Registration for 3D Surfaces with Large Deformations Using Quasi-Conformal Curvature Flow Wei Zeng Computer Science Department Stony Brook University zengwei@cs.sunysb.edu Xianfeng David Gu Computer Science Department Stony Brook University gu@cs.sunysb.edu Abstract A novel method for registering 3D surfaces with large deformations is presented, which is based on quasi- conformal geometry. A general diffeomorphism distorts the conformal structure of the surface, which is represented as the Beltrami coefficient. Inversely, the diffeomorphism can be determined by the Beltrami coefficient in an essentially unique way. Our registration method first extracts the fea- tures on the surfaces, then estimates the Beltrami coeffi- cient, and finally uniquely determines the registration map- ping by solving Beltrami equations using curvature flow. The method is 1) general, it can search the desired reg- istration in the whole space of diffeomorphisms, which in- cludes the conventional searching spaces, such as rigid mo- tions, isometric transformations or conformal mappings; 2) global optimal, the global optimum is determined by the method unique up to a 3 dimensional transformation group; 3) robust, it handles large surfaces with complicated topolo- gies; 4) rigorous, it has solid theoretic foundation. Exper- iments on the real surfaces with large deformations and complicated topologies demonstrate the efficiency, robust- ness of the proposed method. 1. Introduction 3D surface matching and registration have fundamental importance in computer vision. They have wide applica- tions in many computing fields [4, 12, 31], such as track- ing, classification, recognition for deformable objects (e.g., human expression scans) in computer vision, morphologi- cal study for medical data (e.g., human cortex CT scans) in medical imaging, mapping or animation between surfaces (e.g., human body scans) in game industry, and so on. One of the most challenging problems in 3D surface registration is to tackle the non-rigid deformation and substantial dis- tortion due to the anisotropic characteristics of the elasticity property of the material (e.g., human muscles, tissues). This work proposes an algorithm to register surfaces with large deformations using quasi-conformal curvature flow method, which can produce general diffeomorphisms, and the solution is essentially unique. 1.1. Classification of Registration Methods In the vast literature of 3D surface registration, numerous algorithms have been developed. Generally speaking, sur- face registration aims at finding an optimal mapping among surfaces. Different algorithms search in the different map- ping spaces. The common mapping spaces are the follow- ing transformation groups: {Rigd } ⊲ {Isom} ⊲ { Con f } ⊲ {Di f f }, where A ⊲ B represents A is a proper subgroup of B. {Rigd } represents the rigid motions in ℝ 3 ; {Isom}, isometric map- pings; { Con f }, conformal mappings; {Di f f }, all the dif- feomorphisms, which are smooth and bijective. The diffeo- morphism group is the most general; all other transforma- tion groups are its special subgroups. Most existing 3D surface registration methods can be classified by their mapping space. Iterative closest point (ICP) method [2] finds the optimal rigid motion to match two surfaces embedded in ℝ 3 , so it belongs to the {Rigd } category. The isomap algorithm [22] uses geodesic expo- nential map to register two surfaces, the mapping preserving the geodesic distance, therefore it belongs to {Isom} cate- gory. The matching algorithm based on heat diffusion dis- tance is based on the fact that the heat kernel signature de- termines the Riemannian metric [23], therefore the method also belongs to the {Isom} category. Conformal mappings [13, 14, 28, 35] and M ¨ obius transformations [15] have been widely applied for surface registration. These methods map the surfaces onto canonical domains conformally, then find a conformal map between canonical domains. Therefore, these methods belong to the conformal transformation cat- egory { Con f }. Harmonic map methods [27, 30, 33, 34] search for the best harmonic diffeomorphisms. Harmonic 2457