Research Interests

Computational Conformal Geometry
All surfaces are Riemann surfaces, they have intrinsic conformal structures, which are invariants under global conformal transformation groups. Conformal stuctures are geometric characteristics between topology and Euclidean geometry. I will focus on computing conformal structures for general surfaces, and applied them in shape analysis, surface classification, surface matching, geometric compression. Covariant differentiation can be performed by means of conformal structures, this is essential for solving differential equations on surfaces.

Geometric Structures on Surfaces
Surfaces admit different geometric structures, such as spherical, affine, hyperbolic and real projective structures. Different structure allows different geometry to be defined on the surface. Convential planar geometric algorithms can be generalized to the surface domain directly via geometric structures, such as convex hull can be generalized via projective structure, polar form splines can be generalized on to the surface using affine structure.

Computational Ricci Flow
Ricci flow is a powerful tool to compute Riemannian metrics on surfaces. The uniformization metrics induces various geometric structures, which are crucial for engineering applications.

Manifold Spline
Conformal structure offers deeper insight to understand manifold splines. The existence of manifold spline can be interpreted as the existance of special geometric structure on the surfaces, where topological obstruction plays crutial roles. The goal of this research is to build coherent theoretic frame work for splines on manifold, and design spline schemes and subdivision schemes on manifolds.

Harmonic shape analysis
The functional space on a surface is determined by the surface geometric characteristics. The harmonic spectrum can reflect many global geometric information of the surface. The surface matching, shape anylasis can be carried out by examing the behavior of special differential operators on surfaces.

Geometry Image
Geometry Image unifies the representations for geometry and textures. Signal processing techniques for images can be applied to geometry directly. Computer graphics hardware can speed up rendering efficiency by using geometry image.

Brain Mapping & Colon Flattening in Medical Imaging
Conformal brain mapping is important in medical imaging. By mapping brain cortical surfaces onto the spheres, different brains can be matched and quantatively compared, analyzed. Conformal colon flattening is cirtical for fusing colon surfaces reconstructed from MRI images and help for diagnosing potential illness.