Quadratic curve and surface fitting via squared distance minimization

Quadratic curve and surface fitting to a set of data points are fundamental problems in reverse engineering and many other application areas. We develop the fitting methods for quadratic curves and surfaces based on the squared distance minimization technology. The basic idea of squared distance minimization for curve and surface fitting is first presented. Then we devise the corresponding squared distance term for each quadratic curve and surface, and minimize it to obtain its parameters. We repeat the squared distance minimization and update the parameters of the quadratic curve and surface by iterations until convergency. Consequently, the final fitting result is achieved. Experimental results demonstrate the effectiveness of the fitting method.     

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。     

Restricting Voronoi diagrams to meshes using corner validation

Restricted Voronoi diagrams are a fundamental geometric structure used in many applications such as surface reconstruction from point sets or optimal transport. Given a set of sites V = { v _k}ⁿ_k=1 ? ?^d and a mesh X with vertices in ?^d connected by triangles, the restricted Voronoi diagram partitions X by computing for each site the portion of X for which the site is the nearest. The restricted Voronoi diagram is the intersection between the regular Voronoi diagram and the mesh. Depending on the site distribution or the ambient space dimension computing the regular Voronoi diagram may not be feasible using classical algorithms. In this paper, we extend Lévy and Bonneel's approach [ LB12 ] based on nearest neighbor queries. We show that their method is limited when the sites are not located on X . We propose a new algorithm for computing restricted Voronoi which reduces the number of sites considered for each triangle of the mesh and scales smoothly when the sites are far from the surface.     

Improved 2D mass-spring-damper model with unstructured triangular meshes

In this paper, we investigate both the visual realism and the physical accuracy of the 2D mass-spring-damper (MSD) model with general unstructured triangular meshes for the simulation of rigid cloth. For visual realism, the model should, at a minimum, bend smoothly under pure bending load conditions. For physical accuracy, it should bend approximately the same amount and shape as dictated by continuum mechanics. By matching the 2D MSD model with an elastic plate, we obtain a series of constraints on the parameters of the model. We find that for a 2D unstructured MSD model, it is necessary to apply preloads on the springs for accurate modeling of bending resistance. By simultaneously applying the constraints for both visual realism and physical accuracy, we can optimize the parameters of the model to enhance its fidelity. The simulation shows that the deformation of the optimized MSD model with preload is very close to the result obtained by the finite element method (FEM) under either point load condition or pressure load condition. With a much smaller computational burden compared with FEM, the optimized MSD model is especially suitable for real time haptic applications.     

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.     

Removing local irregularities of triangular meshes with highlight line models

The highlight line model is a powerful tool in assessing the quality of a surface. It increases the flexibility of an interactive design environment. In this paper, a method to generate a highlight line model on an arbitrary triangular mesh is presented. Based on the highlight line model, a technique to remove local shape irregularities of a triangular mesh is then presented. The shape modification is done by solving a minimization problem and performing an iterative procedure. The new technique improves not only the shape quality of the mesh surface, but also the shape of the highlight line model. It provides an intuitive and yet suitable method for locally optimizing the shape of a triangular mesh. Supported by National Science Foundation of China (Grant Nos. 60533070, 60625202), National Basic Research Program of China (Grant No. 2004CB719400), National High-Tech Research & Development Program of China (Grant No. 2007AA040401), Fok Ying Tung Education Foundation (Grant No. 111070), National Science Foundation of USA (Grant Nos. DMI-0422126, DMS-0310645), and Kentucky Science & Technology Corporation (Grant No. COMM-Fund-712)     

Numerical Fitting of Planar Photographic Images with Spherical Voronoi Diagrams

There are many phenomena that generate polygonal tessellations on surfaces of 3D objects. One interesting example is the jackfruit, a multiple fruit found in the tropics. A recent study found the best-fit spherical Voronoi diagram from a photo of jackfruit skin, but the optimization was relative to the radius of the sphere and the height of the spikes. In this study, we propose a method for adjusting the position of the center of the sphere in addition to these parameters. Experiments were conducted using both ideal and real data. However, convergence with real data has not been confirmed due to relaxation of the convergence condition.     

Efficient and flexible sampling with blue noise properties of triangular meshes

This paper deals with the problem of taking random samples over the surface of a 3D mesh describing and evaluating efficient algorithms for generating different distributions. We discuss first the problem of generating a Monte Carlo distribution in an efficient and practical way avoiding common pitfalls. Then, we propose Constrained Poisson-disk sampling, a new Poisson-disk sampling scheme for polygonal meshes which can be easily tweaked in order to generate customized set of points such as importance sampling or distributions with generic geometric constraints. In particular, two algorithms based on this approach are presented. An in-depth analysis of the frequency characterization and performance of the proposed algorithms are also presented and discussed.     

Smooth closed surfaces with discrete triangular interpolants

Discrete interpolants which involve cross boundary derivatives in an attempt to form C¹ surfaces have the following major problem: Requiring C¹ joins between patches makes sense only if the patch domains are adjacent in the domain space. This makes it impossible to form C¹ closed surfaces, or indeed any surface which contains more connections than can be achieved in the domain.

This paper develops a method of forming smooth closed (or otherwise complexly connected) surfaces from a discrete triangular interpolant by relaxing the C¹ property of an interpolant to ‘Visually C¹”.

The only constraint on the scheme is that the data to be interpolated define a unique tangent plane at each vertex where several triangles meet. Then each patch can be calculated independently of its neighbors, using only data defined at its vertices, and the domain for each triangular patch can be chosen without regarding the connectivity of the patch with others. This last feature could be of great interest to a designer of a surface since one can choose the domain of each patch to be an equilateral triangle, and give it no further thought.     

利用Voronoi协方差矩阵重建隐式曲面

目的 针对含少量离群点的噪声点云,提出了一种Voronoi协方差矩阵的曲面重建方法。方法 以隐函数梯度在Voronoi协方差矩阵形成的张量场内的投影最大化为目标,构建隐函数微分方程,采用离散外微分形式求解连续微分方程,从而将曲面重建问题转化为广义特征值求解问题。在点云空间离散化过程中,附加最短边约束条件,避免了局部空间过度剖分。并引入概率测度理论定义曲面窄带,提高了算法抵抗离群点能力,通过精细剖分曲面窄带,提高了曲面重建精度。结果 实验结果表明,该算法可以抵抗噪声点和离群点的影响,可以生成不同分辨率的曲面。通过调整拟合参数,可以区分曲面的不同部分。结论 提出了一种新的隐式曲面重建方法,无需点云法向、稳健性较强,生成的三角面纵横比好。     

Generic remeshing of 3D triangular meshes with metric-dependent discrete voronoi diagrams

In this paper, we propose a generic framework for 3D surface remeshing. Based on a metric-driven Discrete Voronoi Diagram construction, our output is an optimized 3D triangular mesh with a user defined vertex budget. Our approach can deal with a wide range of applications, from high quality mesh generation to shape approximation. By using appropriate metric constraints the method generates isotropic or anisotropic elements. Based on point-sampling, our algorithm combines the robustness and theoretical strength of Delaunay criteria with the efficiency of entirely discrete geometry processing . Besides the general described framework, we show experimental results using isotropic, quadric-enhanced isotropic and anisotropic metrics which prove the efficiency of our method on large meshes, for a low computational cost.     

Fast parallel polynomial division via reduction to triangular toeplitz matrix inversion and to polynomial inversion modulo a power

It is known that division with a remainder of two polynomials of degree at most s can be performed over an arbitrary field F of constants using uniform arithmetic and Boolean circuits of depth O(log s log log s) and polynomial size. A new algorithm is presented that yields those bounds via reduction to triangular Toeplitz matrix inversion and to polynomial inversion modulo a power. (If|F| > (s?1)² or if P-uniform computation is allowed, then the depth can be reduced to O(log s).) This approach is new and makes the result conceptually simpler.     

A list-processing approach to compute Voronoi diagrams and theEuclidean distance transform

We propose an efficient Voronoi transform algorithm for constructing Voronoi diagrams using segment lists of rows. A significant feature of the algorithm is that it takes segments rather than pixels as the basic units to represent and propagate the nearest neighbor information. The segment lists are dynamically updated as they are scanned. A distance map can then be easily computed from the segment list representation of the Voronoi diagram. Experimental results have demonstrated its high efficiency. Extension of the algorithm to higher dimensions is also discussed     

Fitting curves and surfaces with constrained implicit polynomials

A problem which often arises while fitting implicit polynomials to 2D and 3D data sets is the following: although the data set is simple, the fit exhibits undesired phenomena, such as loops, holes, extraneous components, etc. Previous work tackled these problems by optimizing heuristic cost functions, which penalize some of these topological problems in the fit. The paper suggests a different approach-to design parameterized families of polynomials whose zero-sets are guaranteed to satisfy certain topological properties. Namely, we construct families of polynomials with star-shaped zero-sets, as well as polynomials whose zero-sets are guaranteed not to intersect an ellipse circumscribing the data or to be entirely contained in such an ellipse. This is more rigorous than using heuristics which may fail and result in pathological zero-sets. The ability to parameterize these families depends heavily on the ability to parameterize positive polynomials. To achieve this, we use some powerful results from real algebraic geometry     

带两个参数的三角多项式曲线曲面构造

目的 为了使扩展的曲线曲面保留传统Bézier方法以及B样条方法良好性质的同时,具备保形性、形状可调性、高阶连续性以及广泛的应用性,本文在拟扩展切比雪夫空间利用开花的性质构造了一组最优规范全正基,并利用该基进行曲线曲面构造。方法 首先构造一组最优规范全正基,并给出该基生成的拟三次TC-Bézier曲线的割角算法;接着利用最优规范全正基的线性组合构造拟三次均匀TC-B样条基,根据曲线的性质假设拟三次均匀B样条基函数具有规范性和C²连续性,进而得到其表达式;然后证明拟三次均匀TC-B样条基具有全正性和高阶连续性;最后定义拟三次均匀TC-B样条曲线曲面,并证明曲线曲面的性质,给出曲线表示整圆和旋转曲面的表示方法,设计出球面和旋转曲面的直接生成方法。结果 实验表明,本文在拟扩展切比雪夫空间构造的具有全正性曲线曲面,不仅能够灵活地进行形状调整,而且具有高阶连续性、保形性。结论 本文在三角函数空间利用两个形状参数进行曲线曲面构造,大量的分析以及案例说明本文构造的曲线曲面不仅保留了传统的Bézier方法以及B样条方法的良好性质,而且具备保形性、形状可调性、高阶连续性以及广泛的应用性,适合用于曲线曲面设计。     

Method for intersecting algebraic surfaces with rational polynomial patches

The paper presents a hybrid algorithm for the computation of the intersection of an algebraic surface and a rational polynomial parametric surface patch. This algorithm is based on analytic representation of the intersection as an algebraic curve expressed in the Bernstein basis; automatic computation of the significant points of the curve using numerical techniques, subdivision and convexity properties of the Bernstein basis; partitioning of the intersection domain at these points; and tracing of the resulting monotonic intersection segments using coarse subdivision and faceting methods coupled with Newton techniques. The algorithm described in the paper treats intersections of arbitrary order algebraic surfaces with rational biquadratic and bicubic patches and introduces efficiency enhancements in the partitioning and tracing parts of the solution process. The algorithm has been tested with up to degree four algebraics and bicubic patches.     

Prediction of rank deficiency in partition of unity-based methods with plane triangular or quadrilateral meshes

In the partition of unity (PU)-based methods, the global approximation is built by multiplying a partition of unity by local approximations. Within this framework, high-order approximations are achieved by directly adopting high-order polynomials as local approximations, and therefore nodes along sides or inside elements, which are usually adopted in the conventional finite element methods, are no more required. However, the PU-based approximation constructed in this way may suffer from rank deficiency due to the linear dependence of the global degrees of freedom. In this paper, the origin of the rank deficiency in the PU-based approximation space is first dissected at an element level, and then an approach to predict the rank deficiency for a mesh is proposed together with the principle of the increase of rank deficiency. Finally, examples are investigated to validate the present approach. The current work indicates such a fact that the rank deficiency is an unrelated issue to the nullity of the global matrix. It can be resolved in its own manner.     

An intrinsic algorithm for computing geodesic distance fields on triangle meshes with holes

As a fundamental concept, geodesics play an important role in many geometric modeling applications. However, geodesics are highly sensitive to topological changes; a small topological shortcut may result in a significantly large change of geodesic distance and path. Most of the existing discrete geodesic algorithms can only be applied to noise-free meshes. In this paper, we present a new algorithm to compute the meaningful approximate geodesics on polygonal meshes with holes. Without the explicit hole filling, our algorithm is completely intrinsic and independent of the embedding space; thus, it has the potential for isometrically deforming objects as well as meshes in high dimensional space. Furthermore, our method can guarantee the exact solution if the surface is developable. We demonstrate the efficacy of our algorithm in both real-world and synthetic models.