Public Talks

Computing Surface Hyperbolic Structure and Real Projective Structure

M. Jin, F. Luo and X. Gu
Solid and Physics Modeling 2006
[Presentation]
Geometric structures are natural structures of surfaces, which enable different geometries to be defined on the surfaces. Algorithms designed for planar domains based on a specific geometry can be systematically generalized to surface domains via the corresponding geometric structure. For example, polar form splines with planar domains are based on affine invariants. Polar form splines can be generalized to manifold splines on the surfaces which admit affine structures and are equipped with affine geometries. Surfaces with negative Euler characteristic numbers admit hyperbolic structures and allow hyperbolic geometry. All surfaces admit real projective structures and are equipped with real projective geometry. Because of their general existence, both hyperbolic structures and real projective structures have the potential to replace the role of affine structures in defining manifold splines. This paper introduces theoretically rigorous and practically simple algorithms to compute hyperbolic structures and real projective structures for general surfaces. The method is based on a novel geometric tool - discrete variational Ricci flow. Any metric surface admits a special uniformization metric, which is conformal to its original metric and induces constant curvature. Ricci flow is an efficient method to calculate the uniformization metric, which determines the hyperbolic structure and real projective structure. The algorithms have been verified on real surfaces scanned from sculptures. The method is efficient and robust in practice. To the best of our knowledge, this is the first work of introducing algorithms based on Ricci flow to compute hyperbolic structure and real projective structure. More importantly, this work introduces the framework of general geometric structures, which enable different geometries to be defined on manifolds and lay down the theoretical foundation for many important applications in geometric modeling.

Manifold Splines

X. Gu, Y. He and H. Qin
Solid and Physics Modeling 2005.
[Presentation] [Web]
Constructing splines whose parametric domain is an arbitrary manifold and effectively computing such splines in real-world applications are of fundamental importance in solid and shape modeling, geometric design, graphics, etc. This paper presents a general theoretical and computational framework, in which spline surfaces defined over planar domains can be systematically extended to manifold domains with arbitrary topology with or without boundaries. We study the affine structure of domain manifolds in depth and prove that the existence of manifold splines is equivalent to the existence of a manifold's affine atlas. Based on our theoretical breakthrough, we also develop a set of practical algorithms to generalize triangular B-spline surfaces from planar domains to manifold domains. We choose triangular B-splines mainly because of its generality and many of its attractive properties. As a result, our new spline surface defined over any manifold is a piecewise polynomial surface with high parametric continuity without the need for any patching and/or trimming operations. Through our extensive experiments, we hope to demonstrate that our novel manifold splines are both powerful and efficient in modeling arbitrarily complicated geometry and representing continuously-varying physical quantities defined over shapes of arbitrary topology.

Global Conformal Surface Parameterization

X. Gu and S.-T. Yau
ACM Symposium on Geometry Processing 2003, 127-137
[Presentation]
We solve the problem of computing global conformal parametrizations for surfaces with arbitrary topologies, with or without boundaries. The parameterization preserves the conformality everywhere except for a few points, and it 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 instead of finding just one. 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. Our algorithm is based on the properties of gradient fields of conformal maps, which are closedness, harmonity, conjugacy, duality and symmetry. These properties can be formulated by sparse linear systems, so the method is easy to implement and the whole process is automatic.

Geometry Images

X. Gu S. Gortler and H. Hoppe
Computer Graphics (SIGGRAPH 2002 Proceedings)
[Presentation]
Surface geometry is often modeled with irregular triangle meshes. The process of remeshing refers to approximating such geometry using a mesh with (semi)-regular connectivity, which has advantages for many graphics applications. However, current techniques for remeshing arbitrary surfaces create only semi-regular meshes. The original mesh is typically decomposed into a set of disk-like charts, onto which the geometry is parametrized and sampled. In this paper, we propose to remesh an arbitrary surface onto a completely regular structure we call a 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 arbitrary mesh along a network of edge paths, and parametrize the resulting single chart onto a square. Geometry images can be encoded using traditional image compression algorithms, such as wavelet-based coders. Although the paper emphasizes the exciting possibilities of resampling mesh geometry into an image, the same parametrization scheme can also be used to construct single-chart parametrizations of irregular meshes, for seam-free texture mapping.

Computing Conformal Structures of Surfaces

Xianfeng Gu and Shing-Tung Yau
[Abstract] and [Presentation]
According to Klein's Erlangen program, different geometry branches study the invariants of a space under different transformation groups. For example, topology studies the topological invariants of a surface, which are the holomogy groups etc. The topological invariants can be computed from triangulated meshes. The Euclidean transformation keeps the induced metric from R^3 on the surface. The curvature, area, and angles can be computed. Between topological transformation group and isometric transformation group (Euclidean group is a subgroup of isometric group), there exists a conformal transformation group, which has not been used in computer graphics and vision before. We present a method to compute the conformal invariants under this group, and to verify if two surfaces can be transformed to each other by conformal maps. The proposed conformal geometry method has many applications including texture mapping, remeshing, geometric morphing in graphics, surface identification and classification in vision.