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 G¹ 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.     

Representing higher-order singularities in vector fields on piecewise linear surfaces

Accurately representing higher-order singularities of vector fields defined on piecewise linear surfaces is a non-trivial problem. In this work, we introduce a concise yet complete interpolation scheme of vector fields on arbitrary triangulated surfaces. The scheme enables arbitrary singularities to be represented at vertices. The representation can be considered as a facet-based "encoding" of vector fields on piecewise linear surfaces. The vector field is described in polar coordinates over each facet, with a facet edge being chosen as the reference to define the angle. An integer called the period jump is associated to each edge of the triangulation to remove the ambiguity when interpolating the direction of the vector field between two facets that share an edge. To interpolate the vector field, we first linearly interpolate the angle of rotation of the vectors along the edges of the facet graph. Then. we use a variant of Nielson's side-vertex scheme to interpolate the vector field over the entire surface. With our representation, we remove the bound imposed on the complexity of singularities that a vertex can represent by its connectivity. This bound is a limitation generally exists in vertex-based linear schemes. Furthermore, using our data structure, the index of a vertex of a vector field can be combinatorily determined. We show the simplicity of the interpolation scheme with a GPU-accelerated algorithm for a LIC-based visualization of the so-defined vector fields, operating in image space. We demonstrate the algorithm applied to various vector fields on curved surfaces.     

Convolution surfaces for arcs and quadratic curves with a varying kernel

A convolution surface is an isosurface in a scalar field defined by convolving a skeleton, comprising of points, curves, surfaces, or volumes, with a potential function. While convolution surfaces are attractive for modeling natural phenomena and objects of complex evolving topology, the analytical evaluation of integrals of convolution models still poses some open problems. This paper presents some novel analytical convolution solutions for arcs and quadratic spline curves with a varying kernel. In addition, we approximate planar higher-degree polynomial spline curves by optimal arc splines within a prescribed tolerance and sum the potential functions of all the arc primitives to approximate the field for the entire spline curve. Published online: November 20, 2002     

Non-Uniform Subdivision Surfaces with Sharp Features

Sharp features are important characteristics in surface modelling. However, it is still a significantly difficult task to create complex sharp features for Non-Uniform Rational B-Splines compatible subdivision surfaces. Current non-uniform subdivision methods produce sharp features generally by setting zero knot intervals, and these sharp features may have unpleasant visual effects. In this paper, we construct a non-uniform subdivision scheme to create complex sharp features by extending the eigen-polyhedron technique. The new scheme allows arbitrarily specifying sharp edges in the initial mesh and generates non-uniform cubic B-spline curves to represent the sharp features. Experimental results demonstrate that the present method can generate visually more pleasant sharp features than other existing approaches.     

A new force-directed graph drawing method based on edge–edge repulsion

The conventional force-directed methods for drawing undirected graphs are based on either vertex–vertex repulsion or vertex–edge repulsion. In this paper, we propose a new force-directed method based on edge–edge repulsion to draw graphs. In our framework, edges are modelled as charged springs, and a final drawing can be generated by adjusting positions of vertices according to spring forces and the repulsive forces, derived from potential fields, among edges. Different from the previous methods, our new framework has the advantage of overcoming the problem of zero angular resolution, guaranteeing the absence of any overlapping of edges incident to the common vertex. Given graph layouts probably generated by previous algorithms as the inputs to our algorithm, experimental results reveal that our approach produces promising drawings not only preserving the original properties of a high degree of symmetry and uniform edge length, but also preventing zero angular resolution and usually having larger average angular resolution. However, it should be noted that exhibiting a higher degree of symmetry and larger average angular resolution does not come without a price, as the new approach might result in the increase in undesirable overlapping of vertices as some of our experimental results indicate. To ease the problem of node overlapping, we also consider a hybrid approach which takes into account both edge–edge and vertex–vertex repulsive forces in drawing a graph.     

Sharpen&Bend: recovering curved sharp edges in triangle meshes produced by feature-insensitive sampling

Various acquisition, analysis, visualization, and compression approaches sample surfaces of 3D shapes in a uniform fashion without any attempt to align the samples with sharp edges or to adapt the sampling density to the surface curvature. Consequently, triangle meshes that interpolate these samples usually chamfer sharp features and exhibit a relatively large error in their vicinity. We present two new filters that improve the quality of these resampled models. EdgeSharpener restores the sharp edges by splitting the chamfer edges and forcing the new vertices to lie on intersections of planes extending the smooth surfaces incident upon these chamfers. Bender refines the resulting triangle mesh using an interpolating subdivision scheme that preserves the sharpness of the recovered sharp edges while bending their polyline approximations into smooth curves. A combined Sharpen&Bend postprocessing significantly reduces the error produced by feature-insensitive sampling processes. For example, we have observed that the mean-squared distortion introduced by the SwingWrapper remeshing-based compressor can often be reduced by 80 percent executing EdgeSharpener alone after decompression. For models with curved regions, this error may be further reduced by an additional 60 percent if we follow the EdgeSharpening phase by Bender.     

Polynomial splines over general T-meshes

The present authors have introduced polynomial splines over T-meshes (PHT-splines) and provided theories and applications for PHT-splines over hierarchical T-meshes. This paper generalizes PHT-splines to arbitrary topology over general T-meshes with any structures (GPT-splines). GPT-spline surfaces can be constructed through a unified scheme to interpolate the local geometric information at the basis vertices of the T-mesh. We also discuss general edge insertion and removal algorithms for GPT-splines. As applications, we present algorithms to construct a GPT-spline surface from a quadrilateral mesh and to simplify a tensor-product B-spline surface into a GPT-spline surface with superfluous edges removal.     

Feature-preserving mesh denoising based on vertices classification

In this paper, we present an effective surface denoising method for noisy surfaces. The two key steps in this method involve feature vertex classification and an iterative, two-step denoising method depending on two feature weighting functions. The classification for feature vertices is based on volume integral invariant. With the super nature of this integral invariant, the features of vertices can be fixed with less influence of noise, and different denoising degrees can be applied to different parts of the pending surface. Compared with other methods, our approach produces better results in feature-preserving.     

Fitting Sharp Features with Loop Subdivision Surfaces

Various 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.     

Generating subdivision surfaces from profile curves

The construction of freeform models has always been a challenging task. A popular approach is to edit a primitive object such that its projections conform to a set of given planar curves. This process is tedious and relies very much on the skill and experience of the designer in editing 3D shapes. This paper describes an intuitive approach for the modeling of freeform objects based on planar profile curves. A freeform surface defined by a set of orthogonal planar curves is created by blending a corresponding set of sweep surfaces. Each of the sweep surfaces is obtained by sweeping a planar curve about a computed axis. A Catmull-Clark subdivision surface interpolating a set of data points on the object surface is then constructed. Since the curve points lying on the computed axis of the sweep will become extraordinary vertices of the subdivision surface, a mesh refinement process is applied to adjust the mesh topology of the surface around the axis points. In order to maintain characteristic features of the surface defined with the planar curves, sharp features on the surface are located and are retained in the mesh refinement process. This provides an intuitive approach for constructing freeform objects with regular mesh topology using planar profile curves.     

Discrete Distortion in Triangulated 3‐Manifolds

We introduce a novel notion, that we call discrete distortion, for a triangulated 3‐manifold. Discrete distortion naturally generalizes the notion of concentrated curvature defined for triangulated surfaces and provides a powerful tool to understand the local geometry and topology of 3‐manifolds. Discrete distortion can be viewed as a discrete approach to Ricci curvature for singular flat manifolds. We distinguish between two kinds of distortion, namely, vertex distortion, which is associated with the vertices of the tetrahedral mesh decomposing the 3‐manifold, and bond distortion, which is associated with the edges of the tetrahedral mesh. We investigate properties of vertex and bond distortions. As an example, we visualize vertex distortion on manifold hypersurfaces in R⁴ defined by a scalar field on a 3D mesh. distance fields.     

Anisotropic mesh adaptation for evolving triangulated surfaces

Dynamic surfaces arise in many applications, such as free surfaces in multiphase flows and moving interfaces in fluid–solid interaction. In many engineering applications, an explicit surface triangulation is often used to represent dynamic surfaces, posing significant challenges in adapting their meshes, especially if large curvatures and sharp features may dynamically emerge or vanish as the surfaces evolve. In this paper, we present an anisotropic mesh adaptation technique to meet these challenges. Our technique strives for optimal aspect ratios of the triangulation to reduce positional errors and to capture geometric features of dynamic surfaces based on a novel extension of the quadrics. Our adaptation algorithm combines the operations of vertex redistribution, edge flipping, edge contraction, and edge splitting. Experimental results demonstrate the effectiveness of our anisotropic adaptation technique for static and dynamic surfaces.     

Robust Feature-Preserving Mesh Denoising Based on Consistent Subneighborhoods

In this paper, we introduce a feature-preserving denoising algorithm. It is built on the premise that the underlying surface of a noisy mesh is piecewise smooth, and a sharp feature lies on the intersection of multiple smooth surface regions. A vertex close to a sharp feature is likely to have a neighborhood that includes distinct smooth segments. By defining the consistent subneighborhood as the segment whose geometry and normal orientation most consistent with those of the vertex, we can completely remove the influence from neighbors lying on other segments during denoising. Our method identifies piecewise smooth subneighborhoods using a robust density-based clustering algorithm based on shared nearest neighbors. In our method, we obtain an initial estimate of vertex normals and curvature tensors by robustly fitting a local quadric model. An anisotropic filter based on optimal estimation theory is further applied to smooth the normal field and the curvature tensor field. This is followed by second-order bilateral filtering, which better preserves curvature details and alleviates volume shrinkage during denoising. The support of these filters is defined by the consistent subneighborhood of a vertex. We have applied this algorithm to both generic and CAD models, and sharp features, such as edges and corners, are very well preserved.     

应用NURBS曲面磨光多面体

本文应用ＮＵＲＢＳ曲面磨光多面体,产生了处处Ｃ′连续的过渡曲面．多面体的磨光分为边的磨光和顶点的磨光两种情形,边的磨光相对较容易,而顶点的磨光则很困难．本文所采用的应用ＮＵＲＢＳ曲面磨光多面体的顶点和边的方法,不仅可以统一实现二者的磨光操作,而且方法简单且统一,产生了Ｃ′连续的过渡面．较之以前的方法,首先,利用ＮＵＲＢＳ曲面可以精确地描述对边磨光所用的柱面（等半径或非等半径）,其次,在对顶点的磨光中,同以往的方法不同,将与该顶点相邻的边的过渡曲面相互分离,并首次引入了“补面”的概念,使得对该点所产生的过渡曲面处处Ｃ′连续．本算法首先构造用以磨光多面体顶点和边的ＮＵＲＢＳ曲面的边界曲线网络图产生边界曲线的控制点及其权值（ＮＵＲＢＳ表示）,然后依据连续性准则,产生ＮＵＲＢＳ曲面的控制信息．     

Locally toleranced surface simplification

We present a technique for simplifying a triangulated surface. Simplifying consists of approximating the surface with another surface of lower triangle count. Our algorithm can preserve the volume of a solid to within machine accuracy; it favors the creation of near-equilateral triangles. We develop novel methods for reporting and representing a bound to the approximation error between a simplified surface and the original, and respecting a variable tolerance across the surface. A different positive error value is reported at each vertex. By linearly blending the error values in between vertices, we define a volume of space, called the error volume, as the union of balls of linearly varying radii. The error volume is built dynamically as the simplification progresses, on top of preexisting error volumes that it contains. We also build a tolerance volume to forbid simplification errors exceeding a local tolerance. The information necessary to compute error values is local to the star of a vertex; accordingly, the complexity of the algorithm is either linear or in O(n log n) in the original number of surface edges, depending on the variant. We extend the mechanisms of error and tolerance volumes to preserve during simplification scalar and vector attributes associated with surface vertices. Assuming a linear variation across triangles, error and tolerance volumes are defined in the same fashion as for positional error. For normals, a corrective term is applied to the error measured at the vertices to compensate for nonlinearities     

基于边折叠和质点-弹簧模型的网格简化优化算法

通过边折叠实现网格曲面简化,提出了保持曲面特征的边折叠基本规则,引入边折叠顺序控制因子λ,给出了折叠点坐标获取方法,简化过程中网格边长度趋于均匀．在曲面简化基础上,利用质点-弹簧模型优化网格形状．将网格顶点邻域参数化到二维域上,在质点-弹簧模型中引入约束弹簧,约束调整网格顶点,并逆映射到三维原始曲面上,局部优化网格顶点的相邻网格;调整曲面上所有网格顶点,在全局上优化网格形状．在曲面简化优化过程中,建立原始模型曲面和简化优化后曲面之间的双向映射关系;曲面的网格顶点始终在原始模型表面上滑动,并以双向Hausdorff距离衡量、控制曲面间的形状误差．应用实例表明：文中算法稳定、高效,适合于任意复杂的二维流形网格．     

Error Compensation in Leaf Power Problems

The k-Leaf Power recognition problem is a particular case of graph power problems: For a given graph it asks whether there exists an unrooted tree—the k-leaf root—with leaves one-to-one labeled by the graph vertices and where the leaves have distance at most k iff their corresponding vertices in the graph are connected by an edge. Here we study "error correction" versions of k-Leaf Power recognition—that is, adding or deleting at most l edges to generate a graph that has a k-leaf root. We provide several NP-completeness results in this context, and we show that the NP-complete Closest 3-Leaf Power problem (the error correction version of 3-Leaf Power) is fixed-parameter tractable with respect to the number of edge modifications or vertex deletions in the given graph. Thus, we provide the seemingly first nontrivial positive algorithmic results in the field of error compensation for leaf power problems with k > 2. To this end, as a result of independent interest, we develop a forbidden subgraph characterization of graphs with 3-leaf roots.     

A direct approach for subdivision surface fitting from a dense triangle mesh

This article presents a new and direct approach for fitting a subdivision surface from an irregular and dense triangle mesh of arbitrary topological type. All feature edges and feature vertices of the original mesh model are first identified. A topology- and feature-preserving mesh simplification algorithm is developed to further simplify the dense triangle mesh into a coarse mesh. A subdivision surface with exactly the same topological structure and sharp features as that of the simplified mesh is finally fitted from a subset of vertices of the original dense mesh. During the fitting process, both the position masks and subdivision rules are used for setting up the fitting equation. Some examples are provided to demonstrate the proposed approach.     

Interpolation over arbitrary topology meshes using a two-phase subdivision scheme

The construction of a smooth surface interpolating a mesh of arbitrary topological type is an important problem in many graphics applications. This paper presents a two-phase process, based on a topological modification of the control mesh and a subsequent Catmull-Clark subdivision, to construct a smooth surface that interpolates some or all of the vertices of a mesh with arbitrary topology. It is also possible to constrain the surface to have specified tangent planes at an arbitrary subset of the vertices to be interpolated. The method has the following features: 1) it is guaranteed to always work and the computation is numerically stable, 2) there is no need to solve a system of linear equations and the whole computation complexity is O(K) where K is the number of the vertices, and 3) each vertex can be associated with a scalar shape handle for local shape control. These features make interpolation using Catmull-Clark surfaces simple and, thus, make the new method itself suitable for interactive free-form shape design.