Optimal Global Conformal Surface Parameterization Miao Jin * Computer Science Department SUNY at Stony Brook Yalin Wang † Math Department UCLA Shing-Tung Yau ‡ Math Department Harvard University Xianfeng Gu § Computer Science Department SUNY at Stony Brook Figure 1: Uniform global conformal parameterization ((a) and (b)) and region emphasized conformal parameterization ((c) and (d)). (a). Least uniform conformal parameterization with energy: 21.208e - 5. (b). Most uniform conformal parameterization with energy: 3.685e - 5. (c). Maximizing the parameter area of the left half surface (with percentage: 83.48%). (d). Maximizing the parameter area of the right half surface (with percentage: 82.58%.) ABSTRACT All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical phenom- ena, such as heat diffusion and electro-magnetic fields on the sur- face. A good parameterization is crucial for simulation and visual- ization. This paper provides an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces. It relies on certain holomorphic differential forms and conformal mappings from differential geometry and Riemann surface theories. Algo- rithms are developed to modify topology, locate zero points, and determine cohomology types of differential forms. The implemen- tation is based on a finite dimensional optimization method. The optimal parameterization is intrinsic to the geometry, preserves an- gular structure, and can play an important role in various applica- tions including texture mapping, remeshing, morphing and simu- lation. The method is demonstrated by visualizing the Riemann surface structure of real surfaces represented as triangle meshes. CR Categories: I.3.5 [Computational Geometry and Object Mod- eling]: Curve, surface, solid, and object representations—Surface Parameterization Keywords: Computational geometry and object modeling; Curve, surface, solid, and object representations; Surface parameterization. 1 I NTRODUCTION * e-mail: mjin@cs.sunysb.edu † e-mail:ylwang@math.ucla.edu ‡ e-mail:yau@math.harvard.edu § e-mail:gu@cs.sunysb.edu Surface parameterization is the process of mapping a surface to a planar domain and has many applications in various fields of sci- ence and engineering, including texture mapping, geometric mor- phing, surface matching, surface remeshing, and surface extrapo- lation. For example, texture mapping can be used to enhance the visual quality and generate different visual results. Geometric mor- phing can be used to generate vivid animation results. Essentially, surface parametrization can convert a 3D geometric problem to 2D, thereby improving the efficiency and simplifying the computation. Conformal surface parameterizations have many merits: they preserve angular structure, are intrinsic to geometry, and are stable with respect to different triangulations and small deformations. It has been widely used for many applications, such as non-distorted texture mapping [23], [16],[20], surface remeshing [1], surface fair- ing [22], surface matching [14], brain mapping [2], [13] etc. It is desirable to parameterize surfaces globally without any seams. The existence of global conformal parameterization is a non-trivial fact. This is equivalent to the fact that all orientable sur- faces are Riemann surfaces. The atlas formed by the global confor- mal parameterization is the so-called conformal structure. Confor- mal structure is a fundamental structure between metric structure and topological structure and governs many natural physical phe- nomena. The abstract concept of a Riemann surface can also be visualized by texture mapping special patterns using global confor- mal parameterizations. This is the only means of visually conveying conformal information of surfaces. The early work of global conformal parameterization has been done in [14, 15], where the basis for all possible global conformal parameterizations are computed. Because global conformal param- eterization is non-unique, the problem of finding the optimal one remains open. This paper introduces an explicit method to find the optimal global conformal parameterizations of arbitrary surfaces. First, the metrics for measuring the quality of conformal parameterizations are designed. Second, the major factors affecting the quality of the parameterization are summarized. Then, algorithms are developed to modify the topology, locate the zero points, and determine the cohomology types of the differential forms. The method is based October 10-15, Austin, Texas, USA IEEE Visualization 2004 0-7803-8788-0/04/$20.00 ©2004 IEEE 267