Computing discrete geodesic distance over triangle meshes is one of the fundamental problems in computational geometry and computer graphics. In this problem, an effective window pruning strategy can significantly affect the actual running time. Due to its importance, we conduct an in-depth study of window pruning operations in this paper, and produce an exhaustive list of scenarios where one window can make another window partially or completely redundant. To identify a maximal number of redundant windows using such pairwise cross checking, we propose a set of procedures to synchronize local window propagation within the same triangle by simultaneously propagating a collection of windows from one triangle edge to its two opposite edges. On the basis of such synchronized window propagation, we design a new geodesic computation algorithm based on a triangle-oriented region growing scheme. Our geodesic algorithm can remove most of the redundant windows at the earliest possible stage, thus significantly reducing computational cost and memory usage at later stages. In addition, by adopting triangles instead of windows as the primitive in propagation management, our algorithm significantly cuts down the data management overhead. As a result, it runs 4-15 times faster than MMP and ICH algorithms, 2-4 times faster than FWP-MMP and FWP-CH algorithms, and also incurs the least memory usage.

Surface parameterizations have been widely applied to computer graphics and digital geometry processing. In this paper, we propose a novel stretch energy minimization (SEM) algorithm for the computation of equiareal parameterizations of simply connected open surfaces with very small area distortions and highly improved computational efficiencies. In addition, the existence of nontrivial limit points of the SEM algorithm is guaranteed under some mild assumptions of the mesh quality. Numerical experiments indicate that the accuracy, effectiveness, and robustness of the proposed SEM algorithm outperform the other state-of-the-art algorithms. Applications of the SEM on surface remeshing, registration and morphing for simply connected open surfaces are demonstrated thereafter. Thanks to the SEM algorithm, the computation for these applications can be carried out efficiently and reliably.

Automatic hexahedral mesh generation plays a fundamental role in CAD/CAE fields, especially for IGA (isogeometric analysis). Recently, an automatic hex-mesh generation method has been proposed, which is based on the equivalence among three key concepts: colorable quadrilateral meshes, finite measured foliations and Strebel differentials. This work focuses on the computational aspect of Strebel differentials on high genus surfaces, which is based on graph-valued harmonic mapping. The algorithmic pipeline is as follows: first, the user inputs an admissible curve system, which induces a cylindric-decomposition graph; then the user specifies the lengths of the edges of the graph; third, the algorithm finds the unique harmonic map from the surface to the metric graph by a non-linear heat flow; finally, the harmonic map induces a Strebel differential, which can be further utilized to generate the hex-mesh. The method has solid theoretic foundation, which guarantees the existence and the uniqueness of the graph-valued harmonic map. The algorithm is capable of handling surfaces with complicated topologies, and producing all possible Strebel differentials on the surface. The user has full control of the combinatorial type and the geometry of the Strebel differential. The computational pipeline is automatic. The experimental results show the efficiency and efficacy of the algorithm, and demonstrate the great potential for automating the structured hexahedral mesh generation.

In the past decades, many methods for computing conformal mesh parameterizations have been developed in response to demand of numerous applications in the field of geometry processing. Spectral conformal parameterization (SCP) (Mullen et al. in Proceedings of the symposium on geometry processing, SGP ’08. Eurographics Association, Aire-la-Ville, Switzerland, pp 1487–1494, 2008) is one of these methods used to compute a quality conformal parameterization based on the spectral techniques. SCP focuses on a generalized eigenvalue problem (GEP) $L_{C} \bf{f} = \lambda B \bf{f}$ whose eigenvector(s) associated with the smallest positive eigenvalue(s) provide the conformal parameterization result. This paper is devoted to studying a novel eigensolver for this GEP. Based on structures of the matrix pair $(L_{C}, B)$, we show that this GEP can be transformed into a small-scale compressed and deflated standard eigenvalue problem with a symmetric positive definite skew-Hamiltonian operator. We then propose a symmetric skew-Hamiltonian isotropic Lanczos algorithm (SHILA) to solve the reduced problem. Numerical experiments show that our compressed deflating technique can exclude the impact of convergence from the kernel of $L_{C}$ and transform the original problem to a more robust system. The novel SHILA method can effectively avoid the disturbance of duplicate eigenvalues. As a result, based on the spectral model of SCP, our numerical eigensolver can compute the conformal parameterization accurately and efficiently.

Ricci flow deformsthe Riemannian metric proportionallyto the curvature, such that the curvatureevolves accordingto a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far. This work introduces the unified theoretic framework for discrete Surface Ricci Flow, including all the common schemes: Tangential Circle Packing, Thurston’s Circle Packing, Inversive Distance Circle Packing and Discrete Yamabe Flow. Furthermore, this work also introduces a novel schemes, Virtual Radius Circle Packing and the Mixed Type schemes, under the unified framework. This work gives explicit geometric interpretation to the discrete Ricci energies for all the schemes with all back ground geometries, and the corresponding Hessian matrices. The unified frame work deepens our understanding to the the discrete surface Ricci flow theory, and has inspired us to discover the new schemes, improved the flexibility and robustness of the algorithms, greatly simplified the implementation and improved the efficiency. Experimental results show the unified surface Ricci flow algorithms can handle general surfaces with different topologies, and is robust to meshes with different qualities, and is effective for solving real problems.