Xianfeng David Gu

Discrete surface Ricci flow: theory and applications

Conformal geometry is at the core of pure mathematics. Conformal structure is more flexible than Riemaniann metric but more rigid than topology. Conformal geometric methods have played important roles in engineering fields. This work introduces a theoretically rigorous and practically efficient method for computing Riemannian metrics with prescribed Gaussian curvatures on discrete surfaces--discrete surface Ricci flow, whose continuous counter part has been used in the proof of Poincare conjecture. Continuous Ricci flow conformally deforms a Riemannian metric on a smooth surface such that the Gaussian curvature evolves like a heat diffusion process. Eventually, the Gaussian curvature becomes constant and the limiting Riemannian metric is conformal to the original one. In the discrete case, surfaces are represented as piecewise linear triangle meshes. Since the Riemannian metric and the Gaussian curvature are discretized as the edge lengths and the angle deficits, the discrete Ricci flow can be defined as the deformation of edge lengths driven by the discrete curvature. The existence and uniqueness of the solution and the convergence of the flow process are theoretically proven, and numerical algorithms to compute Riemannian metrics with prescribed Gaussian curvatures using discrete Ricci flow are also designed. Discrete Ricci flow has broad applications in graphics, geometric modeling, and medical imaging, such as surface parameterization, surface matching, manifold splines, and construction of geometric structures on general surfaces.

Registration for 3D surfaces with large deformations using quasi-conformal curvature flow

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 features on the surfaces, then estimates the Beltrami coefficient, and finally uniquely determines the registration mapping by solving Beltrami equations using curvature flow. The method is 1) general, it can search the desired registration in the whole space of diffeomorphisms, which includes the conventional searching spaces, such as rigid motions, 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 topologies; 4) rigorous, it has solid theoretic foundation. Experiments on the real surfaces with large deformations and complicated topologies demonstrate the efficiency, robustness of the proposed method.

Area-Preservation Mapping using Optimal Mass Transport

We present a novel area-preservation mapping/flattening method using the optimal mass transport technique, based on the Monge-Brenier theory. Our optimal transport map approach is rigorous and solid in theory, efficient and parallel in computation, yet general for various applications. By comparison with the conventional Monge-Kantorovich approach, our method reduces the number of variables from O(n2) to O(n), and converts the optimal mass transport problem to a convex optimization problem, which can now be efficiently carried out by Newtons method. Furthermore, our framework includes the area weighting strategy that enables users to completely control and adjust the size of areas everywhere in an accurate and quantitative way. Our method significantly reduces the complexity of the problem, and improves the efficiency, flexibility and scalability during visualization. Our framework, by combining conformal mapping and optimal mass transport mapping, serves as a powerful tool for a broad range of applications in visualization and graphics, especially for medical imaging. We provide a variety of experimental results to demonstrate the efficiency, robustness and efficacy of our novel framework.

Surface Quasi-Conformal Mapping by Solving Beltrami Equations

We consider the problem of constructing quasi-conformal mappings between surfaces by solving Beltrami equations. This is of great importance for shape registration. In the physical world, most surface deformations can be rigorously modeled as quasi-conformal maps. The local deformation is characterized by a complex-value function, Beltrami coefficient , which describes the deviation from conformality of the deformation at each point. We propose an effective algorithm to solve the quasi-conformal map from the Beltrami coefficient. The major strategy is to deform the conformal structure of the original surface to a new conformal structure by the Beltrami coefficient, such that the quasi-conformal map becomes a conformal map. By using holomorphic differential forms, conformal maps under the new conformal structure are calculated, which are the desired quasi-conformal maps. The efficiency and efficacy of the algorithms are demonstrated by experimental results. Furthermore, the algorithms are robust for surfaces scanned from real life, and general for surfaces with different topologies.

Covering space for in-network sensor data storage

For in-network storage schemes, one maps data, indexed in a logical space, to the distributed sensor locations. When the physical sensor network has an irregular shape and possibly holes, the mapping of data to sensors often creates unbalanced storage load with high data concentration on nodes near network boundaries. In this paper we propose to map data to a covering space, which is a tiling of the plane with copies of the sensor network, such that the sensors receive uniform storage load and traffic. We propose distributed algorithms to construct the covering space with Ricci flow and Möbius transforms. The use of the covering space improves the performance of many in-network storage and retrieval schemes such as geographical hash tables (GHTs) or the double rulings (quorum based schemes), and provides better load balanced routing.

Topology-driven surface mappings with robust feature alignment

Topological concepts and techniques have been broadly applied in computer graphics and geometric modeling. However, 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 triangle 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 correspondence, we articulate a new technique that robustly satisfies hard feature constraints, without the use of heuristics 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.

An automatic 3D expression recognition framework based on sparse representation of conformal images

We propose a general and fully automatic framework for 3D facial expression recognition by modeling sparse representation of conformal images. According to Riemann Geometry theory, a 3D facial surface S embedded in ℝ3, which is a topological disk, can be conformally mapped to a 2D unit disk D through the discrete surface Ricci Flow algorithm. Such a conformal mapping induces a unique and intrinsic surface conformal representation denoted by a pair of functions defined on D, called conformal factor image (CFI) and mean curvature image (MCI). As facial expression features, CFI captures the local area distortion of S induced by the conformal mapping; MCI characterizes the geometry information of S. To model sparse representation of conformal images for expression classification, both CFI and MCI are further normalized by a Mobius transformation. This transformation is defined by the three main facial landmarks (i.e. nose tip, left and right inner eye corners) which can be detected automatically and precisely. Expression recognition is carried out by the minimal sparse expression-class-dependent reconstruction error over the conformal image based expression dictionary. Extensive experimental results on the BU-3DFER dataset demonstrate the effectiveness and generalization of the proposed framework.

Numerical Computation of Surface Conformal Mappings

We report recent progress in the computation of conformal mappings from surfaces with arbitrary topologies to canonical domains. Two major computational methodologies are emphasized; one is holomorphic differentials based on Riemann surface theory and the other is surface Ricci flow from geometric analysis. The applications of surface conformal mapping in the field of engineering are briefly reviewed.

Conformal Magnifier: A Focus+Context Technique with Local Shape Preservation

We present the conformal magnifier, a novel interactive focus+context visualization technique that magnifies a region of interest (ROI) using conformal mapping. Our framework supports the arbitrary shape design of magnifiers for the user to enlarge the ROI while globally deforming the context region without any cropping. By using the mathematically well-defined conformal mapping theory and algorithm, the ROI is magnified with local shape preservation (angle distortion minimization), while the transition area between the focus and context regions is deformed smoothly and continuously. After the selection of a specified magnifier shape, our system can automatically magnify the ROI in real time with full resolution even for large volumetric data sets. These properties are important for many visualization applications, especially for the computer aided detection and diagnosis (CAD). Our framework is suitable for diverse applications, including the map visualization, and volumetric visualization. Experimental results demonstrate the effectiveness, robustness, and efficiency of our framework.

Automatic non-rigid registration of 3D dynamic data for facial expression synthesis and transfer

Automatic non-rigid registration of 3D time-varying data is fundamental in many vision and graphics applications such as facial expression analysis, synthesis, and recognition. Despite many research advances in recent years, it still remains to be technically challenging, especially for 3D dynamic, densely-sampled facial data with a large number of degrees of freedom (necessarily used to represent rich and subtle facial expressions). In this paper, we present a new method for automatic non-rigid registration of 3D dynamic facial data using least-squares conformal maps, and based on this registration method, we also develop a new framework of facial expression synthesis and transfer. Nowadays more and more 3D dynamic, densely-sampled data become prevalent with the advancement of novel 3D scanning techniques. To analyze and utilize such huge 3D data, an efficient non-rigid registration algorithm is needed to establish one-to-one inter frame correspondences. Towards this goal, a non-rigid registration algorithm of 3D dynamic facial data is developed by using least-squares conformal maps with additional feature correspondences detected by employing active appearance models (AAM). The proposed method with additional, interior feature constraints guarantees that the non-rigid data will be accurately registered. The least-squares conformal maps between two 3D surfaces are globally optimized with the least angle distortion and the resulting 2D maps are stable and one-to-one. Furthermore, by using this non-rigid registration method, we develop a new system of facial expression synthesis and transfer. Finally, we perform a series of experiments to evaluate our non-rigid registration method and demonstrate its efficacy and efficiency in the applications of facial expression synthesis and transfer.