Geometric segmentation of 3D scanned surfaces

The geometric segmentation of a discrete geometric model obtained by the scanning of real objects is affected by various problems that make the segmentation difficult to perform without uncertainties. Certain factors, such as point location noise (coming from the acquisition process) and the coarse representation of continuous surfaces due to triangular approximations, introduce ambiguity into the recognition process of the geometric shape. To overcome these problems, a new method for geometric point identification and surface segmentation is proposed.The point classification is based on a fuzzy parameterization using three shape indexes: the smoothness indicator, shape index and flatness index. A total of 11 fuzzy domain intervals have been identified and comprise sharp edges, defective zones and 10 different types of regular points. For each point of the discrete surface, the related membership functions are dynamically evaluated to be adapted to consider, point by point, those properties of the geometric model that affects uncertainty in point type attribution.The methodology has been verified in many test cases designed to represent critical conditions for any method in geometric recognition and has been compared with one of the most robust methods described in the related literature.     

A discrete geometry approach for dominant point detection

We propose two fast methods for dominant point detection and polygonal representation of noisy and possibly disconnected curves based on a study of the decomposition of the curve into the sequence of maximal blurred segments [2]. Starting from results of discrete geometry [3] and [4], the notion of maximal blurred segment of width ν[2] has been proposed, well adapted to possibly noisy curves. The first method uses a fixed parameter that is the width of considered maximal blurred segments. The second method is deduced from the first one based on a multi-width approach to obtain a non-parametric method that uses no threshold for working with noisy curves. Comparisons with other methods in the literature prove the efficiency of our approach. Thanks to a recent result [5] concerning the construction of the sequence of maximal blurred segments, the complexity of the proposed methods is O(n log n). An application of vectorization is also given in this paper.     

A robust and accurate geometric model for automated design of drawbeads in sheet metal forming

A robust and accurate geometric model of real drawbeads that can be used for the automated design of drawbeads is presented in the paper. A three-dimensional geometric drawbead is a lofted surface, of which the section curves are constructed parallel to the stamping direction on the control points. Adaptive control point interpolation is introduced to simplify the management of the drawbead geometry and avoid unexpected shapes. Given primitive control points on a drawbead curve, dominant control points are adaptively obtained with the shapes of both the drawbead curve and the binder considered. An a priori heuristic parameter adjustment strategy is proposed to correct the parameter errors of section curves, which improves the accuracy and consistency of the drawbead geometry. By incorporating the proposed geometric drawbead with a previously developed intelligent drawbead optimization algorithm, a fully automated design process for drawbeads is realized that includes geometric modeling, finite element analysis, intelligent optimization of the drawbead geometry, and die manufacturing. Finally, a fender example is presented to verify the feasibility and validity of the fully automated drawbead design process. The simulation results with the optimized geometric drawbeads and equivalent drawbeads are compared with the experimental results. The proposed geometric drawbead shows remarkable practicability and accuracy in the automated design of drawbeads in sheet metal forming and demonstrates good consistency with the experimental results while the equivalent drawbead model introduces unneglectable deviations.     

Robust estimation of continuous nonlinear plants with discrete measurements

In this work the problem of designing a state estimator for completely or partially observable continuous nonlinear plants with discrete measurements is addressed. The combination of a geometric approach with a stability analysis yields an estimator design methodology with a nonlinear detectability condition susceptible of testing, a systematic estimator construction, a robust convergence criterion coupled with a simple tuning scheme, as well as a rationale to explain the interplay between sampling time, estimator gains, and estimator functioning. Comparing with the continuous measurement case where the convergence is attained by tuning the gain above a low limit, in the discrete measurement case the loss of information due to the measurement sampling increases the size of the lower gain limit, and imposes sampling time and high gain limits. The proposed methodology is applied to address the estimation problem of a class of solution homopolymerization reactors, and is tested with a methyl-methacrylate polymerization run taken from a previous extended Kalman filter implementation study with experimental data.     

Estimating differential quantities from point cloud based on a linear fitting of normal vectors

Estimation of differential geometric properties on a discrete surface is a fundamental work in computer graphics and computer vision. In this paper, we present an accurate and robust method for estimating differential quantities from unorganized point cloud. The principal curvatures and principal directions at each point are computed with the help of partial derivatives of the unit normal vector at that point, where the normal derivatives are estimated by fitting a linear function to each component of the normal vectors in a neighborhood. This method takes into account the normal information of all neighboring points and computes curvatures directly from the variation of unit normal vectors, which improves the accuracy and robustness of curvature estimation on irregular sampled noisy data. The main advantage of our approach is that the estimation of curvatures at a point does not rely on the accuracy of the normal vector at that point, and the normal vectors can be refined in the process of curvature estimation. Compared with the state of the art methods for estimating curvatures and Darboux frames on both synthetic and real point clouds, the approach is shown to be more accurate and robust for noisy and unorganized point cloud data. Supported in part by the National Natural Science Foundation of China (Grant Nos. 60672148, 60872120), the National High-Tech Research & Development Program of China (Grant Nos. 2006AA01Z301, 2008AA01Z301), and Beijing Municipal Natural Science Foundation (Grant No. 4062033)     

离散广义区间动力系统的稳定性

利用范数理论,研究离散广义区间动力系统的稳定性问题,给出其区间动力系统的正则、因果且稳定的充分条件.在此基础上,进一步考虑该系统的区间极点集的配置问题,给出了使离散广义区间动力系统的极点落在预先给定园盘内的判定定理.数值算例表明了该方法的可行性.     

Analysis of Noisy Digital Contours with Adaptive Tangential Cover

The notion of tangential cover, based on maximal segments, is a well-known tool to study the geometrical characteristics of a discrete curve. However, it is not robust to noise, while extracted contours from digital images typically contain noise, and this makes the geometric analysis tasks on such contours difficult. To tackle this issue, we investigate in this paper a discrete structure, named adaptive tangential cover (ATC), which is based on the notion of tangential cover and on a local noise estimator. More specifically, the ATC is composed of maximal segments with different widths deduced from the local noise values estimated at each point of the contour. Furthermore, a parameter-free algorithm is also presented to compute ATC. This study leads to the proposal of several applications of ATC on noisy digital contours: dominant point detection, contour length estimator, tangent/normal estimator, detection of convex and concave parts. An extension of ATC to 3D curves is also proposed in this paper. The experimental results demonstrate the efficiency of this new notion.     

A robust observer for discrete time nonlinear systems

This paper extends the results developed in (Ciccarella et al., 1993) and presents a robust observer for discrete time nonlinear systems. A simple, robust and easy to implement algorithm is given whose convergence properties are guaranteed for autonomous and forced systems. Combined parameter and state estimation is made for a numerical example, which compares the robust observer to the observer given in (Ciccarella et al., 1993).     

Robust egomotion estimation from the normal flow using searchsubspaces

We address the problem of egomotion estimation for a monocular observer moving under arbitrary translation and rotation, in an unknown environment. The method we propose is uniquely based on the spatio-temporal image derivatives, or the normal flow. We introduce a search paradigm which is based on geometric properties of the normal flow field, and consists in considering a family of search subspaces to estimate the egomotion parameters. Various algorithms are proposed within this framework. In order to decrease the noise sensitivity of the estimation methods, we use statistical tools, based on robust regression theory. Finally, we present and discuss a wide variety of experiments with synthetic and real images, for various kinds of camera motion     

等弧长原则的NURBS曲线离散算法

NURBS曲线广泛应用于工业产品复杂曲线曲面设计中,但在实际应用中常遇到曲线离散的几何处理问题。针对NURBS曲线离散问题,提出了一种按等弧长原则对NURBS曲线进行离散的方法。该方法引入步长函数控制离散曲线段的弧长,采用积分法和迭代法调整步长函数以控制曲线的离散精度,通过误差检验方法校验曲线离散的逼近精度。通过实际算例,验证了NURBS曲线等弧长离散算法的合理性和有效性。     

Discrete derivative approximations with scale-space properties: A basis for low-level feature extraction

This article shows how discrete derivative approximations can be defined so thatscale-space properties hold exactly also in the discrete domain. Starting from a set of natural requirements on the first processing stages of a visual system,the visual front end, it gives an axiomatic derivation of how a multiscale representation of derivative approximations can be constructed from a discrete signal, so that it possesses analgebraic structure similar to that possessed by the derivatives of the traditional scale-space representation in the continuous domain. A family of kernels is derived that constitutediscrete analogues to the continuous Gaussian derivatives.The representation has theoretical advantages over other discretizations of the scale-space theory in the sense that operators that commute before discretizationcommute after discretization. Some computational implications of this are that derivative approximations can be computeddirectly from smoothed data and that this will giveexactly the same result as convolution with the corresponding derivative approximation kernel. Moreover, a number ofnormalization conditions are automatically satisfied.The proposed methodology leads to a scheme of computations of multiscale low-level feature extraction that is conceptually very simple and consists of four basic steps: (i)large support convolution smoothing, (ii)small support difference computations, (iii)point operations for computing differential geometric entities, and (iv)nearest-neighbour operations for feature detection.Applications demonstrate how the proposed scheme can be used for edge detection and junction detection based on derivatives up to order three.     

Sampling framework for accurate curvature estimation in discrete surfaces

Accurate curvature estimation in discrete surfaces is an important problem with numerous applications. Curvature is an indicator of ridges and can be used in applications such as shape analysis and recognition, object segmentation, adaptive smoothing, anisotropic fairing of irregular meshes, and anisotropic texture mapping. In this paper, a new framework is proposed for accurate curvature estimation in discrete surfaces. The proposed framework is based on a local directional curve sampling of the surface where the sampling frequency can be controlled. This local model has a large number of degrees of freedoms compared with known techniques and, so, can better represent the local geometry. The proposed framework is quantitatively evaluated and compared with common techniques for surface curvature estimation. In order to perform an unbiased evaluation in which smoothing effects are factored out, we use a set of randomly generated Bezier surface patches for which the curvature values can be analytically computed. It is demonstrated that, through the establishment of sampling conditions, the error in estimations obtained by the proposed framework is smaller and that the proposed framework is less sensitive to low sampling density, sampling irregularities, and sampling noise.     

离散点云原始形状及边界曲线提取算法

大规模离散点云包含多种类型的扫描缺陷:噪声、异常数据、孔洞及不规则的各向异性采样,大部分现有的算法不能够很好地处理这些缺陷,这对点云拓扑关系的恢复及特征提取带来了困难.针对此问题,提出了一种健壮有效的点云重构算法,首先,计算每个数据点的局部属性;然后利用局部属性探测点云中包含的原始形状;最后利用统计优化方法对原始形状中...     

三角网格曲面上离散曲率改进算法

计算三角网格离散曲面曲率的Meyer方法几何意义简明,计算量较小,但其计算效果仍有进一步提高的潜力。通过对Meyer方法的深入分析,提出了平均曲率构造向量和Gauss曲率构造角的概念,并指出了它们的几何意义,在此基础上构造了对Meyer方法的改进算法。经分析,提出的改进算法精简了各个主要计算步骤,避免了不必要的计算误差。仿真计算结果表明,改进算法是有效的,提高了三角网格离散曲率的计算精度和计算效率。     

Differential quadrature element analysis using extended differential quadrature

The extended differential quadrature (EDQ) has been proposed. A certain order derivative or partial derivative of the variable function with respect to the coordinate variables at an arbitrary discrete point is expressed as a weighted linear sum of the values of function and/or its possible derivatives at all grid nodes. The grid pattern can be fixed while the selection of discrete points for defining discrete fundamental relations is flexible. This method can be used to the differential quadrature element and generalized differential quadrature element analyses.     

Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects

We introduce techniques for the processing of motion and animations of non‐rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in ?ⁿ to the processing of motion of non‐rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced‐order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non‐linear “Bézier curves” by executing de Casteljau's algorithm in shape space.     

Robust LQG optimal controller design for multivariable discrete saturating systems with noise spectral uncertainties and non-linear time-varying unmodelled dynamics

A design procedure is proposed for robust linear-quadratic-gaussian (LQG) optimal controller synthesis against noise spectral uncertainties, non-linear time-varying (NLTV) unmodelled dynamics in discrete saturating systems. A robust stability criterion is derived for multivariable stochastic discrete-time systems with NLTV unmodelled dynamics and constrained actuators. An algorithm based on the robust stabilization creterion is presented for synthesing a robust controller not only to minimize the least favourable cost functional but also to satisfy the robust stabilization criterion by specifying an appropriate weighting scalar in the cost functional. A necessary and sufficient condition for the solvability of such a robust stabilization problem is derived by means of the Nevanlinna-Pick interpolation theory. The Wiener Z-domain solution for controller synthesis, the saddle point theory, and the properties of Schur operator (Class S) are employed to treat this problem. Finally, a numerical example is given to illustrate the results.     

一种修改NURBS曲线形状的新方法

曲线曲面的形状修改是计算机几何造型过程中的重要部分．文章提出了一种修改NURBS曲线的新方法，使得修改后的曲线在多个参数点处满足用户给定的几何约束(如点约束、切矢约束)，首先引入了一些新的概念如：局部曲线、总曲线、多余约束和多余曲线等．对于每个参数点分别计算出一系列满足该点处几何约束的局部曲线，并由此构造了总曲线．接着插值一条满足多余约束的多余曲线．最后运用构造Coons曲面的思想，计算出最终的修改曲线，它等于总曲线减去多余曲线．同时我们发现两种现存的修改NURBS曲线的方法是一样的．实例表明此方法适用于CAD软件系统。