We propose a novel method to robustly and efficiently compute the maximum allowable step sizes so that the 3D high-order finite elements continuously preserve geometric validity when moving along the given directions with positive step sizes smaller than the computed ones. We transform the problem of finding the maximum allowable step sizes to one of solving roots of cubic polynomials. To use interval arithmetic to avoid numerical issues in cubic equation solving, we completely enumerate the roots of cubic polynomials and apply the interval version of the Newton-Raphson iteration. The effectiveness of our algorithm is demonstrated through extensive testing. Compared to the state-of-the-art method, our algorithm achieves higher efficiency.

We propose a novel method to exactly and efficiently resolve intersections and self-intersections in triangle meshes. Our method contains two key components. First, we present a new concept of geometric predicates, called indirect offset predicates , to represent all intersection points through a new formulation and establish all necessary geometric predicates. Consequently, we reduce numerical errors in floating-point evaluations and improve the success rate of early stages of arithmetic filtering. Second, we develop localization and dimension reduction techniques for sorting, deduplicating, and locating the intersection points, thereby boosting efficiency and parallelism while maintaining accuracy. Rigorous testing confirms the robustness of our algorithm and consistency with previous methods. Comprehensive testing across diverse datasets further highlights the speed improvement achieved by our method, which is one order of magnitude faster than the state-of-the-art methods.

We propose a new structure called a higher-order shell, which is composed of a set of triangular prisms. Each triangular prism is enveloped by three Bézier triangles (top, middle, and bottom) and three side surfaces, each of which is trimmed from a bilinear surface. Moreover, we define a continuous vector field to smoothly and bijectively transfer attributes between two surfaces inside the shell. Since the higher-order shell has several hard construction constraints, we apply an interior-point strategy to robustly and automatically construct a high-order shell for an input mesh. Specifically, the strategy starts from a valid linear shell with a small thickness. Then, the shell is optimized until the specified thickness is reached, where explicit checks ensure that the constraints are always satisfied. We extensively test our method on more than 8300 models, demonstrating its robustness and performance. Compared to state-of-the-art methods, our bijective projection is smoother, and the space between the shell and input mesh is more uniform.

We propose a robust and automatic method to construct manifold cages for 3D triangular meshes. The cage contains hundreds of triangles to tightly enclose the input mesh without self-intersections. To generate such cages, our algorithm consists of two phases: (1) construct manifold cages satisfying the tightness, enclosing, and intersection-free requirements and (2) reduce mesh complexities and approximation errors without violating the enclosing and intersection-free requirements. To theoretically make the first stage have those properties, we combine the conformal tetrahedral meshing and tetrahedral mesh subdivision. The second step is a constrained remeshing process using explicit checks to ensure that the enclosing and intersection-free constraints are always satisfied. Both phases use a hybrid coordinate representation, i.e., rational numbers and floating point numbers, combined with exact arithmetic and floating point filtering techniques to guarantee the robustness of geometric predicates with a favorable speed. We extensively test our method on a data set of over 8500 models, demonstrating robustness and performance. Compared to other state-of-the-art methods, our method possesses much stronger robustness.

Abstract We propose a novel image triangulation method to reduce the complexity of image triangulation under the color error‐bounded constraint and the triangle quality constraint. Meanwhile, we realize a variety of visual effects by supporting different types of triangles (e.g., linear or curved) and color approximation functions (e.g., constant, linear, or quadratic). To adapt to these discontinuous and combinatorial objectives and constraints, we formulate it as a constrained optimization problem that is solved by a series of tailored local remeshing operations. The feasibility and practicability of our method are demonstrated over various types of images, such as organisms, landscapes, portraits and cartoons. Compared to state‐of‐the‐art methods, our method generates far fewer triangles for the same color error or much smaller color errors using the same number of triangles.

Abstract We propose an interactive method to edit a discrete Chebyshev net, which is a quad mesh with edges of the same length. To ensure that the edited mesh is always a discrete Chebyshev net, the maximum difference of all edge lengths should be zero during the editing process. Hence, we formulate an objective function using ℓp‐norm (p > 2) to force the maximum length deviation to approach zero in practice. To optimize the nonlinear and non‐convex objective function interactively and efficiently, we develop a novel second‐order solver. The core of the solver is to construct a new convex majorizer for our objective function to achieve fast convergence. We present two acceleration strategies to further reduce the optimization time, including adaptive p change and adaptive variables reduction. A large number of experiments demonstrate the capability and feasibility of our method for interactively editing complex discrete Chebyshev nets.

Abstract We propose a simple yet effective method to perform surface remeshing with hard constraints, such as bounding approximation errors and ensuring Delaunay conditions. The remeshing is formulated as a constrained optimization problem, where the variables contain the mesh connectivity and the mesh geometry. To solve it effectively, we adopt traditional local operations, including edge split, edge collapse, edge flip, and vertex relocation, to update the variables. Central to our method is an evolutionary vertex optimization algorithm, which is derivative‐free and robust. The feasibility and practicability of our method are demonstrated in two applications, including error‐bounded Delaunay mesh simplification and error‐bounded angle improvement with a given number of vertices, over many models. Compared to state‐of‐the‐art methods, our method achieves higher remeshing quality.