Xianfeng David Gu


Assistant Professor
Department of Computer Science
State University of New York at Stony Brook

Room 2425 Computer Science Building
State University of New York at Stony Brook
Stony Brook, New York 11794-4400

Phone:  (631) 632-1828 (Office) 
Fax: (631) 632-8334
gu at cs.sunysb.edu

http://www.cs.sunysb.edu/~gu
Center of Visual Computing

Research Summary

Conformal Structure is a natural geometric structure on surfaces, which governs many physics phenomena, such as heat diffusion, electric-magnetic fields, etc. Conformal field theory plays fundamental role in string theory.

In mathematics, conformal means "angle preserving". Conformal structure is a specail atlas of the surface, such that angles among tangent vectors can be coherently defined on different local coordinate systems. Furthermore, concepts in complex anylasis can be defined on the surface via conformal structure. Conformal geometry is the intersection of algebraic geometry, differential geometry, complex anylasis and algebraic topology.

In engineering, conformal structure is between topological structure and geometric structure, which is more rigid than topology and more flexible than geometry. Therefore, conformal structure leads to canonical non-rigid deformation, which is important for engineering applications, especially for shape anylasis, classification and registration.

The goal of computational conformal geometry is to convert concepts and theorems from Riemann surface theory to practical algorithms, and implement them for engineering applications.

A surface with a conformal structure is called a Riemann surface. All metric surfaces are Riemann surfaces. The top figure illustrates the conformal structure using isothermal coordinates. The algorithm is based on computing holomorphic one-forms. According to Riemann uniformization theorem, all metric surfaces can be conformally mapped to three canonical shapes, the sphere, the plane and the hyperbolic disk. The mappings are periodic and reflect the intrinsic symmetries of the surfaces. The mappings can be computed using Ricci flow.

Ricci flow is a powerful geometric analytic tool, which has been applied to prove Poincare conjecture. Ricci flow is a parabolic system of partial differential equations which acts like the heat equation to spread the curvature of a Riemannian metric evenly over the surface to produce a metric of constant curvature. Computational Ricci flow has been invented and applied for computing hyperbolic structures and conformal surface parameterizations, it is expected to play important roles in both mathematics and engineering fields.

Algorithms for computing conformal structure can be summaried as
  1. For genus zero surface in the first column, the mapping can be commputed using spherical harmonic maps, in paper Genus Zero Surface Conformal Mapping and Its Application to Brain Surface Mapping. Spherical geometry can be defined on the surface.
  2. For genus one surface in the second column, the mapping can be commputed using holomorphic one forms, in paper Global Conformal Surface Parameterization. Another algorithm is to use Euclidean Ricci flow, in paper Conformal Surface Parameterization Using Euclidean Ricci Flow. Euclidean geometry can be defined on the surface.
  3. For higher genus surfaces in the third column, the mapping can be commputed using hyperbolic Ricci flow, in paper Computing Surface Hyperbolic Structure and Real Projective Structure. Hyperbolic geometry can be defined on the surface.
Applications of conformal geometry in engineering fields are innumerous, the followings are the most directly related ones,