Topology-driven Surface Mappings with Robust Feature Alignment Christopher Carner, Miao Jin, Xianfeng Gu, and Hong Qin Stony Brook University Figure 1: Surface mapping between horse and lizard. The color-coding shows the mapping of each region, guided by eight user-specified feature curves. Our topology-driven method provides mappings of different homotopy type between the two surfaces as shown in (c) and (d). We show feature curves in red. Abstract Topological concepts and techniques have been broadly ap- plied in computer graphics and geometric modeling. How- ever, the homotopy type of a mapping between two surfaces has not been addressed before. In this paper, we present a novel solution to the problem of computing continuous maps with different homotopy types between two arbitrary trian- gle meshes with the same topology. Inspired by the rich theory of topology as well as the existing body of work on surface mapping, our newly-developed mapping techniques are both fundamental and unique, offering many attractive advantages. First, our method allows the user to change the homotopy type or global structure of the mapping with minimal intervention. Moreover, to locally affect shape cor- respondence, we articulate a new technique that robustly satisfies hard feature constraints, without the use of heuris- tics to ensure validity. In addition to acting as a useful tool for computer graphics applications, our method can be used as a rigorous and practical mechanism for the visualization of abstract topological concepts such as homotopy type of surface mappings, homology basis, fundamental domain, and universal covering space. At the core of our algorithm is a procedure for computing the canonical homology basis and using it as a common cut graph for any surface with the same topology. We demonstrate our results by applying our algorithm to shape morphing in this paper. 1 Introduction Surface mapping is of prime significance in many graphics applications including shape analysis, texture mapping, an- imation transfer, shape morphing, feature registration, and many other digital geometry processing methods. In prin- ciple, parameterization-based surface mapping methods can be classified by the topology of the parametric domain. Typ- ically, surfaces that are homeomorphic to a disk can be easily parameterized over the plane. For topologically more com- plicated shapes, the parameter domain can be an arbitrary surface in R 3 , and so a parameterization will essentially be a mapping between two objects. In such a case, a continuous, meaningful mapping requires that the two surfaces share the same topological attributes, such as genus and number of boundaries. In addition to the topological factors, a desir- able surface mapping also hinges upon the specific applica- tion demands. For example, texture mapping frequently re- quires a planar parameterization because most textures are acquired/synthesized as 2D images. However, applications such as remeshing, morphing, and medical model registration are better suited to a surface mapping between topologically equivalent manifolds in 3D. Furthermore, topological concepts and techniques have been broadly applied in computer graphics and geometric model- ing ([6]). However, the homotopy type of a mapping between two surfaces has not been addressed before. While other sur- face mapping methods focus on a single homotopy class, in this paper, we articulate a theoretically rigorous method that produces many continuous maps of different homotopy type between two arbitrary triangle meshes with the same topol- ogy (see Figures 2 and 10). The uniqueness of our method- ology results from applying the rich mathematical theory of topology to surface classification, rather than relying solely on the embedded geometry. In a nutshell, we first compute a special set of curves, called the canonical homology basis, that will cut any two homeomorphic surfaces in the same way into a topological disk (see Figure 6). Then, we can parameterize each sliced surface over a planar domain, us- ing a metric to reduce the distortion. Next, we can create a mapping between the planar domains of two surfaces and extract the final mapping between the original surfaces from the shared planar domain. In addition, our novel, topology-based approach has sev- eral other advantages. First, current methods require the use of heuristics to avoid bad path-tracing while partitioning the surface into multiple regions between features. On the other hand, our technique enables an elegant feature map- ping mechanism that can robustly satisfy user-specified, hard constraints without relying on ad-hoc approaches to ensure validity. Moreover, unlike some methods, we do not require a minimum number of features to be inserted. The features (including points and line segments) are always guaranteed to be parts of cutting curves mapped to hard, planar bound-