Initial Control Mesh Research Articles

Stochastic Geometric Iterative Method for Loop Subdivision Surface Fitting

In this paper, we propose a stochastic geometric iterative method (S-GIM) to approximate the high-resolution 3D models by finite loop subdivision surfaces. Given an input mesh as the fitting target, the initial control mesh is generated using the mesh simplification algorithm. Then, our method adjusts the control mesh iteratively to make its finite loop subdivision surface approximate the input mesh. In each geometric iteration, we randomly select part of points on the subdivision surface to calculate the difference vectors and distribute the vectors to the control points. Finally, the control points are updated by adding the weighted average of these difference vectors. We prove the convergence of S-GIM and verify it by demonstrating error curves in the experiment. In addition, compared with existing geometric iterative methods, S-GIM has a shorter running time under the same number of iteration steps.

An Interactive Modeling Method for Complex Hexahedral Mesh Generation with Volumetric Subdivision

The existing methods are difficult to construct hexahedral meshes. To address this problem, an interactive construction method of complex hexahedral mesh model based on volumetric subdivision is presented. The user firstly needs to interactively construct the model skeleton, by placing cubes at the nodes of the skeleton structure, and performing interactive operations such as rotation, translation, and scaling of the node cubes. Then, through the connection between the nodes and the topological split operation, the initial control mesh can be constructed. Further, interpolatory Catmull-Clark volumetric subdivision method is used to generate hexahedral meshes with different resolutions. Finally, the padding operation is used to eliminate the degraded elements at the boundary and improve the quality of the hexahedral mesh elements to obtain the final hexahedral mesh. The results of numerical example show that the method can easily and efficiently generate hexahedral meshes interactively, and eliminates the intermediate steps of generating volume meshes from surface meshes. It has some practical applications in finite element analysis, isogeometric analysis and geometric modeling in animation.

Non-uniform interpolatory subdivision schemes with improved smoothness

Subdivision schemes are used to generate smooth curves or surfaces by iteratively refining an initial control polygon or mesh. We focus on univariate, linear, binary subdivision schemes, where the vertices of the refined polygon are computed as linear combinations of the current neighbouring vertices. In the classical stationary setting, there are just two such subdivision rules, which are used throughout all subdivision steps to construct the new vertices with even and odd indices, respectively. These schemes are well understood and many tools have been developed for deriving their properties, including the smoothness of the limit curves. For non-stationary schemes, the subdivision rules are not fixed and can be different in each subdivision step. Non-uniform schemes are even more general, as they allow the subdivision rules to be different for every new vertex that is generated by the scheme. The properties of non-stationary and non-uniform schemes are usually derived by relating the scheme to a corresponding stationary scheme and then exploiting the fact that the properties of the stationary scheme carry over under certain proximity conditions. In particular, this approach can be used to show that the limit curves of a non-stationary or non-uniform scheme are as smooth as those of a corresponding stationary scheme. In this paper we show that non-uniform subdivision schemes have the potential to generate limit curves that are smoother than those of stationary schemes with the same support size of the subdivision rule. For that, we derive interpolatory 2-point and 4-point schemes that generate C1 and C2 limit curves, respectively. These values of smoothness exceed the smoothness of classical interpolating schemes with the same support size by one.

Isogeometric finite element approximation of minimal surfaces based on extended loop subdivision

In this paper, we investigate the formulation of isogeometric analysis for minimal surface models on planar bounded domains by extended Loop surface subdivision approach. The exactness of the physical domain of interest is fixed on the coarsest level of the triangular discretization with any topological structure, which is thought of as the initial control mesh of Loop subdivision. By performing extended Loop subdivision, the control mesh can be repeatedly refined, and the geometry is described as an infinite set of quartic box-spline while maintaining its original exactness. The limit function representation of extended Loop subdivision forms our finite element space, which possesses C1 smoothness and the flexibility of mesh topology. We establish its inverse inequalities which resemble the ones of general finite element spaces. We develop the approximation estimate with the aid of H1 convergence property of the corresponding linear models. It enables us to overcome the difficulty of proving the boundedness of the gradient of finite element solutions appearing in the coefficient of minimal surface models. Numerical examples are given with the comparison to the classical linear finite element method which is consistent with our theoretical results.

Error bounds for Loop subdivision surfaces

In this paper, we obtain the error bounds on the distance between a Loop subdivision surface and its control mesh. Both local and global bounds are derived by means of computing and analysing the control meshes with two rounds of refinement directly. The bounds can be expressed with the maximum edge length of all triangles in the initial control mesh. Our results can be used as posterior estimates and also can be used to predict the subdivision depth for any given tolerance.

Slicing Contour Generation of Z-Constant Contour Machining for Subdivision Surface

For the Z-constant contour machining, a slicing contour generation procedure for subdivision surface is presented. Subdivision surface defines a smooth surface as the limit of a sequence of successive refinements starting from an initial control mesh. It can easily generate smooth surfaces of complex arbitrary topology without intricate patch stitching and curve trimming. Instead of randomly intersecting each facet with a slicing plane and having to construct the slicing contour later, this paper takes full use of the topological information of subdivision surface to construct the slicing contour. The contour information is collected as marching from one effective edge to neighboring effective edge. Example shows our method is obviously easier and the complexity has been reduced

2215 メッシュデータに対する高品質細分割曲面の当てはめ(OS4-2 自由曲面モデリング,OS4 デジタルエンジニアリング)

We present a method to construct a control mesh of a high quality subdivision surface. Our method assumes that the given triangular mesh is a control net of a Loop subdivision surface. The initial mesh is simplified using Quadric Error Metrics (QEM) algorithm. The initial control mesh is obtained by fitting simplified mesh using weights for each valence of points. The control points are iteratively updated using coefficients which are obtained by calculating weights for one control point of Box Spline. We demonstrate effectiveness of the presented method for closed objects.

Fitting Sharp Features with Loop Subdivision Surfaces

AbstractVarious methods have been proposed for fitting subdivision surfaces to different forms of shape data (e.g., dense meshes or point clouds), but none of these methods effectively deals with shapes with sharp features, that is, creases, darts and corners. We present an effective method for fitting a Loop subdivision surface to a dense triangle mesh with sharp features. Our contribution is a new exact evaluation scheme for the Loop subdivision with all types of sharp features, which enables us to compute a fitting Loop subdivision surface for shapes with sharp features in an optimization framework. With an initial control mesh obtained from simplifying the input dense mesh using QEM, our fitting algorithm employs an iterative method to solve a nonlinear least squares problem based on the squared distances from the input mesh vertices to the fitting subdivision surface. This optimization framework depends critically on the ability to express these distances as quadratic functions of control mesh vertices using our exact evaluation scheme near sharp features. Experimental results are presented to demonstrate the effectiveness of the method.

Surface interpolation of meshes by geometric subdivision

Subdivision surfaces are generated by repeated approximation or interpolation from initial control meshes. In this paper, two new non-linear subdivision schemes, face based subdivision scheme and normal based subdivision scheme, are introduced for surface interpolation of triangular meshes. With a given coarse mesh more and more details will be added to the surface when the triangles have been split and refined. Because every intermediate mesh is a piecewise linear approximation to the final surface, the first type of subdivision scheme computes each new vertex as the solution to a least square fitting problem of selected old vertices and their neighboring triangles. Consequently, sharp features as well as smooth regions are generated automatically. For the second type of subdivision, the displacement for every new vertex is computed as a combination of normals at old vertices. By computing the vertex normals adaptively, the limit surface is G^1 smooth. The fairness of the interpolating surface can be improved further by using the neighboring faces. Because the new vertices by either of these two schemes depend on the local geometry, but not the vertex valences, the interpolating surface inherits the shape of the initial control mesh more fairly and naturally. Several examples are also presented to show the efficiency of the new algorithms.

Subdivision surfaces for CAD—an overview

Subdivision surfaces refer to a class of modelling schemes that define an object through recursive subdivision starting from an initial control mesh. Similar to B-splines, the final surface is defined by the vertices of the initial control mesh. These surfaces were initially conceived as an extension of splines in modelling objects with a control mesh of arbitrary topology. They exhibit a number of advantages over traditional splines. Today one can find a variety of subdivision schemes for geometric design and graphics applications. This paper provides an overview of subdivision surfaces with a particular emphasis on schemes generalizing splines. Some common issues on subdivision surface modelling are addressed. Several key topics, such as scheme construction, property analysis, parametric evaluation and subdivision surface fitting, are discussed. Some other important topics are also summarized for potential future research and development. Several examples are provided to highlight the modelling capability of subdivision surfaces for CAD applications.

Subdivision Surfaces for CAD

AbstractSubdivision surfaces refer to a class of modelling schemes that define an object through recursive subdivision starting from an initial control mesh. Similar to B-splines, the final surface is defined by the vertices of the initial control mesh. These surfaces were initially conceived as an extension of splines in modelling objects with a control mesh of arbitrary topology. They exhibit a number of advantages over traditional splines. Today one can find a variety of subdivision schemes for geometric design and graphics applications. This paper provides an overview of subdivision surfaces with a particular emphasis on schemes generalizing splines. Some common issues on subdivision surfaces modelling are addressed. Several key topics, such as scheme construction, property analysis and parametric evaluation, are discussed. Some other important topics are also summarized for potential future research and development.

A novel FEM-based dynamic framework for subdivision surfaces

Recursive subdivision on an initial control mesh generates a visually pleasing smooth surface in the limit. Nevertheless, users must carefully specify the initial mesh and/or painstakingly manipulate the control vertices at different levels of subdivision hierarchy to satisfy a diverse set of functional requirements and aesthetic criteria in the limit shape. This modeling drawback results from the lack of direct manipulation tools for the limit geometric shape. To improve the efficiency of interactive geometric modeling and engineering design, in this paper we integrate novel physics-based modeling techniques with powerful geometric subdivision principles, and develop a unified finite element method (FEM)-based methodology for arbitrary subdivision schemes. Strongly inspired by the recent research on Dynamic Non-Uniform Rational B-Splines (D-NURBS), we formulate and develop a dynamic framework that permits users to directly manipulate the limit surface obtained from any subdivision procedure via simulated “force” tools. The most significant contribution of our unified approach is the formulation of the limit surface of an arbitrary subdivision scheme as being composed of a single type of novel finite element. The specific geometric and dynamic features of our subdivision-based finite elements depend on the subdivision scheme used. We present our novel FEM for the modified butterfly and Catmull–Clark subdivision schemes, and generalize our dynamic framework to be applicable to other subdivision schemes. Our FEM-based approach significantly advances the state-of-the-art in physics-based geometric modeling since it provides a universal physics-based framework for any subdivision scheme. In addition, we systematically devise a mechanism that allows users to directly (not via control meshes) deform any subdivision surface; finally, we represent the limit surface of any subdivision scheme using a collection of subdivision-based novel finite elements. Our experiments demonstrate that the new unified FEM-based framework not only promises a greater potential for subdivision techniques in solid modeling, finite element analysis, and engineering design, but that it will further foster the applicability of subdivision geometry in a wide range of visual computing applications such as visualization, virtual reality, computer graphics, computer vision, robotics, and medical imaging as well.