Parametrization for surfaces with arbitrary topologies
Xianfeng Gu
Thesis, Computer Science, Harvard University, 2002
Surface parametrization is a fundamental problem in computer
graphics. It is essential for operations such as texture mapping, texture
synthesis on surface, interactive 3D painting, remeshing and multi-resolution
analyis and mesh compression. Conformal parametrization, which preserves angles,
has many nice properties such as having no local distortion on textures, and
being independent of trangulation or resolution. Existing conformal
parametrization methods partition the mesh into several charts, each of which is
then parametrized and packed to an atlas. These methods surffer from limitations
such as difficulty in segmenting the mesh and artifacts caused by discontinuites
between charts. This paper presents a method to compute global conformal
parametrization for triangulated surfaces with arbitrary topologies. Our method
is boundary free, hence eliminating the need to chartify the mesh. We compute
the natural conformal structure of the surface, which is determined solely by
its geometry. The parametrization is stable in the sense that if the geometries
are similar, then the parametrizations on canonical domain are also close. The
parametrization is conformal everywhere except on $2g-2$ number of points, where
$g$ is the number of genus. We first study special tangential vector fileds, the
so called holomorphic differentials, which are the gradients of local conformal
maps. It is proven that all discrete holomorphic differentials form a $2g$
dimensional linear space. We derive practical algorithms of computing the bases
of this linear space from the constructive proofs. Then by integrating the
linear combination of a set of bases, we can construct any global conformal
parametrization of the surface. The algorithms only involve solving linear
systems and is easy to implement. In this work, we also propose to remesh an
arbitrary surface onto a completely regular structure, we call \emph{geometry
image}. It captures geometry as a simple 2D array of quantized points. Surface
signals like normals and colors are stored in similar 2D arrays using the same
implicit surface parametrization-texture coordinates are absent. To create a
geometry image, we cut an arbitary mesh along a network of edge paths, and
parametrize the resulting single chart onto a square. Geometry images unifies
the geometry and image format. Many techinques in image processing field can be
applied to geometry directly. The methods introduced in this paper are very
general and can be applied to many other topologic and geomeric problems in
computer graphics.