基于边长的三维形状插值

形状插值在计算机图形学和几何处理中是一个极其重要而基础的问题,在计算机动画等领域有 着广泛应用。注意到在平面三角网格和三维四面体网格插值问题中,对边长平方插值等价于对回拉度量进行插 值,因此具有等距扭曲和共形扭曲同时有界的良好性质。通过将其推广至曲面三角网格,提出了一种完全基于 边长的曲面三角网格插值算法。给定边长,在重建网格阶段,使用牛顿法对边长误差能量进行优化。并且给出 了其海森矩阵的解析正定化形式,从而避免了高代价的特征值分解步骤。注意到四面体网格的边长平方插值结 果具有极低曲率,意味着只需少许修改即可将其压平从而嵌入三维空间。因此提出先将曲面三角网格四面体化, 再从四面体网格的插值结果提取表面。然后将这表面作为初始化用于边长误差能量的牛顿迭代,从而使得收敛 结果更加接近全局最优。在一系列三角网格上进行了实验,结果说明了本文方法比之前方法的边长误差更小, 且得到的结果还是有界扭曲的。     

Automatic mesh generation scheme for a two- or three-dimensional triangular curved surface

Two different methods of automatic mesh generation—the area system and the isoparametric coordinate system—both using quadratic shape function, have been introduced to generate meshes for curved triangular surfaces. Depending on geometrical and material variations, part of the region to be discretized is manually divided into a number of triangular zones. An algorithm is given to generate meshes of triangular elements with three or six nodes automatically using quadratic shape function and area-coordinate system for these zones. Any node number can be assigned to the vertices of a triangular zone. Different zones will be tied together automatically by using a merge algorithm. The application of the volume-coordinate system and quadratic shape function in generating solid elements for triangular prisms will also be discussed.     

三角网格上五次齐次代数曲面的重构

提出三角网格上重建代数曲面的一种方法,利用三次控制曲面来构造五次具有"齐次"形式的GC<'1>光滑曲面,所构造的代数曲面具有2次精度、局部性好、计算量低、自由参数几何意义明确的优点;而且这个五次代数曲面在与一簇特殊的平面相交时,交线为一个四次代数曲线和一条直线,从而化简了这类曲面参数化的计算量.     

基于Java Servlet方式的等值线生成系统的设计与实现

本文研究了webGIS下基于Servlet方式等值线的绘制方法。在讨论了等值线生成的基本算法，包括离散点的网络化、等值点的确定与搜索、未知点的插值以及使用矩形网格进行边界裁剪等方法的基础上，设计和实现了基于Servlet方式的等值线生成系统。     

A Fictitious Domain Method with Distributed Lagrange Multiplier for Parabolic Problems With Moving Interfaces

In this paper, we study the fictitious domain method with distributed Lagrange multiplier for the jump-coefficient parabolic problems with moving interfaces. The equivalence between the fictitious domain weak form and the standard weak form of a parabolic interface problem is proved, and the uniform well-posedness of the full discretization of fictitious domain finite element method with distributed Lagrange multiplier is demonstrated. We further analyze the convergence properties for the fully discrete finite element approximation in the norms of \(L^2\), \(H^1\) and a new energy norm. On the other hand, we introduce a subgrid integration technique in order to allow the fictitious domain finite element method to be performed on the triangular meshes without doing any interpolation between the authentic domain and the fictitious domain. Numerical experiments confirm the theoretical results, and show the good performances of the proposed schemes.     

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.     

Approximate on‐Surface Distance Computation using Quasi‐Developable Charts

There is a vast number of applications that require distance field computation over triangular meshes. State‐of‐the‐art algorithms have quadratic or sub‐quadratic worst‐case complexity, making them impractical for interactive applications. While most of the research on this subject has been focused on reducing the computation complexity of the algorithms, in this work we propose an approximate algorithm that achieves similar results working in lower resolutions of the input meshes. The creation of lower resolution meshes is the essence of our proposal. The idea is to identify regions on the input mesh that can be unfolded into planar regions with minimal area distortion (i.e. quasi‐developable charts). Once charts are computed, their interior is re‐triangulated to reduce the number of triangles, which results in a collection of simplified charts that we call a base mesh. Due to the properties of quasi‐developable regions, we are able to compute distance fields over the base mesh instead of over the input mesh. This reduces the memory footprint and data processed for distance computations, which is the bottleneck of these algorithms. We present results that are one order of magnitude faster than current exact solutions, with low approximation errors.     

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.     

Development of feature segmentation algorithms for quadratic surfaces

Segmentation is a process of partitioning the data in a triangular model to extract the feature regions for use in surface reconstruction. Quadratic surfaces are among the common entities in typical CAD models and should be reconstructed accurately. The purpose of this study is to develop a method for segmenting quadratic features from triangular meshes. The proposed process is primarily composed of two steps. In the first step, a region growing is developed to search for a small area near a seed point to determine the feature type, which can either be a plane, a spherical surface, a cylindrical surface or a conical surface. In the second step, a re-growing procedure is employed to search for the points of the same feature type. Moreover, an automatic algorithm is proposed to extract all planar regions for complex triangular models. The feasibility and limitations of the proposed method are demonstrated by real range data with various quadratic surfaces.     

Automatic construction of quadrilateral network of curves from triangular meshes

The aim of this study is to propose a method for building quadrilateral network of curves automatically from a huge number of triangular meshes. The curve net can be served as the framework of automatic surface reconstruction. The proposed method mainly includes three stages: mesh simplification, quadrangulation and curve net generation. Mesh simplification is employed to reduce the number of meshes in accordance with a quadratic error metric for each vertex. Additional post-processing criteria are also employed to improve the shape of the reduced meshes. For quadrangulation, a front composed of a sequence of edges is introduced. An algorithm is proposed to combine each pair of triangles along the front. A new front is then formed and quadrangulation is continued until all triangles are combined or converted. For curve net generation, each edge of quadrilateral meshes is projected onto the triangular meshes to acquire a set of slicing points first. A constrained curve fitting is then employed to convert all sets of slicing points into B-spline curves, with appropriate continuity conditions across adjacent curves. Several examples have been presented to demonstrate the feasibility of the proposed method and its application in automatic surface reconstruction.     

Metamorphosis of arbitrary triangular meshes

Three-dimensional metamorphosis (or morphing) establishes a smooth transition from a source object to a target object. The primary issue in 3D metamorphosis is to establish surface correspondence between the source and target objects, by which each point on the surface of the source object maps to a point on the surface of the target object. Having established this correspondence, we can generate a smooth transition by interpolating corresponding points from the source to the target positions. We handle 3D geometric metamorphosis between two objects represented as triangular meshes. To improve the quality of 3D morphing between two triangular meshes, we particularly consider the following two issues: 1) metamorphosis of arbitrary meshes; 2) metamorphosis with user control. We can address the first issue using our recently proposed method based on harmonic mapping (T. Kanai et al., 1998). In that earlier work, we developed each of the two meshes (topologically equivalent to a disk and having geometrically complicated shapes), into a 2D unit circle by harmonic mapping. Combining those two embeddings produces surface correspondence between the two meshes. However, this method doesn't consider the second issue: how to let the user control surface correspondence. The article develops an effective method for 3D morphing between two arbitrary meshes of the same topology. We extend our previously proposed method to achieve user control of surface correspondence     

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。     

Accuracy enhancement for non-isoparametric finite-element simulations in curved domains; application to fluid flow

Among a few known techniques the isoparametric version of the finite element method for meshes consisting of curved triangles or tetrahedra is the one most widely employed to solve PDEs with essential conditions prescribed on curved boundaries. It allows to recover optimal approximation properties that hold for elements of order greater than one in the energy norm for polytopic domains. However, besides a geometric complexity, this technique requires the manipulation of rational functions and the use of numerical integration. We consider a simple alternative to deal with Dirichlet boundary conditions that bypasses these drawbacks, without eroding qualitative approximation properties. In the present work we first recall the main principle this technique is based upon, by taking as a model the solution of the Poisson equation with quadratic Lagrange finite elements. Then we show that it extends very naturally to viscous incompressible flow problems. Although the technique applies to any higher order velocity–pressure pairing, as an illustration a thorough study thereof is conducted in the framework of the Stokes system solved by the classical Taylor–Hood method.     

Mesh massage

We present a general framework for post-processing and optimizing surface meshes with respect to various target criteria. On the one hand, the framework allows us to control the shapes of the mesh triangles by applying simple averaging operations; on the other hand we can control the Hausdorff distance to some reference geometry by minimizing a quadratic energy. Due to the simplicity of this setup, the framework is efficient and easy to implement, yet it also constitutes an effective and versatile tool with a variety of possible applications. In particular, we use it to reduce the texture distortion in animated mesh sequences, to improve the results of cross-parameterizations, and to minimize the distance between meshes and their remeshes.     

Multiresolution analysis on irregular surface meshes

Wavelet-based methods have proven their efficiency for visualization at different levels of detail, progressive transmission, and compression of large data sets. The required core of all wavelet-based methods is a hierarchy of meshes that satisfies subdivision-connectivity. This hierarchy has to be the result of a subdivision process starting from a base mesh. Examples include quadtree uniform 2D meshes, octree uniform 3D meshes, or 4-to-1 split triangular meshes. In particular, the necessity of subdivision-connectivity prevents the application of wavelet-based methods on irregular triangular meshes. In this paper, a “wavelet-like” decomposition is introduced that works on piecewise constant data sets over irregular triangular surface meshes. The decomposition/reconstruction algorithms are based on an extension of wavelet-theory allowing hierarchical meshes without property. Among others, this approach has the following features: it allows exact reconstruction of the data set, even for nonregular triangulations, and it extends previous results on Haar-wavelets over 4-to-1 split triangulations     

An Unconditionally Stable Quadratic Finite Volume Scheme over Triangular Meshes for Elliptic Equations

In this note, we present and analyze a special quadratic finite volume scheme over triangular meshes for elliptic equations. The scheme is designed with the second degree Gauss points on the edges and the barycenters of the triangle elements. With a novel from-the-trial-to-test-space mapping, the inf–sup condition of the scheme is shown to hold independently of the minimal angle of the underlying mesh. As a direct consequence, the \(H^1\) norm error of the finite volume solution is shown to converge with the optimal order.     

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.     

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.     

A new intrinsic numerical method for PDE on surfaces

In this note, we introduce a simple, effective numerical method, the local tangential lifting method, for solving partial differential equations for scalar- and vector-valued data defined on surfaces. Even though we follow the traditional way to approximate the regular surfaces under consideration by triangular meshes, the key idea of our algorithm is to develop an intrinsic and unified way to compute directly the partial derivatives of functions defined on triangular meshes. We present examples in computer graphics and image processing applications.     

Stable Real-Time Surgical Cutting Simulation of Deformable Objects Embedded with Arbitrary Triangular Meshes

Surgical simulators need to simulate deformation and cutting of deformable objects. Adaptive octree mesh based cutting methods embed the deformable objects into octree meshes that are recursively refined near the cutting tool trajectory. Deformation is only applied to the octree meshes; thus the deformation instability problem caused by degenerated elements is avoided. Biological tissues and organs usually contain complex internal structures that are ignored by previous work. In this paper the deformable objects are modeled as voxels connected by links and embedded inside adaptive octree meshes. Links swept by the cutting tool are disconnected and object surface meshes are reconstructed from disconnected links. Two novel methods for embedding triangular meshes as internal structures are proposed. The surface mesh embedding method is applicable to arbitrary triangular meshes, but these meshes have no physical properties. The material sub-region embedding method associates the interiors enclosed by the triangular meshes with physical properties, but requires that these meshes are watertight, and have no self-intersections, and their smallest features are larger than a voxel. Some local features are constructed in a pre-calculation stage to increase simulation performance. Simulation tests show that our methods can cut embedded structures in a way consistent with the cutting of the deformable objects. Cut fragments can also deform correctly along with the deformable objects.