Artifact analysis on triangular box-splines and subdivision surfaces defined by triangular polyhedra

Surface artifacts are features in a surface which cannot be avoided by movement of control points. They are present in B-splines, box splines and subdivision surfaces. We showed how the subdivision process can be used as a tool to analyse artifacts in surfaces defined by quadrilateral polyhedra ( [Sabin et al., 2005] and [Augsd?rfer et al., 2011]).In this paper we are utilising the subdivision process to develop a generic expression which can be employed to determine the magnitude of artifacts in surfaces defined by any regular triangular polyhedra. We demonstrate the method by analysing box-splines and regular regions of subdivision surfaces based on triangular meshes: Loop subdivision, Butterfly subdivision and a novel interpolating scheme with two smoothing stages. We compare our results for surfaces defined by triangular polyhedra to those for surfaces defined by quadrilateral polyhedra.     

Diffusion equations over arbitrary triangulated surfaces for filtering and texture applications

In computer graphics, triangular mesh representations of surfaces have become very popular. Compared with parametric and implicit forms of surfaces, triangular mesh surfaces have many advantages, such as easy to render, convenient to store and the ability to model geometric objects with arbitrary topology. In this paper, we are interested in data processing over triangular mesh surfaces through PDEs (partial differential equations). We study several diffusion equations over triangular mesh surfaces, and present corresponding numerical schemes to solve them. Our methods work for triangular mesh surfaces with arbitrary geometry (the angles of each triangle are arbitrary) and topology (open meshes or closed meshes of arbitrary genus). Besides the flexibility, our methods are efficient due to the implicit/semi-implicit time discretization. We finally apply our methods to several filtering and texture applications such as image processing, texture generating and regularization of harmonic maps over triangular mesh surfaces. The results demonstrate the flexibility and effectiveness of our methods.     

Empirical mode decomposition on surfaces

Empirical Mode Decomposition (EMD) is a powerful tool for analysing non-linear and non-stationary signals, and has drawn a great deal of attentions in various areas. In this paper, we generalize the classical EMD from Euclidean space to the setting of surfaces represented as triangular meshes. Inspired by the EMD, we also propose a feature-preserving smoothing method based on extremal envelopes. The core of our generalized EMD on surfaces is an envelope computation method that solves a bi-harmonic field with Dirichlet boundary conditions. Experimental results show that the proposed generalization of EMD on surfaces works well. We also demonstrate that the generalized EMD can be effectively utilized in filtering scalar functions defined over surfaces and surfaces themselves.     

Adaptive quasi-interpolating quartic splines

We present an adaptive quasi-interpolating quartic spline construction for regularly sampled surface data. The method is based on a uniform quasi-interpolating scheme, employing quartic triangular patches with C ¹-continuity and optimal approximation order within this class. Our contribution is the adaption of this scheme to surfaces of varying geometric complexity, where the tiling resolution can be locally defined, for example driven by approximation errors. This way, the construction of high-quality spline surfaces is enhanced by the flexibility of adaptive pseudo-regular triangle meshes. Numerical examples illustrate the use of this method for adaptive terrain modeling, where uniform schemes produce huge numbers of patches.     

Shape aware normal interpolation for curved surface shading from polyhedral approximation

Independent interpolation of local surface patches and local normal patches is an efficient way for fast rendering of smooth curved surfaces from rough polyhedral meshes. However, the independently interpolating normals may deviate greatly from the analytical normals of local interpolating surfaces, and the normal deviation may cause severe rendering defects when the surface is shaded using the interpolating normals. In this paper we propose two novel normal interpolation schemes along with interpolation of cubic Bézier triangles for rendering curved surfaces from rough triangular meshes. Firstly, the interpolating normal is computed by a Gregory normal patch to each Bézier triangle by a new definition of quadratic normal functions along cubic space curves. Secondly, the interpolating normal is obtained by blending side-vertex normal functions along side-vertex parametric curves of the interpolating Bézier surface. The normal patches by these two methods can not only interpolate given normals at vertices or boundaries of a triangle but also match the shape of the local interpolating surface very well. As a result, more realistic shading results are obtained by either of the two new normal interpolation schemes than by the traditional quadratic normal interpolation method for rendering rough triangular meshes.     

A local tangential lifting differential method for triangular meshes

In this note we present a local tangential lifting (LTL) algorithm to compute differential quantities for triangular meshes obtained from regular surfaces. First, we introduce a new notation of the local tangential polygon and lift functions and vector fields on a triangular mesh to the local tangential polygon. Then we use the centroid weights proposed by Chen and Wu [4] to define the discrete gradient of a function on a triangular mesh. We also use our new method to define the discrete Laplacian operator acting on functions on triangular meshes. Higher order differential operators can also be computed successively. Our approach is conceptually simple and easy to compute. Indeed, our LTL method also provides a unified algorithm to estimate the shape operator and curvatures of a triangular mesh and derivatives of functions and vector fields. We also compare three different methods : our method, the least square method and Akima’s method to compute the gradients of functions.     

Polynomial surfaces interpolating arbitrary triangulations

Triangular Bezier patches are an important tool for defining smooth surfaces over arbitrary triangular meshes. The previously introduced 4-split method interpolates the vertices of a 2-manifold triangle mesh by a set of tangent plane continuous triangular Bezier patches of degree five. The resulting surface has an explicit closed form representation and is defined locally. In this paper, we introduce a new method for visually smooth interpolation of arbitrary triangle meshes based on a regular 4-split of the domain triangles. Ensuring tangent plane continuity of the surface is not enough for producing an overall fair shape. Interpolation of irregular control-polygons, be that in 1D or in 2D, often yields unwanted undulations. Note that this undulation problem is not particular to parametric interpolation, but also occurs with interpolatory subdivision surfaces. Our new method avoids unwanted undulations by relaxing the constraint of the first derivatives at the input mesh vertices: The tangent directions of the boundary curves at the mesh vertices are now completely free. Irregular triangulations can be handled much better in the sense that unwanted undulations due to flat triangles in the mesh are now avoided.     

A local tangential lifting differential method for triangular meshes

In this note we present a local tangential lifting (LTL) algorithm to compute differential quantities for triangular meshes obtained from regular surfaces. First, we introduce a new notation of the local tangential polygon and lift functions and vector fields on a triangular mesh to the local tangential polygon. Then we use the centroid weights proposed by Chen and Wu [4] to define the discrete gradient of a function on a triangular mesh. We also use our new method to define the discrete Laplacian operator acting on functions on triangular meshes. Higher order differential operators can also be computed successively. Our approach is conceptually simple and easy to compute. Indeed, our LTL method also provides a unified algorithm to estimate the shape operator and curvatures of a triangular mesh and derivatives of functions and vector fields. We also compare three different methods : our method, the least square method and Akima’s method to compute the gradients of functions.     

三角网格上五次组合形式的代数曲面重构

提出了三角网格上代数曲面重构的一种方法。构造三个与任意两条边界GC1光滑拼接,与另一条边界GC0拼接的四次代数曲面,将这三个四次代数曲面分别与相应截面相乘并作线性组合,即可得到与三条边界光滑拼接的一个具有组合形式的五次代数曲面。所构造代数曲面具有二次精度、较好局部性、计算复杂度低、较大灵活性等优点。     

General triangular midpoint subdivision

In this paper, we introduce triangular subdivision operators which are composed of a refinement operator and several averaging operators, where the refinement operator splits each triangle uniformly into four congruent triangles and in each averaging operation, every vertex will be replaced by a convex combination of itself and its neighboring vertices. These operators form an infinite class of triangular subdivision schemes including Loop's algorithm with a restricted parameter range and the midpoint schemes for triangular meshes. We analyze the smoothness of the resulting subdivision surfaces at their regular and extraordinary points by generalizing an established technique for analyzing midpoint subdivision on quadrilateral meshes. General triangular midpoint subdivision surfaces are smooth at all regular points and they are also smooth at extraordinary points under certain conditions. We show some general triangular subdivision surfaces and compare them with Loop subdivision surfaces.     

在任意拓扑三角形网格上的光滑样条曲面

介绍了一种在控制三角形网格上创建光滑样条曲面的算法,该控制网格能够刻画具有或没有边界的任意自由曲面.生成的曲面有一个4次参数多项式表示并且被表示成一个切平面连续的三角形Bézier片网.曲面对网格的逼近程度受到一个混合比控制,当混合比为0时,产生的曲面插值网格.该算法是一种局部方法,简单且效率高,适合于外形设计.     

Bilateral recovering of sharp edges on feature-insensitive sampled meshes

A variety of computer graphics applications sample surfaces of 3D shapes in a regular grid without making the sampling rate adaptive to the surface curvature or sharp features. Triangular meshes that interpolate or approximate these samples usually exhibit relatively big error around the insensitive sampled sharp features. This paper presents a robust general approach conducting bilateral filters to recover sharp edges on such insensitive sampled triangular meshes. Motivated by the impressive results of bilateral filtering for mesh smoothing and denoising, we adopt it to govern the sharpening of triangular meshes. After recognizing the regions that embed sharp features, we recover the sharpness geometry through bilateral filtering, followed by iteratively modifying the given mesh's connectivity to form single-wide sharp edges that can be easily detected by their dihedral angles. We show that the proposed method can robustly reconstruct sharp edges on feature-insensitive sampled meshes.     

G1连续几何偏微分方程Bézier曲面的构造

基于三角形和四边形网格上Laplace-Beltrami算子、高斯曲率和平均曲率的离散及其收敛性分析,提出了一种使用四阶几何流构造几何偏微分方程Bézier曲面的方法.使用该方法构造出的Bézier曲面既具有几何偏微分方程曲面的最优性质,同时又满足G1连续性.算法收敛性的数值实验表明该方法是有效的.     

Detection of Salient Curvature Features on Polygonal Surfaces

We develop an approach for stable detection of perceptually salient curvature features on surfaces approximated by dense triangle meshes. The approach explores an "area degenerating" effect of the focal surface near its singularities and combines together a new approximations of the mean and Gaussian curvatures, nonlinear averaging of curvature maps, histogram-based curvature extrema filtering, and an image processing skeletonization procedure adapted for triangular meshes. Finally we use perceptually significant curvature extrema triangles to enhance the Garland-Heckbert mesh decimation method.     

Boundary Handling at Cloth–Fluid Contact

We present a robust and efficient method for the two‐way coupling between particle‐based fluid simulations and infinitesimally thin solids represented by triangular meshes. Our approach is based on a hybrid method that combines a repulsion force approach with a continuous intersection handling to guarantee that no penetration occurs. Moreover, boundary conditions for the tangential component of the fluid's velocity are implemented to model the different slip conditions. The proposed method is particularly useful for dynamic surfaces, like cloth and thin shells. In addition, we demonstrate how standard fluid surface reconstruction algorithms can be modified to prevent the calculated surface from intersecting close objects. For both the two‐way coupling and the surface reconstruction, we take into account that the fluid can wet the cloth. We have implemented our approach for the bidirectional interaction between liquid simulations based on Smoothed Particle Hydrodynamics (SPH) and standard mesh‐based cloth simulation systems.     

接触面的一致性三角形网格生成算法*

为了简化数值模拟中三角形网格模型之间的接触问题,设计了在接触面上生成一致性三角形网格的算法,基本步骤为构造网格拓扑关系、搜索重叠网格、完善重叠网格、拷贝网格、修补边界空隙。采用面向配对顶点的搜索方法、点投影的判定方法、基于边界环的孔洞搜索方法,准确地确定了接触区域,利用最短对角线的方法修补了边界空隙,较好地生成了一致性网格,并采用三轴分块排序表提高了效率。实验结果表明,该算法准确、高效地实现了接触面上网格的一致性。     

A computational approach to joint line detection on triangular meshes

In this paper, we present formulae for evaluating differential quantities at vertices of triangular meshes that may approximate potential piecewise smooth surfaces with discontinuous normals or discontinuous curvatures at the joint lines. We also define the C ¹ and C ² discontinuity measures for surface meshes using changing rates of one-sided curvatures or changing rates of curvatures across mesh edges. The curvatures are computed discretely as of local interpolating surfaces that lie within a tolerance to the mesh. Together with proper estimation of local shape parameters, the obtained discontinuity measures own properties like sensitivity to salient joint lines and being scale invariant. A simple algorithm is finally developed for detection of C ¹ or C ² discontinuity joint lines on triangular meshes with even highly non-uniform triangulations. Several examples are provided to demonstrate the effectiveness of the proposed method.     

渐进插值的LOOP曲面细分*

目前很多细分方法都存在不能用同一种方法处理封闭网格和开放网格的问题。对此,一种新的基于插值技术的LOOP曲面细分方法,其主要思想就是给定一个初始三角网格M,反复生成新的顶点,新顶点是通过其相邻顶点的约束求解得到的,从而构造一个新的控制网格M,在取极限的情况下,可以证明插值过程是收敛的;因为生成新顶点使用的是与其相连顶点的约束求解得到的,本质上是一种局部方法,所以,该方法很容易定义。它在本地方法和全局方法中都有优势,能处理任意顶点数量和任意拓扑结构的网格,从而产生一个光滑的曲面并忠实于给定曲面的形状,其控制     

三角网格上的代数曲面重建

提出一种用分片代数曲面构造三角曲面片的方法,利用具有公共边的2个三角形区域的4个顶点的函数值以及公共边2个端点的外法向量来构造一个二次曲面V(g)和一个截面V(h),其交V(g,h)即为2个三角曲面片的公共边界曲线.对每个已确定了边界条件的三角片内部进一步划分成3部分,每部分各自定义一个三次代数曲面.这3个三次代数曲面不仅在其交线处光滑拼接,而且分别沿三角形的边界与V(g)光滑拼接,从而构成一个具有GC1连续性的分片代数曲面.对于只属于一个三角片的边界留有一个自由度,可对曲面形状加以控制.     

Mesh generation for spherical and conical surfaces using transfinite interpolation

A computer oriented method is presented which generates meshes of quadrilateral and triangular elements for spherical and conical surfaces using the transfinite interpolation technique. The input data required are very simple. Examples are given to illustrate the range of meshes that can be generated. It is demonstrated that the present method gives better apsect ratios of the elements and avoids the assumption of fictitious radius near the poles.