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 novel method to generate a high‐order triangular mesh for an input 2D domain with two key characteristics: (1) the mesh precisely conforms to a set of input piecewise rational domain curves, and (2) the geometric map on each curved triangle is injective. Central to the algorithm is a new sufficient condition for placing control points of a rational Bézier triangle to guarantee that the conformance and injectivity constraints are theoretically satisfied. Taking advantage of this condition, we provide an explicit construct that robustly creates higher‐order 2D meshes satisfying the two characteristics. We demonstrate the robustness and effectiveness of our algorithm over a data set containing 2200 examples.

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.

Abstract Research on tooth surface design based on deep neural networks has recently achieved progress in terms of both accuracy and execution efficiency. However, unrealistic outputs are still a challenging issue, partially because of (1) the lack of semantic guidance, (2) the inability to discover and rectify false results, and (3) the lack of exploration of structural coherence in intermediate layers. In this paper, we present an approach to predict depth images for designed teeth based on a conditional generative adversarial network (CGAN) by incorporating semantic guidance. Moreover, the uncertainty of semantic inference is employed to improve the model outputs, and a structural coherence loss is proposed for adversarial learning to enhance the discrimination capability of the network in intermediate layers. We evaluate the performance of our approach with the Shining3D tooth dataset. The experimental results show that our method produces better results than the other available approaches in terms of accuracy.

Low isometric distortion is often required for mesh parameterizations. A configuration of some vertices, where the distortion is concentrated, provides a way to mitigate isometric distortion, but determining the number and placement of these vertices is non-trivial. We call these vertices distortion points. We present a novel and automatic method to detect distortion points using a voting strategy. Our method integrates two components: candidate generation and candidate voting. Given a closed triangular mesh, we generate candidate distortion points by executing a three-step procedure repeatedly: (1) randomly cut an input to a disk topology; (2) compute a low conformal distortion parameterization; and (3) detect the distortion points. Finally, we count the candidate points and generate the final distortion points by voting. We demonstrate that our algorithm succeeds when employed on various closed meshes with a genus of zero or higher. The distortion points generated by our method are utilized in three applications, including planar parameterization, semi-automatic landmark correspondence, and isotropic remeshing. Compared to other state-of-the-art methods, our method demonstrates stronger practical robustness in distortion point detection.

Abstract We present a novel method to construct short cuts for parameterizations with low isometric distortion. The algorithm contains two steps: (i) detect feature points, where the distortion is usually concentrated; and (ii) construct a cut by connecting the detected feature points. Central to each step is a greedy method. After generating a redundant feature point set, a greedy filtering process is performed to identify the feature points required for low isometric distortion parameterizations. This filtering process discards the feature points that are useless for distortion reduction while still enabling us to obtain low isometric distortion. Next, we formulate the process of connecting the detected feature points as a Steiner tree problem. To find an approximate solution, we first successively and greedily produce a collection of auxiliary points. Then, a cut is constructed by connecting the feature points and auxiliary points. In the 26,299 test cases in which an exact solution to the Steiner tree problem is available, the length of the cut obtained by our method is on average 0.17% longer than optimal. Compared to state‐of‐the‐art cut construction methods, our method is one order of magnitude faster and generates shorter cuts while achieving similar isometric distortion.

We present a novel algorithm to refine an input atlas with bounded packing efficiency. Central to this method is the use of the axis-aligned structure that converts the general polygon packing problem to a rectangle packing problem, which is easier to achieve high packing efficiency. Given a parameterized mesh with no flipped triangles, we propose a new angle-driven deformation strategy to transform it into a set of axis-aligned charts, which can be decomposed into rectangles by the motorcycle graph algorithm. Since motorcycle graphs are not unique, we select the one balancing the trade-off between the packing efficiency and chart boundary length, while maintaining bounded packing efficiency. The axis-aligned chart often contains greater distortion than the input, so we try to reduce the distortion while bounding the packing efficiency and retaining bijection. We demonstrate the efficacy of our method on a data set containing over five thousand complex models. For all models, our method is able to produce packed atlases with bounded packing efficiency; for example, when the packing efficiency bound is set to 80%, we elongate the boundary length by an average of 78.7% and increase the distortion by an average of 0.0533%. Compared to state-of-the-art methods, our method is much faster and achieves greater packing efficiency.

Compatible remeshing provides meshes with common connectivity structures. The existing compatible remeshing methods usually suffer from high computational cost or poor quality. In this paper, we present a fast method for computing compatible meshes with high quality. Given two closed, oriented, and topologically equivalent surfaces and a sparse set of corresponding landmarks, we first compute a bijective inter-surface mapping, from which compatible meshes are generated. We then improve the remeshing quality by using a volume-enhanced optimization. In contrast to previous work, our method designs a fast volume-enhanced improvement procedure that directly reduces the isometric distortion of the map between the compatible meshes. Our method also tries to preserve the shapes of the input meshes by projecting the vertices of the compatible meshes onto the input surfaces. Central to this approach is the use of the monotone preconditioned conjugate gradient method, which minimizes the energies effectively and efficiently. Compared with state-of-the-art methods, our method performs about one order of magnitude faster with better remeshing quality. We demonstrate the efficiency and efficacy of our method using various model pairs.