Reduce the stretch in surface flattening by finding cutting paths to the surface boundary

This paper presents a method for finding cutting paths on a 3D triangular mesh surface to reduce the stretch in the flattened surface. The cutting paths link the surface boundary and the nodes where the Gaussian curvature is high, and their total length is minimized. First, a linear algorithm for computing an approximate boundary geodesic distance map is introduced; the map encapsulates the undirected geodesic distance from every triangular node to the surface boundary approximately. This is followed by determining the undirected shortest paths passing through all the nodes where the Gaussian curvature is larger than a threshold. The cutting paths walk along the triangular edges of the given surface. Compared with other similar approaches, our method reaches a faster speed, and can deal with surfaces with widely distributed curvatures.     

Normal Vector Voting: Crease Detection and Curvature Estimation on Large,Noisy Meshes

This paper describes a robust method for crease detection and curvature estimation on large, noisy triangle meshes. We assume that these meshes are approximations of piecewise-smooth surfaces derived from range or medical imaging systems and thus may exhibit measurement or even registration noise. The proposed algorithm, which we call normal vector voting, uses an ensemble of triangles in the geodesic neighborhood of a vertex—instead of its simple umbrella neighborhood—to estimate the orientation and curvature of the original surface at that point. With the orientation information, we designate a vertex as either lying on a smooth surface, following a crease discontinuity, or having no preferred orientation. For vertices on a smooth surface, the curvature estimation yields both principal curvatures and principal directions while for vertices on a discontinuity we estimate only the curvature along the crease. The last case for no preferred orientation occurs when three or more surfaces meet to form a corner or when surface noise is too large and sampling density is insufficient to determine orientation accurately. To demonstrate the capabilities of the method, we present results for both synthetic and real data and compare these results to the G. Taubin (1995, in Proceedings of the Fifth International Conference on Computer Vision, pp. 902–907) algorithm. Additionally, we show practical results for several large mesh data sets that are the motivation for this algorithm.     

A curvature-based approach to terrain recognition

The authors describe an algorithm which uses a Gaussian and mean curvature profile for extracting special points on terrain and then use these points for recognition of particular regions of the terrain. The Gaussian and mean curvatures are chosen because they are invariant under rotation and translation. In the Gaussian and mean curvature image, the points of maximum and minimum curvature are extracted and used for matching. The stability of the position of those points in the presence of noise and with resampling is investigated. The input for this algorithm consists of 3-D digital terrain data. Curvature values are calculated from the data by fitting a quadratic surface over a square window and calculating directional derivatives of this surface. A method of surface fitting which is invariant to coordinate system transformation is suggested and implemented. The algorithm is tested with and without the presence of noise, and its performance is described     

Exact and interpolatory quadratures for curvature tensor estimation

The computation of the curvature of smooth surfaces has a long history in differential geometry and is essential for many geometric modeling applications such as feature detection. We present a novel approach to calculate the mean curvature from arbitrary normal curvatures. Then, we demonstrate how the same method can be used to obtain new formulae to compute the Gaussian curvature and the curvature tensor. The idea is to compute the curvature integrals by a weighted sum by making use of the periodic structure of the normal curvatures to make the quadratures exact. Finally, we derive an approximation formula for the curvature of discrete data like meshes and show its convergence if quadratically converging normals are available.     

三维模型表面特征的水墨效果渲染算法

提出一种对三维模型实现水墨效果渲染的两步二维纹理映射方法.该方法首先通过模型顶点的视线向量和法线向量的点乘积计算其夹角的余弦值,作为第一步的纹理坐标进行映射;再通过欧拉公式计算径向曲率及其方向导数值,作为第二步映射的纹理坐标.为达到实时的交互浏览,在第二步映射时还可采用基于视点无关的曲率特征的映射策略(如平均曲率、高斯曲率、主曲率等).在快速生成水墨画效果的同时也显示出更多的特征细节.     

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。     

离散三角网格模型顶点法矢量估算

提出一种新的三角网格模型顶点法矢估算方法,采用以三角网格顶点一阶邻域三角形的形状因子与顶点到三角形质心距进行综合加权的方法。同时指出：在同等三角网格曲面,随着三角网格划分精度的提升,网格顶点法矢估算精度有增大趋势;在同等网格划分精度条件下,对于平均曲率小以及平均曲率变化率小的三角网格模型,其网格顶点法矢估算精度也有增大趋势。实例计算和误差分析表明,该方法的计算结果更为精确合理。     

Improving three-dimensional point reconstruction from image correspondences using surface curvatures

Recovering three-dimensional (3D) points from image correspondences is an important and fundamental task in computer vision. Traditionally, the task is completed by triangulation whose accuracy has its limitation in some applications. In this paper, we present a framework that incorporates surface characteristics such as Gaussian and mean curvatures into 3D point reconstruction to enhance the reconstruction accuracy. A Gaussian and mean curvature estimation scheme suitable to the proposed framework is also introduced in this paper. Based on this estimation scheme and the proposed framework, the 3D point recovery from image correspondences is formulated as an optimization problem with the surface curvatures modeled as soft constraints. To analyze the performance of proposed 3D reconstruction approach, we generated some synthetic data, including the points on the surfaces of a plane, a cylinder and a sphere, to test the approach. The experimental results demonstrated that the proposed framework can indeed improve the accuracy of 3D point reconstruction. Some real-image data were also tested and the results also confirm this point.     

Discrete surface Ricci flow

This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by user-defined Gaussian curvatures. Furthermore, the target metrics are conformal (angle-preserving) to the original metrics. A Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newton's method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient. Our experimental results demonstrate the efficiency, accuracy and flexibility of the algorithms. They have the potential for a wide range of applications in graphics, geometric modeling, and medical imaging. We demonstrate their practical values by global surface parameterizations.     

Contouring controller design based on iterative contour error estimation for three-dimensional machining

Recently, there has been a growth of interest in high precision machining in multi-axis feed drive systems, subjected to problems such as friction, cutting force and incompatibility of individual driving axis dynamics. To guarantee high precision machining in modern computer numerical controlled (CNC) machines, CNC's controllers do its control efforts to reduce contour error. One of the common approaches is to design a controller based on the estimation of contour error in real time. However, for complex contours with severe curvatures, there is a lack of effective algorithms to calculate contour errors accurately. To address this problem, this paper proposes an accurate contour error estimation procedure for three-dimensional machining tasks. The proposed method is based on an iterative estimation of the instantaneous curvature of the reference trajectory and coordinates transformation approach, and hence, it is effective for complex reference trajectories with high curvatures. In addition, contour error controller is presented to reduce the estimated contour error. The feasibility and superiority of the proposed model as well as contour error controller are demonstrated through experimental system using a desk-top three-axis CNC machine.     

Point fingerprint: a new 3-D object representation scheme

This paper proposes a new, efficient surface representation method for surface matching. A feature carrier for a surface point, which is a set of two-dimensional (2-D) contours that are the projections of geodesic circles on the tangent plane, is generated. The carrier is named point fingerprint because its pattern is similar to human fingerprints and plays a role in discriminating surface points. Corresponding points on surfaces from different views are found by comparing their fingerprints. The point fingerprint is able to carry curvature, color, and other information which can improve matching accuracy, and the matching process is faster than 2-D image comparison. A novel candidate point selection method based on the fingerprint irregularity is introduced. Point fingerprint is successfully applied to pose estimation of real range data.     

Robust estimation of adaptive tensors of curvature by tensor voting

Although curvature estimation from a given mesh or regularly sampled point set is a well-studied problem, it is still challenging when the input consists of a cloud of unstructured points corrupted by misalignment error and outlier noise. Such input is ubiquitous in computer vision. In this paper, we propose a three-pass tensor voting algorithm to robustly estimate curvature tensors, from which accurate principal curvatures and directions can be calculated. Our quantitative estimation is an improvement over the previous two-pass algorithm, where only qualitative curvature estimation (sign of Gaussian curvature) is performed. To overcome misalignment errors, our improved method automatically corrects input point locations at subvoxel precision, which also rejects outliers that are uncorrectable. To adapt to different scales locally, we define the RadiusHit of a curvature tensor to quantify estimation accuracy and applicability. Our curvature estimation algorithm has been proven with detailed quantitative experiments, performing better in a variety of standard error metrics (percentage error in curvature magnitudes, absolute angle difference in curvature direction) in the presence of a large amount of misalignment noise.     

Surface parameterization in volumetric images for curvature-based feature classification

Curvature-based surface features are well suited for use in multimodal medical image registration. The accuracy of such feature-based registration techniques is dependent upon the reliability of the feature computation. The computation of curvature features requires second derivative information that is best obtained from a parametric surface representation. We present a method of explicitly parameterizing surfaces from volumetric data. Surfaces are extracted, without a global thresholding, using active contour models. A monge/spl acute/ basis for each surface patch is estimated and used to transform the patch into local, or parametric, coordinates. Surface patches are fit to a bicubic polynomial in local coordinates using least squares solved by singular value decomposition. We tested our method by reconstructing surfaces from the surface model and analytically computing Gaussian and mean curvatures. The model was tested on analytical and medical data.     

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     

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     

Anisotropic Filtering of Non-Linear Surface Features

A new method for noise removal of arbitrary surfaces meshes is presented which focuses on the preservation and sharpening of non‐linear geometric features such as curved surface regions and feature lines. Our method uses a prescribed mean curvature flow (PMC) for simplicial surfaces which is based on three new contributions: 1. the definition and efficient calculation of a discrete shape operator and principal curvature properties on simplicial surfaces that is fully consistent with the well‐known discrete mean curvature formula, 2. an anisotropic discrete mean curvature vector that combines the advantages of the mean curvature normal with the special anisotropic behaviour along feature lines of a surface, and 3. an anisotropic prescribed mean curvature flow which converges to surfaces with an estimated mean curvature distribution and with preserved non‐linear features. Additionally, the PMC flow prevents boundary shrinkage at constrained and free boundary segments.     

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.