Spectral Processing of Tangential Vector Fields

We propose a framework for the spectral processing of tangential vector fields on surfaces. The basis is a Fourier‐type representation of tangential vector fields that associates frequencies with tangential vector fields. To implement the representation for piecewise constant tangential vector fields on triangle meshes, we introduce a discrete Hodge–Laplace operator that fits conceptually to the prominent cotan discretization of the Laplace–Beltrami operator. Based on the Fourier representation, we introduce schemes for spectral analysis, filtering and compression of tangential vector fields. Moreover, we introduce a spline‐type editor for modelling of tangential vector fields with interpolation constraints for the field itself and its divergence and curl. Using the spectral representation, we propose a numerical scheme that allows for real‐time modelling of tangential vector fields.     

Designing 2D Vector Fields of Arbitrary Topology

We introduce a scheme of control polygons to design topological skeletons for vector fields of arbitrary topology. Based on this we construct piecewise linear vector fields of exactly the topology specified by the control polygons. This way a controlled construction of vector fields of any topology is possible. Finally we apply this method for topology‐preserving compression of vector fields consisting of a simple topology.     

Discrete 2‐Tensor Fields on Triangulations

Geometry processing has made ample use of discrete representations of tangent vector fields and antisymmetric tensors (i.e., forms) on triangulations. Symmetric 2‐tensors, while crucial in the definition of inner products and elliptic operators, have received only limited attention. They are often discretized by first defining a coordinate system per vertex, edge or face, then storing their components in this frame field. In this paper, we introduce a representation of arbitrary 2‐tensor fields on triangle meshes. We leverage a coordinate‐free decomposition of continuous 2‐tensors in the plane to construct a finite‐dimensional encoding of tensor fields through scalar values on oriented simplices of a manifold triangulation. We also provide closed‐form expressions of pairing, inner product, and trace for this discrete representation of tensor fields, and formulate a discrete covariant derivative and a discrete Lie bracket. Our approach extends discrete/finite‐element exterior calculus, recovers familiar operators such as the weighted Laplacian operator, and defines discrete notions of divergence‐free, curl‐free, and traceless tensors–thus offering a numerical framework for discrete tensor calculus on triangulations. We finally demonstrate the robustness and accuracy of our operators on analytical examples, before applying them to the computation of anisotropic geodesic distances on discrete surfaces.     

A greedy Delaunay-based surface reconstruction algorithm

In this paper, we present a new greedy algorithm for surface reconstruction from unorganized point sets. Starting from a seed facet, a piecewise linear surface is grown by adding Delaunay triangles one by one. The most plausible triangles are added first and in such a way as to prevent the appearance of topological singularities. The output is thus guaranteed to be a piecewise linear orientable manifold, possibly with boundary. Experiments show that this method is very fast and achieves topologically correct reconstruction in most cases. Moreover, it can handle surfaces with complex topology, boundaries, and nonuniform sampling.     

Loop subdivision surfaces interpolating -spline curves

This paper presents a novel method for defining a Loop subdivision surface interpolating a set of popularly-used cubic B-spline curves. Although any curve on a Loop surface corresponding to a regular edge path is usually a piecewise quartic polynomial curve, it is found that the curve can be reduced to a single cubic B-spline curve under certain constraints of the local control vertices. Given a set of cubic B-spline curves, it is therefore possible to define a Loop surface interpolating the input curves by enforcing the interpolation constraints. In order to produce a surface of local or global fair effect, an energy-based optimization scheme is used to update the control vertices of the Loop surface subjecting to curve interpolation constraints, and the resulting surface will exactly interpolate the given curves. In addition to curve interpolation, other linear constraints can also be conveniently incorporated. Because both Loop subdivision surfaces and cubic B-spline curves are popularly used in engineering applications, the curve interpolation method proposed in this paper offers an attractive and essential modeling tool for computer-aided design.     

A simple method for interpolating meshes of arbitrary topology by Catmull–Clark surfaces

Interpolating an arbitrary topology mesh by a smooth surface plays important role in geometric modeling and computer graphics. In this paper we present an efficient new algorithm for constructing Catmull–Clark surface that interpolates a given mesh. The control mesh of the interpolating surface is obtained by one Catmull–Clark subdivision of the given mesh with modified geometric rule. Two methods—push-back operation based method and normal-based method—are presented for the new geometric rule. The interpolation method has the following features: (1) Efficiency: we obtain a generalized cubic B-spline surface to interpolate any given mesh in a robust and simple manner. (2) Simplicity: we use only simple geometric rule to construct control mesh for the interpolating subdivision surface. (3) Locality: the perturbation of a given vertex only influences the surface shape near this vertex. (4) Freedom: for each edge and face, there is one degree of freedom to adjust the shape of the limit surface. These features make interpolation using Catmull–Clark surfaces very simple and thus make the method itself suitable for interactive free-form shape design.     

A Recursive Subdivision Algorithm for Piecewise Circular Spline

We present an algorithm for generating a piecewise G ¹ circular spline curve from an arbitrary given control polygon. For every corner, a circular biarc is generated with each piece being parameterized by its arc length. This is the first subdivision scheme that produces a piecewise biarc curve that can interpolate an arbitrary set of points. It is easily adopted in a recursive subdivision surface scheme to generate surfaces with circular boundaries with pieces parameterized by arc length, a property not previously available. As an application, a modified version of Doo–Sabin subdivision algorithm is outlined making it possible to blend a subdivision surface with other surfaces having circular boundaries such as cylinders.     

基于顶点法向量约束的Catmull-Clark细分插值方法

提出一种基于顶点法向量约束实现插值的两步Catmull-Clark细分方法.第一步,通过改造型Catmull-Clark细分生成新网格.第二步,通过顶点法向量约束对新网格进行调整.两步细分分别运用渐进迭代方法和拉格朗日乘子法,使得极限曲面插值于初始控制顶点和法向量.实验结果证明了该方法可同时实现插值初始控制顶点和法向量,极限曲面具有较好的造型效果.     

基于时态边缘算子的时间序列分段线性表示

时间序列的分段线性表示算法通常基于单一的启发式规则,难以适用于不同数据特征的时间序列。借鉴了边缘算子的思想来提取时间序列的边缘点,提出了一种基于时态边缘算子的时间序列分段线性表示算法。在来自不同领域的公开数据集上进行的实验结果表明：与两种主要的分段线性表示算法相比,该算法具有更好的拟合性能,并且更为稳定,能够适用于各类不同数据特征的时间序列。     

A Second-Order Difference Scheme for the Penalized Black–Scholes Equation Governing American Put Option Pricing

In this paper we present a stable finite difference scheme on a piecewise uniform mesh along with a power penalty method for solving the American put option problem. By adding a power penalty term the linear complementarity problem arising from pricing American put options is transformed into a nonlinear parabolic partial differential equation. Then a finite difference scheme is proposed to solve the penalized nonlinear PDE, which combines a central difference scheme on a piecewise uniform mesh with respect to the spatial variable with an implicit time stepping technique. It is proved that the scheme is stable for arbitrary volatility and arbitrary interest rate without any extra conditions and is second-order convergent with respect to the spatial variable. Furthermore, our method can efficiently treats the singularities of the non-smooth payoff function. Numerical results support the theoretical results.     

Segmentation of discrete vector fields

In this paper, we propose an approach for 2D discrete vector field segmentation based on the Green function and normalized cut. The method is inspired by discrete Hodge decomposition such that a discrete vector field can be broken down into three simpler components, namely, curl-free, divergence-free, and harmonic components. We show that the Green function method (GFM) can be used to approximate the curl-free and the divergence-free components to achieve our goal of the vector field segmentation. The final segmentation curves that represent the boundaries of the influence region of singularities are obtained from the optimal vector field segmentations. These curves are composed of piecewise smooth contours or streamlines. Our method is applicable to both linear and nonlinear discrete vector fields. Experiments show that the segmentations obtained using our approach essentially agree with human perceptual judgement.     

Knot placement for piecewise polynomial approximation of curves

For piecewise polynomial representation of curves, an algorithm to create knots is presented. The aim is to minimize the interpolation error for a given number of knots or, conversely, the number of knots needed to interpolate within a tolerance. The method used is a modification of de Boor's knot placement scheme. The algorithm described in this paper has been realized in the CADCAM system SYRKO, a Daimler-Benz development for car body design and manufacturing.     

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.     

Learning Probabilistic Models for Contour Completion in Natural Images

Using a large set of human segmented natural images, we study the statistics of region boundaries. We observe several power law distributions which likely arise from both multi-scale structure within individual objects and from arbitrary viewing distance. Accordingly, we develop a scale-invariant representation of images from the bottom up, using a piecewise linear approximation of contours and constrained Delaunay triangulation to complete gaps. We model curvilinear grouping on top of this graphical/geometric structure using a conditional random field to capture the statistics of continuity and different junction types. Quantitative evaluations on several large datasets show that our contour grouping algorithm consistently dominates and significantly improves on local edge detection.     

一种B样条表面插值控制点的缩减方法

研究了从给定节点向量中选择节点进行B样条曲线插值的方法，并将此方法应用到行数据点不相同的B样条曲面插值，得到了一个通过对行节点矢量调整传递的曲面插值方法，理论分析和实验表明该方法可大量减少曲面控制点的数目．     

Neural Representation of Open Surfaces

Neural implicit surfaces have emerged as an effective, learnable representation for shapes of arbitrary topology. However, representing open surfaces remains a challenge. Different methods, such as unsigned distance fields (UDF), have been proposed to tackle this issue, but a general solution remains elusive. The generalized winding number (GWN), which is often used to distinguish interior points from exterior points of 3D shapes, is arguably the most promising approach. The GWN changes smoothly in regions where there is a hole in the surface, but it is discontinuous at points on the surface. Effectively, this means that it can be used in lieu of an implicit surface representation while providing information about holes, but, unfortunately, it does not provide information about the distance to the surface necessary for e.g. ray tracing, and special care must be taken when implementing surface reconstruction. Therefore, we introduce the semi-signed distance field (SSDF) representation which comprises both the GWN and the surface distance. We compare the GWN and SSDF representations for the applications of surface reconstruction, interpolation, reconstruction from partial data, and latent vector analysis using two very different data sets. We find that both the GWN and SSDF are well suited for neural representation of open surfaces.     

细分曲面拟合的局部渐进插值方法

逼近型细分方法生成的细分曲面其品质要优于插值型细分方法生成的细分曲面.然而,逼近型细分方法生成的细分曲面不能插值于初始控制网格顶点.为使逼近型细分曲面具有插值能力,一般通过求解全局线性方程组,使其插值于网格顶点.当网格顶点较多时,求解线性方程组的计算量很大,因此,难以处理稠密网格.与此不同,在不直接求解线性方程组的情况下,渐进插值方法通过迭代调整控制网格顶点,最终达到插值的效果.渐进插值方法可以处理稠密的任意拓扑网格,生成插值于初始网格顶点的光滑细分曲面.并且经证明,逼近型细分曲面渐进插值具有局部性质,也就是迭代调整初始网格的若干控制顶点,且保持剩余顶点不变,最终生成的极限细分曲面仍插值于初始网格中被调整的那些顶点.这种局部渐进插值性质给形状控制带来了更多的灵活性,并且使得自适应拟合成为可能.实验结果验证了局部渐进插值的形状控制以及自适应拟合能力.     

Radiance caching for efficient global illumination computation

In this paper, we present a ray tracing-based method for accelerated global illumination computation in scenes with low-frequency glossy BRDFs. The method is based on sparse sampling, caching, and interpolating radiance on glossy surfaces. In particular, we extend the irradiance caching scheme proposed by Ward et al. (1988) to cache and interpolate directional incoming radiance instead of irradiance. The incoming radiance at a point is represented by a vector of coefficients with respect to a hemispherical or spherical basis. The surfaces suitable for interpolation are selected automatically according to the roughness of their BRDF. We also propose a novel method for computing translational radiance gradient at a point.     

Flexible and Accurate Transparent‐Object Matting and Compositing Using Refractive Vector Field

In digital image editing, environment matting and compositing are fundamental and interesting operations that can capture and simulate the refraction and reflection effects of light from an environment. The state‐of‐the‐art real‐time environment matting and compositing method is short of flexibility, in the sense that it has to repeat the entire complex matte acquisition process if the distance between the object and the background is different from that in the acquisition stage, and also lacks accuracy, in the sense that it can only remove noises but not errors. In this paper, we introduce the concept of refractive vector and propose to use a refractive vector field as a new representation for environment matte. Such refractive vector field provides great flexibility for transparent‐object environment matting and compositing. Particularly, with only one process of the matte acquisition and the refractive vector field extraction, we are able to composite the transparent object into an arbitrary background at any distance. Furthermore, we introduce a piecewise vector field fitting algorithm to simultaneously remove both noises and errors contained in the extracted matte data. Experimental results show that our method is less sensitive to artefacts and can generate perceptually good composition results for more general scenarios.     

Control Points for Multivariate B-Spline Surfaces over Arbitrary Triangulations

This paper describes first results of a test implementation that implements the new multivariate B-splines as recently developed by Dahmen et al. 10for quadratics and cubics. The surface scheme is based on blending functions and control points and allows the modelling of   C ^{k − 1}  -continuous piecewise polynomial surfaces of degree k over arbitrary triangulations of the parameter plane. The surface scheme exhibits both affine invariance and the convex hull property, and the control points can be used to manipulate the shape of the surface locally. Additional degrees of freedom in the underlying knot net allow for the modelling of discontinuities. Explicit formulas are given for the representation of polynomials and piecewise polynomials as linear combinations of B-splines.