Exploration of local surface geometry with minimum number of contact points and surface normal information

In this paper, we propose a method of exploring the surface geometry of an unknown object by touch. The method is based on the idea that a three-dimensional surface geometry can be reconstructed from two principal curvatures of the object which are estimated from three concurrent curves. First, the process to minimize the number of contact points is addressed for the approximation of an arbitrary curve, which uses normal vectors at the contact points. Then, an algorithm for reconstructing a three-dimensional local surface from four contact points, two of which can be used to compute a normal curvature, is presented. Lastly, our method is applied to cylindrical, spherical and planar objects in simulation and experiments for validation.     

Extraction of ridge and valley lines from unorganized points

Given an unstructured point set, we use an MLS (moving least-squares) approximation to estimate the local curvatures and their derivatives at a point by means of an approximating surface. Then, we compute neighbor information using a Delaunay tessellation. Ridge and valley points can then be detected as zero-crossings, and connected using curvature directions. We demonstrate our method on several large point-sampled models, rendered by point-splatting, on which the ridge and valley lines are rendered with line width determined from curvatures.     

Exploration of unknown object by active touch of robot hand

This paper proposes a method of exploring the local shape of an unknown object using the force and torque information obtained from active touch. In the first, we present a method to estimate an unknown curvature, using rolling and sliding motion with a force/torque sensor attached to the fingertip of the hand. Then, the normal curvature equation from 2D curvatures is obtained. Finally we present a reconstruction algorithm of local geometry by using a normal curvature equation, which is composed of principal curvatures and principal directions. The method is tested by using a hand-arm system consisting of an industrial robot arm and an anthropomorphic robot hand with 6-axis force/torque sensor. The feasibility of the proposed method is experimentally validated for objects with simple geometries such as cylinder, spheres etc.     

An adaptive method for detecting dominant points

In this paper, we propose an adaptive method for the polygonal approximation of a digitized curve. Instead of setting a fixed length of support region in advance, the new method will compute the suitable length of support region for each point to find the best approximated curvature. The dominant points are identified as the points with local maximum curvatures. In addition, the break point detection is conducted to reduce the computations. The experimental results show that the proposed method can approximate the curves effectively.     

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)     

Rendering stylized highlights using projective textures

We present a method to render and control stylized highlights using projective textures for real-time applications. For each vertex of a given 3D model, we compute principal directions and their corresponding principal curvatures in the preprocessing step. We find the maximum specular intensity point on the surface of the model and compute the principal directions and curvatures at the point by bilinear interpolation. The position, orientation, and projection frustum of a texture projector are determined by the principal directions and curvatures at the point. Using a simple culling method, we reduce the number of triangles to compute the principal directions and curvatures. We can control the shape, size, position, and orientation of a stylized highlight in an easy, fast, and intuitive manner. We discuss the limitations of our method and also give approximate solutions for some limitations.     

Curvature estimation for meshes based on vertex normal triangles

The estimation of surface curvature is essential for a variety of applications in computer graphics because of its invariance with respect to rigid transformations. In this article, we describe a curvature estimation method for meshes by converting each planar triangular facet into a curved patch using the vertex positions and the normals of three vertices of each triangle. Our method interpolates three end points and the corresponding normal vectors of each triangle to construct a curved patch. Then, we compute the per triangle curvature of the neighboring triangles of a mesh point of interest. Similar to estimating per vertex normal from the adjacent per triangle normal, we compute the per vertex curvature by taking a weighted average of per triangle curvature. Through some examples, we demonstrate that our method is efficient and its accuracy is comparable to that of the existing methods.     

基于模糊极大似然估计聚类的点云数据分块

对散乱点云数据采用微切平面法进行法矢估计,对法矢方向进行全局协调性调整。采用稳定性较好的二次曲面拟合法估算点云数据的高斯曲率和平均曲率。将点的坐标、法矢和曲率合并为八维特征向量,通过模糊极大似然估计聚类技术,将具有类似几何特征的向量聚为一类,从而实现点云数据的分块。实验证明该方法有效。     

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.     

三角形网格模型顶点曲率的求解算法*

推导了一般三角形网格模型顶点的平均曲率、高斯曲率和主曲率的计算方法,考虑到经常遇到粗糙三角形网格模型,为提高其曲率计算方法的精度,结合Loop细分曲面算法,进一步拓展了该曲率计算方法.该算法用于具有特征保持的网格模型简化取得了良好的效果.     

Curvature-Domain Shape Processing

We propose a framework for 3D geometry processing that provides direct access to surface curvature to facilitate advanced shape editing, filtering, and synthesis algorithms. The central idea is to map a given surface to the curvature domain by evaluating its principle curvatures, apply filtering and editing operations to the curvature distribution, and reconstruct the resulting surface using an optimization approach. Our system allows the user to prescribe arbitrary principle curvature values anywhere on the surface. The optimization solves a nonlinear least‐squares problem to find the surface that best matches the desired target curvatures while preserving important properties of the original shape. We demonstrate the effectiveness of this processing metaphor with several applications, including anisotropic smoothing, feature enhancement, and multi‐scale curvature editing.     

Characterizing three-dimensional surface structures from visualimages

A new technique for computing intrinsic surface properties is presented. Intrinsic surface properties are those properties of a surface that are not affected by the choice of the coordinate system, the position of the viewer relative to the surface, and the particular parametric representation used to describe the imaged surface. Since intrinsic properties are characteristics of a surface, they are ideal for the purposes of representation and recognition. The intrinsic properties of interest are the principal curvatures, the Gaussian curvatures, and the lines of curvature. It is proposed that a structured-light sensing configuration where a grid pattern is projected to encode the imaged surfaces for analysis be adopted. At each stripe junction, the curvatures of the projected stripes on the imaged surface are computed and related to those of the normal sections that share the same tangential directional as the projected curves. The principal curvatures and their directions at the stripe junction under consideration are then recovered using Euler's theorem. Results obtained using both synthetic and real images are presented     

Surface curvature as a measure of image texture

Methods from the theory of surfaces in differential geometry are applied to the gray tone intensity and absorbance surfaces defined by a digital image. In particular, the first fundamental form, the second fundamental form, the two principal curvatures, the Gaussian curvature, and mean curvature from classical differential geometry are presented. New curvatures are introduced which appear to be more appropriate for the purposes of image texture. For each curvature, a texture feature is generated. These features are applied to biological cell nuclei and it is found that they are useful for purposes of discrimination.     

A Riemannian approach to graph embedding

In this paper, we make use of the relationship between the Laplace-Beltrami operator and the graph Laplacian, for the purposes of embedding a graph onto a Riemannian manifold. To embark on this study, we review some of the basics of Riemannian geometry and explain the relationship between the Laplace-Beltrami operator and the graph Laplacian. Using the properties of Jacobi fields, we show how to compute an edge-weight matrix in which the elements reflect the sectional curvatures associated with the geodesic paths on the manifold between nodes. For the particular case of a constant sectional curvature surface, we use the Kruskal coordinates to compute edge weights that are proportional to the geodesic distance between points. We use the resulting edge-weight matrix to embed the nodes of the graph onto a Riemannian manifold. To do this, we develop a method that can be used to perform double centring on the Laplacian matrix computed from the edge-weights. The embedding coordinates are given by the eigenvectors of the centred Laplacian. With the set of embedding coordinates at hand, a number of graph manipulation tasks can be performed. In this paper, we are primarily interested in graph-matching. We recast the graph-matching problem as that of aligning pairs of manifolds subject to a geometric transformation. We show that this transformation is Pro-crustean in nature. We illustrate the utility of the method on image matching using the COIL database.     

Ricci flow embedding for rectifying non-Euclidean dissimilarity data

Pairwise dissimilarity representations are frequently used as an alternative to feature vectors in pattern recognition. One of the problems encountered in the analysis of such data is that the dissimilarities are rarely Euclidean, while statistical learning algorithms often rely on Euclidean dissimilarities. Such non-Euclidean dissimilarities are often corrected or a consistent Euclidean geometry is imposed on them via embedding. This paper commences by reviewing the available algorithms for analysing non-Euclidean dissimilarity data. The novel contribution is to show how the Ricci flow can be used to embed and rectify non-Euclidean dissimilarity data. According to our representation, the data is distributed over a manifold consisting of patches. Each patch has a locally uniform curvature, and this curvature is iteratively modified by the Ricci flow. The raw dissimilarities are the geodesic distances on the manifold. Rectified Euclidean dissimilarities are obtained using the Ricci flow to flatten the curved manifold by modifying the individual patch curvatures. We use two algorithmic components to implement this idea. Firstly, we apply the Ricci flow independently to a set of surface patches that cover the manifold. Second, we use curvature regularisation to impose consistency on the curvatures of the arrangement of different surface patches. We perform experiments on three real world datasets, and use these to determine the importance of the different algorithmic components, i.e. Ricci flow and curvature regularisation. We conclude that curvature regularisation is an essential step needed to control the stability of the piecewise arrangement of patches under the Ricci flow.     

Cooperative exploration of level surfaces of three dimensional scalar fields

We develop strategies for a group of mobile sensing agents to cooperatively explore level surfaces of an unknown 3D scalar field. A cooperative Kalman filter is constructed to combine sensor readings from all agents and give estimates of the field value and gradient at the center of the formation formed by the sensing agents. The formation formed by the agents is controlled to track curves on a level surface in the field under steering control laws. We prove that the formation center can move to a desired level surface and can follow a curve with known frame and curvatures. In particular, we present results on tracking lines of curvature on a desired level surface, revealing the 3D geometry of the scalar field. Taubin’s algorithm is modified and applied to detect and estimate principal curvatures and principal directions for lines of curvature. We prove the sufficient and necessary conditions that ensure reliable estimates using Taubin’s algorithm. We also theoretically justify the minimum number of agents that can be utilized to accomplish the exploration tasks. Simulation results demonstrate that a line of curvature on a desired level surface can be detected and traced successfully.     

Marr-Hildreth边界检测器定位性能分析

本文分析了Marr-Hildreth边界检测器(-△2G)在检测图象中的阶跃(step)边界时产 生定位误差的三个原因,即由于侧向曲率、高斯滤波和噪声干扰带来的影响.首先导出了拉普 拉斯、二阶梯度方向导数和侧向曲率三者之间的关系式,指出由于侧向曲率的影响,拉普拉斯 在检测弯曲边界时必然产生定位误差.依据按曲率分段逼近阶跃边界的模型,给出了误差的 定量分析.最后解释了高斯滤波和噪声干扰对定位的影响.     

Curvature-augmented tensor voting for shape inference from noisy 3Ddata

Improves the basic tensor voting formalism to infer the sign and direction of principal curvatures at each input site from noisy 3D data. Unlike most previous approaches, no local surface fitting, partial derivative computation, nor oriented normal vector recovery is performed in our method. These approaches are known to be noise-sensitive, since accurate partial derivative information is often required, which is usually unavailable from real data. Also, unlike approaches that detect signs of Gaussian curvature, we can handle points with zero Gaussian curvature uniformly, without first localizing them in a separate process. The tensor-voting curvature estimation is non-iterative, does not require initialization, and is robust to a considerable amount of outlier noise, as its effect is reduced by collecting a large number of tensor votes. Qualitative and quantitative results on synthetic and real complex data are presented     

Mesh Statistics for Robust Curvature Estimation

While it is usually not difficult to compute principal curvatures of a smooth surface of sufficient differentiability, it is a rather difficult task when only a polygonal approximation of the surface is available, because of the inherent ambiguity of such representation. A number of different approaches has been proposed in the past that tackle this problem using various techniques. Most papers tend to focus on a particular method, while an comprehensive comparison of the different approaches is usually missing. We present results of a large experiment, involving both common and recently proposed curvature estimation techniques, applied to triangle meshes of varying properties. It turns out that none of the approaches provides reliable results under all circumstances. Motivated by this observation, we investigate mesh statistics, which can be computed from vertex positions and mesh connectivity information only, and which can help in deciding which estimator will work best for a particular case. Finally, we propose a meta‐estimator, which makes a choice between existing algorithms based on the value of the mesh statistics, and we demonstrate that such meta‐estimator, despite its simplicity, provides considerably more robust results than any existing approach.