Estimation of Error in Curvature Computation on Multi-Scale Free-Form Surfaces

A novel technique for multi-scale curvature computation on a free-form 3-D surface is presented. This is achieved by convolving local parametrisations of the surface with 2-D Gaussian filters iteratively. In our technique, semigeodesic coordinates are constructed at each vertex of the mesh. Smoothing results are shown for 3-D surfaces with different shapes indicating that surface noise is eliminated and surface details are removed gradually. A number of evolution properties of 3-D surfaces are described. Next, the surface Gaussian and mean curvature values are estimated accurately at multiple scales which are then mapped to colours and displayed directly on the surface. The performance of the technique when selecting different directions as an arbitrary direction for the geodesic at each vertex are also presented. The results indicate that the error observed for the estimation of Gaussian and mean curvatures is quite low after only one iteration. Furthermore, as the surface is smoothed iteratively, the error is further reduced. The results also show that the estimation error of Gaussian curvature is less than that of mean curvature. Our experiments demonstrate that estimation of smoothed surface curvatures are very accurate and not affected by the arbitrary direction of the first geodesic line when constructing semigeodesic coordinates. Our technique is independent of the underlying triangulation and is also more efficient than volumetric diffusion techniques since 2-D rather than 3-D convolutions are employed. Finally, the method presented here is a generalisation of the Curvature Scale Space method for 2-D contours. The CSS method has outperformed comparable techniques within the MPEG-7 evaluation framework. As a result, it has been selected for inclusion in the MPEG-7 package of standards.     

Axis estimation and grouping of rotationally symmetric object segments

This paper proposes a new method for estimating the symmetric axis of a pottery from its small fragment using surface geometry. Also, it provides a scheme for grouping such fragments into shape categories using distribution of surface curvature. For automatic assembly of pot from broken sherds, axis estimation is an important task and when a fragment is small, it is difficult to estimate axis orientation since it looks like a patch of a sphere and conventional methods mostly fail. But the proposed method provides fast and robust axis estimation by using multiple constraints. The computational cost is also too lowered. To estimate the symmetric axis, the proposed algorithm uses three constraints: (1) The curvature is constant on a circumference C_H. (2) The curvature is invariant in any scale. (3) Also the principal curvatures does not vary on C_H. C_H is a planar circle which is one of all the possible circumferences of a pottery or sherd. A hypothesis test for axis is performed using maximum likelihood. The variance of curvature, multi-scale curvature and principal curvatures is computed in the likelihood function. We also show that the principal curvatures can be used for grouping of sherds. The grouping of sherds will reduce the computation significantly by omitting impossible configurations in broken pottery assembly process.     

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.     

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)     

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     

基于矩形网格追踪法的曲面主曲率等值线生成算法*

针对自由曲面在展开平面上的主曲率等值线生成问题,提出一种基于矩形网格追踪算法的等值线生成算法。利用面积坐标方法将曲面上的点及其极值曲率映射到优化展开平面上,在展开平面上采用遍历法得出等值点序列;利用矩形网格追踪法将等值点序列划分为连续无交叉的开等值线或连续封闭的闭等值线。最后以直纹面、锥形面和自由曲面为算例分别对算法进行计算验证,并与线性插值三角网格法进行对比。计算结果表明,基于矩形网格追踪的等值线生成算法具有计算速度快、精度高的特点。     

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.     

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.     

Modelling fingerprint ridge orientation using Legendre polynomials

The estimation of fingerprint ridge orientation is an essential step in every automatic fingerprint verification system. The importance of ridge orientation can be deflected from the fact that it is inevitably used for detecting, describing and matching fingerprint features such as minutiae and singular points. In this paper we propose a novel method for fingerprint ridge orientation modelling using Legendre polynomials. One of the main problems it addresses is smoothing orientation data while preserving details in high curvature areas, especially singular points. We show that singular points, which result in a discontinuous orientation field, can be modelled by the zero-poles of Legendre polynomials. The models parameters are obtained in a two staged optimization procedure. Another advantage of the proposed method is a very compact representation of the orientation field, using only 56 coefficients. We have carried out extensive experiments using a state-of-the-art fingerprint matcher and a singular point detector. Moreover, we compared the proposed method with other state-of-the-art fingerprint orientation estimation algorithms. We can report significant improvements in both singular point detection and matching rates.     

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.     

Range image segmentation based on differential geometry: a hybridapproach

The authors describe a hybrid approach to the problem of image segmentation in range data analysis, where hybrid refers to a combination of both region- and edge-based considerations. The range image of 3-D objects is divided into surface primitives which are homogeneous in their intrinsic differential geometric properties and do not contain discontinuities in either depth of surface orientation. The method is based on the computation of partial derivatives, obtained by a selective local biquadratic surface fit. Then, by computing the Gaussian and mean curvatures, an initial region-gased segmentation is obtained in the form of a curvature sign map. Two additional initial edge-based segmentations are also computed from the partial derivatives and depth values, namely, jump and roof-edge maps. The three image maps are then combined to produce the final segmentation. Experimental results obtained for both synthetic and real range data of polyhedral and curved objects are given     

Robust Feature-Preserving Mesh Denoising Based on Consistent Subneighborhoods

In this paper, we introduce a feature-preserving denoising algorithm. It is built on the premise that the underlying surface of a noisy mesh is piecewise smooth, and a sharp feature lies on the intersection of multiple smooth surface regions. A vertex close to a sharp feature is likely to have a neighborhood that includes distinct smooth segments. By defining the consistent subneighborhood as the segment whose geometry and normal orientation most consistent with those of the vertex, we can completely remove the influence from neighbors lying on other segments during denoising. Our method identifies piecewise smooth subneighborhoods using a robust density-based clustering algorithm based on shared nearest neighbors. In our method, we obtain an initial estimate of vertex normals and curvature tensors by robustly fitting a local quadric model. An anisotropic filter based on optimal estimation theory is further applied to smooth the normal field and the curvature tensor field. This is followed by second-order bilateral filtering, which better preserves curvature details and alleviates volume shrinkage during denoising. The support of these filters is defined by the consistent subneighborhood of a vertex. We have applied this algorithm to both generic and CAD models, and sharp features, such as edges and corners, are very well preserved.     

三角网格曲面上离散曲率估算方法的比较与分析

对国际上近几年提出的三角网格曲面上估算平均曲率的7种方法和估算高斯曲率的4种方法,进行了系统的总结与大量的实验,并给出误差统计和分析比较,给出了对高斯曲率和平均曲率的估算效果最优的方法,以及较稳定和误差较小的几个新公式．     

A new feature-preserving mesh-smoothing algorithm

We present a novel mesh denoising and smoothing method in this paper. Our approach starts by estimating the principal curvatures and mesh saliency value for each vertex. Then, we calculate the uniform principal curvature of each vertex based on the weighted average of local principal curvatures. After that, we use the weighted bi-quadratic Bézier surface to fit the neighborhood of each vertex using the least-square method and obtain the new vertex position by adjusting the parameters of the fitting surface. Experiments show that our smoothing method preserves the geometric feature of the original mesh model efficiently. Our approach also prevents the volume shrinkage of the input mesh and obtains smooth boundaries for non-closed mesh models.     

Interpretation of image flow: Rigid curved surfaces in motion

A new method is described for interpreting image flow (or optical flow) in a small field of view produced by a rigidly moving curved surface. The equations relating the shape and motion of the surface to the image flow are formulated. These equations are solved to obtainexplicit analytic expressions for the motion, orientation, and curvatures of the surface in terms of the spatial derivatives (up to second order) of the image flow. We state and prove somenew theoretical results concerning the existence of multiple interpretations. Numerical examples are given for some interesting cases where multiple solutions exist. The solution method described here is simpler and more direct than previous methods. The method and the representation described here are part of a unified approach for the interpretation of image motion in a variety of cases (e.g., planar/curved surfaces, constant/accelerated motion, etc.). Thus the representation and the method of analysis adopted here have some advantages in comparison with previous approaches.     

Numerical investigation of the effect of spiral curvature on the flow field in a spiral channel viscous micropump

This paper investigates the effect of spiral curvature on the flow field within a newly introduced spiral channel viscous micropump. Increasing the spiral curvature increases the angle between the channel axis and the direction of upper plate motion, and more fluid is dragged normal to the axis of the channel, which changes the flow characteristics. A number of 3D models for the pump geometry with different design parameters are built and analyzed using the finite volume method. Numerical visualization of the flow field contours through the spiral channel is presented. It has been found numerically that the flow rate varies linearly with both the pressure difference and boundary velocity for a wide range of design parameters, which supports the validity of the linear lubrication model for this problem for the full range of spiral curvatures studied. Further, it is found that the error resulting from ignoring the spiral curvature exceeds 5% for spiral curvature ratio above 2. Obtained results depict a complete representation of the effect of spiral channel width, height, length and curvature on the flow field through a spiral-grooved micropump.     

散乱点云数据曲率估计方法

针对带有强噪声离散点云数据曲率计算问题,提出一种基于稳健统计的曲率估计方法。首先,用一个二次曲面拟合三维空间采样点处的局部形状;其次,随机地选择该采样点邻域内的子集,多次执行这样的拟合过程,通过变窗宽的最大核密度估计,就得到了最优拟合曲面;最后,将采样点投影到该曲面上,计算投影点曲率信息,就得到采样点曲率。实验结果表明,所提方法对噪声和离群点是稳健的,特别是随着噪声方差的增大,要明显好于传统的抛物拟合方法。     

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。     

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.