Surface mapping plays an important role in geometric processing. They induce both area and angular distortions. If the angular distortion is bounded, the mapping is called a quasi-conformal map. Many surface maps in our physical world are quasi-conformal. The angular distortion of a quasi-conformal map can be represented by Beltrami differentials. According to quasi-conformal Teichmüller theory, there is an 1-1 correspondence between the set of Beltrami differentials and the set of quasi-conformal surface maps. Therefore, every quasi-conformal surface map can be fully determined by the Beltrami differential and can be reconstructed by solving the so-called Beltrami equation. In this work, we propose an effective method to solve the Beltrami equation on general Riemann surfaces. The solution is a quasi-conformal map associated with the prescribed Beltrami differential. We firstly formulate a discrete analog of quasi-conformal maps on triangular meshes. Then, we propose an algorithm to compute discrete quasi-conformal maps. The main strategy is to define a discrete auxiliary metric of the source surface, such that the original quasi-conformal map becomes conformal under the newly defined discrete metric. The associated map can then be obtained by using the discrete Yamabe flow method. Numerically, the discrete quasi-conformal map converges to the continuous real solution as the mesh size approaches to 0. We tested our algorithm on surfaces scanned from real life with different topologies. Experimental results demonstrate the generality and accuracy of our auxiliary metric method.

3D surface classification is a fundamental problem in computer vision and computational geometry. Surfaces can be classified by different transformation groups. Traditional classification methods mainly use topological transformation groups and Euclidean transformation groups. This paper introduces a novel method to classify surfaces by conformal transformation groups. Conformal equivalent class is refiner than topological equivalent class and coarser than isometric equivalent class, making it suitable for practical classification purposes. For general surfaces, the gradient fields of conformal maps form a vector space, which has a natural structure invariant under conformal transformations. We present an algorithm to compute this conformal structure, which can be represented as matrices, and use it to classify surfaces. The result is intrinsic to the geometry, invariant to triangulation and insensitive to resolution. To the best of our knowledge, this is the first paper to classify surfaces with arbitrary topologies by global conformal invariants. The method introduced here can also be used for surface matching problems.

This paper presents a method of obtaining geometric registrations between high-genus (g≥1) surfaces. Surface registration between simple surfaces, such as simply connected open surfaces, has been well studied. However, very few works have been carried out for the registration of high-genus surfaces. The high-genus topology of the surface poses a great challenge for surface registration. A possible approach is to partition surfaces into simply connected patches and registration can be done in a patch-by-patch manner. Consistent cuts are required, which are usually difficult to obtain and prone to error. In this work, we propose an effective way to obtain geometric registration between high-genus surfaces without introducing consistent cuts. The key idea is to conformally parameterize the surface into its universal covering space, which is either the Euclidean plane or the hyperbolic disk embedded in R2. Registration can then be done on the universal covering space by iteratively minimizing a shape mismatching energy measuring the geometric dissimilarity between the two surfaces. The Beltrami coefficient of the mapping is considered and adjusted in order to control the bijectivity of the mappings in each iteration. Our proposed algorithm effectively computes a smooth registration between high-genus surfaces that matches geometric information as much as possible. The algorithm can also be applied to find a smooth registration minimizing any general energy functionals. Numerical experiments on high-genus surface data show that our proposed method is effective for registering high-genus surfaces with geometric matching. We also applied the method to register anatomical structures for medical imaging, which demonstrates the usefulness of the proposed algorithm.

A numerical method for computing the extremal Teichmüller map between multiply-connected domains is presented. Given two multiply-connected domains, there exists a unique Teichmüller map (T-Map) between them minimizing the conformality distortion. The extremal T-Map can be considered as the ‘most conformal’ map between multiply-connected domains. In this paper, we propose an iterative algorithm to compute the extremal T-Map using the Beltrami holomorphic flow (BHF). The BHF procedure iteratively adjusts the initial map based on a sequence of Beltrami coefficients, which are complex-valued functions defined on the source domain. It produces a sequence of quasi-conformal maps, which converges to the T-Map minimizing the conformality distortion. We test our method on synthetic data together with real human face data. Results show that our algorithm computes the extremal T-Map between two multiply-connected domains of the same topology accurately and efficiently.

This paper is motivated by the problem of subdividing a prismatic mesh to a tetrahedral mesh (without inserting Steiner points) so as to not only match arbitrarily prescribed boundary conditions but also allow arbitrary topologies in the base mesh. We explore all possible combinations of these two factors, and propose a complete solution to this 3D problem by converting it to an equivalent 2D graph problem, called cutting flow problem. For each case, we not only prove the sufficient and necessary condition for the existence of solutions, but also provide linear and provable algorithms to compute a solution whenever there is one.