomes more and more detailed, there is an increased need for fast and robust techniques to automatically compute least-distorted parameterizations of large meshes. Parameterization of discrete surfaces is a fundamental and widely-used operation in graphics, required, for in- stance, for texture mapping or remeshing.
![compute method map compute method map](https://venturebeat.com/wp-content/uploads/2019/04/sol_still_one1.png)
Furthermore, the algorithms are robust for surfaces scanned from real life, and general for surfaces with different topologies. The efficiency and efficacy of the algorithms are demonstrated by experimental results. By using holomorphic differential forms, conformal maps under the new conformal structure are calculated, which are the desired quasi-conformal maps.
![compute method map compute method map](https://i.stack.imgur.com/zu0gO.png)
The major strategy is to deform the conformal structure of the original surface to a new conformal structure by the Beltrami coefficient, such that the quasi-conformal map becomes a conformal map. We propose an effective algorithm to solve the quasi-conformal map from the Beltrami coefficient. The local deformation is characterized by a complex-value function, Beltrami coefficient, which describes the deviation from conformality of the deformation at each point. In the physical world, most surface deformations can be rigorously modeled as quasi-conformal maps. This is of great importance for shape registration. We consider the problem of constructing quasi-conformal mappings between surfaces by solving Beltrami equations. Integral method and its implementation, and present the numerical experiments Paper, we explain the derivation of the integral equations, describe the finite Makes the integral equation easy to be approximated from point cloud. In FIM, the key idea is toĭerive the integral equations which approximates the Poisson-type equations andĬontains no derivatives but only the values of the unknown function. Method called finite integral method (FIM) to solve the Poisson-type equationsįrom point clouds with convergence guarantees. The complicate geometrical structure of the manifold, it is difficult to getĮfficient numerical method to solve PDE on manifold. Poisson-type equations including the standard Poisson equation and the relatedĮigenproblem of the Laplace-Beltrami operator are of the most important. Including mathematics and many applied fields. Partial differential equations (PDE) on manifolds arise in many areas, Based on the global conformal parameterization of a surface, we can con-struct a conformal atlas and use it to build conformal geometry images which have very accurate reconstructed normals. We also introduce a novel topological modification method to improve the uniformity of the parameterization. These properties can be formulated by sparse linear systems, so the method is easy to im-plement and the entire process is automatic.
![compute method map compute method map](https://i.stack.imgur.com/G7wip.png)
Our algo-rithm is based on the properties of gradient fields of conformal maps, which are closedness, harmonity, conjugacy, duality and symmetry. This space has a natural structure solely determined by the surface geometry, so our computing result is independent of connectivity, insensitive to resolution, and independent of the algorithms to discover it. We analyze the structure of the space of all global conformal parameteri-zations of a given surface and find all possible solutions by constructing a basis of the underlying linear solution space. The parameterization is global in the sense that it preserves the conformality everywhere except for a few points, and has no boundary of discontinuity. We solve the problem of computing global conformal parameterizations for surfaces with nontrivial topologies.