Quading triangular meshes with certain topological constraints

AbstractIn computer graphics and geometric modeling, shapes are often represented by triangular meshes (also called 3D meshes or manifold triangulations). The quadrangulation of a triangular mesh has wide applications. In this paper, we present a novel method of quading a closed orientable triangular mesh into a quasi-regular quadrangulation, i.e., a quadrangulation that only contains vertices of degree four or five. The quasi-regular quadrangulation produced by our method also has the property that the number of quads of the quadrangulation is the smallest among all the quasi-regular quadrangulations. In addition, by constructing the so-called orthogonal system of cycles our method is more effective to control the quality of the quadrangulation

An interactive analysis of harmonic and diffusion equations on discrete 3D shapes

AbstractRecent results in geometry processing have shown that shape segmentation, comparison, and analysis can be successfully addressed through the spectral properties of the Laplace–Beltrami operator, which is involved in the harmonic equation, the Laplacian eigenproblem, the heat diffusion equation, and the definition of spectral distances, such as the bi-harmonic, commute time, and diffusion distances. In this paper, we study the discretization and the main properties of the solutions to these equations on 3D surfaces and their applications to shape analysis. Among the main factors that influence their computation, as well as the corresponding distances, we focus our attention on the choice of different Laplacian matrices, initial boundary conditions, and input shapes. These degrees of freedom motivate our choice to address this study through the executable paper, which allows the user to perform a large set of experiments and select his/her own parameters. Finally, we represent these distances in a unified way and provide a simple procedure to generate new distances on 3D shapes

Adaptive Geometry Images for Remeshing

Geometry images are a kind of completely regular remeshing methods for mesh representation. Traditional geometry images have difficulties in achieving optimal reconstruction errors and preserving manually selected geometric details, due to the limitations of parametrization methods. To solve two issues, we propose two adaptive geometry images for remeshing triangular meshes. The first scheme produces geometry images with the minimum Hausdorff error by finding the optimization direction for sampling points based on the Hausdorff distance between the original mesh and the reconstructed mesh. The second scheme produces geometry images with higher reconstruction precision over the manually selected region-of-interest of the input mesh, by increasing the number of sampling points over the region-of-interest. Experimental results show that both schemes give promising results compared with traditional parametrization-based geometry images

Automatic and Interactive Mesh to T-Spline Conversion

In Geometry Processing, and more specifically in surface approximation, one of the most important issues is the automatic generation of a quad-dominant control mesh from an arbitrary shape (e.g. a scanned mesh). One of the first fully automatic solutions was proposed by Eck and Hoppe in 1996. However, in the industry, designers still use manual tools (see e.g. cyslice). The main difference between a control mesh constructed by an automatic method and the one designed by a human user is that in the second case, the control mesh follows the features of the model. More precisely, it is well known from approximation theory that aligning the edges with the principal directions of curvature improves the smoothness of the reconstructed surface, and this is what designers intuitively do. In this paper, our goal is to automatically construct a control mesh driven by the anisotropy of the shape, mimicking the mesh that a designer would create manually. The control mesh generated by our method can be used by a wide variety of representations (splines, subdivision surfaces...). We demonstrate our method applied to the automatic conversion from a mesh of arbitrary topology into a T-Spline surface. Our method first extracts an initial mesh from a PGP (Periodic Global Parameterization). To facilitate user-interaction, we extend the PGP method to take into account optional user-defined information. This makes it possible to locally tune the orientation and the density of the control mesh. The user can also interactively remove edges or sketch additional ones. Then, from this initial control mesh, our algorithm generates a valid T-Spline control mesh by enforcing some validity constraints. The valid T-Spline control mesh is finally fitted to the original surface, using a classic regularized optimization procedure. To reduce the L-infinity approximation error below a user-defined threshold, we iteratively use the T-Spline adaptive local refinement

Harmonic Functions for Quadrilateral Remeshing of Arbitrary Manifolds

In this paper, we propose a new quadrilateral remeshing method for manifolds of arbitrary genus that is at once general, flexible, and efficient. Our technique is based on the use of smooth harmonic scalar fields defined over the mesh. Given such a field, we compute its gradient field and a second vector field that is everywhere orthogonal to the gradient. We then trace integral lines through these vector fields to sample the mesh. The two nets of integral lines together are used to form the polygons of the output mesh. Curvature-sensitive spacing of the lines provides for anisotropic meshes that adapt to the local shape. Our scalar field construction allows users to exercise extensive control over the structure of the final mesh. The entire process is performed without computing a parameterization of the surface, and is thus applicable to manifolds of any genus without the need for cutting the surface into patches