Loop Subdivision Surface Research Articles

Loop Subdivision Surface Based Progressive Interpolation

A new method for constructing interpolating Loop subdivision surfaces is presented. The new method is an extension of the progressive interpolation technique for B-splines. Given a triangular mesh M, the idea is to iteratively upgrade the vertices of M to generate a new control mesh M such that limit surface of M would interpolate M. It can be shown that the iterative process is convergent for Loop subdivision surfaces. Hence, the method is well-defined. The new method has the advantages of both a local method and a global method, i.e., it can handle meshes of any size and any topology while generating smooth interpolating subdivision surfaces that faithfully resemble the shape of the given meshes. The meshes considered here can be open or closed.

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.

Bounding the distance between a loop subdivision surface and its limit mesh

Given a control mesh of a Loop subdivision surface, by pushing the control vertices to their limit positions, a limit mesh of the Loop surface is obtained. Compared with the control mesh, the limit mesh is a tighter linear approximation in general, which inscribes the limit surface. We derive an upper bound on the distance between a Loop subdivision surface patch and its limit triangle in terms of the maximum norm of the mixed second differences of the initial control vertices and a constant that depends only on the valence of the patch's extraordinary vertex. A subdivision depth estimation formula for the limit mesh approximation is also proposed.

Curvature Computation for Triangular Meshes based on Local Parametrization

AbstractA local parametrization based curvature computation technique for triangular meshes is presented. The computation process starts with interpolating the region-of-interest of the given mesh with a Loop subdivision surface. The interpolation technique guarantees that the resulting surface reflects the local shape of the mesh, including features such as edges and corners; no data simplification is necessary. A blending technique is then applied to vicinities of extra-ordinary vertices to ensure continuity and boundedness of curvature at each extra-ordinary vertex. This blending process does not change the value of the surface at the extra-ordinary points. Curvatures for each given point are subsequently computed based on standard parametrization for Loop surfaces. Advantages of the new technique include that curvature can be computed in any direction for any point of the given mesh and higher accuracy of the computed results due to precise representation obtained by the interpolation process. Test re...

Interpolation by geometric algorithm

We present a novel geometric algorithm to construct a smooth surface that interpolates a triangular or a quadrilateral mesh of arbitrary topological type formed by n vertices. Although our method can be applied to B-spline surfaces and subdivision surfaces of all kinds, we illustrate our algorithm focusing on Loop subdivision surfaces as most of the meshes are in triangular form. We start our algorithm by assuming that the given triangular mesh is a control net of a Loop subdivision surface. The control points are iteratively updated globally by a simple local point-surface distance computation and an offsetting procedure without solving a linear system. The complexity of our algorithm is O ( m n ) where n is the number of vertices and m is the number of iterations. The number of iterations m depends on the fineness of the mesh and accuracy required.