Computing Conformal Structure of Surfaces

Conformal Structures of Surfaces ∗

This paper solves the problem of computing conformal structures of general 2manifolds represented as triangular meshes. We approximate the De Rham cohomology by simplicial cohomology and represent the Laplace-Beltrami operator, the Hodge star operator by linear systems. A basis of holomorphic one-forms is constructed explicitly. We then obtain a period matrix by integrating holomorphic differen...

A Characterization of the Infinitesimal Conformal Transformations on Tangent Bundles

Surface Quasi-Conformal Mapping by Solving Beltrami Equations

We consider the problem of constructing quasi-conformal mappings between surfaces by solving Beltrami equations. This is of great importance for shape registration. In the physical world, most surface deformations can be rigorously modeled as quasi-conformal maps. The local deformation is characterized by a complex-value function, Beltrami coefficient, which describes the deviation from conform...

Global Conformal Surface Parameterization

We solve the problem of computing global conformal parameterizations for surfaces with nontrivial topologies. 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 analyze the structure of the space of all global conformal parameterizations of a given surface and find all possible solutions by ...

Brain Surface Parameterization Using Riemann Surface Structure

We develop a general approach that uses holomorphic 1-forms to parameterize anatomical surfaces with complex (possibly branching) topology. Rather than evolve the surface geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and admit conformal structures, which induce special curvilinear coordinate systems on the surfaces. Based on Riemann sur...