Curvature estimation scheme for triangle meshes using biquadratic Bézier patches

When dealing with triangle meshes, it is often important to compute curvature information for the purposes of feature recognition, segmentation, or shape analysis. Since a triangle mesh is a piecewise linear surface, curvature has to be estimated. Several different schemes have been proposed, both discrete and continuous, i.e. based on fitting surfaces locally. This paper compares commonly used discrete and continuous curvature estimation schemes. We also present a novel method which uses biquadratic Bézier patches as a local surface fitting technique.     

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.     

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 Fairing towards Regular Principal Curvature Line Networks

Freeform surfaces whose principal curvature line network is regularly distributed, are essential to many real applications like CAD modeling, architecture design, and industrial fabrication. However, most designed surfaces do not hold this nice property because it is hard to enforce such constraints in the design process. In this paper, we present a novel method for surface fairing which takes a regular distribution of the principal curvature line network on a surface as an objective. Our method first removes the high‐frequency signals from the curvature tensor field of an input freeform surface by a novel rolling guidance tensor filter, which results in a more regular and smooth curvature tensor field, then deforms the input surface to match the smoothed field as much as possible. As an application, we solve the problem of approximating freeform surfaces with regular principal curvature line networks, discretized by quadrilateral meshes. By introducing the circular or conical conditions on the quadrilateral mesh to guarantee the existence of discrete principal curvature line networks, and minimizing the approximate error to the original surface and improving the fairness of the quad mesh, we obtain a regular discrete principal curvature line network that approximates the original surface. We evaluate the efficacy of our method on various freeform surfaces and demonstrate the superiority of the rolling guidance tensor filter over other tensor smoothing techniques. We also utilize our method to generate high‐quality circular/conical meshes for architecture design and cyclide spline surfaces for CAD modeling.     

Gaussian curvature using fundamental forms for binary voxel data

Local curvature characterizes every point of a surface and measures its deviation from a plane, locally. One application of local curvature measures within the field of image and geometry processing is object segmentation. Here, we present and evaluate a novel algorithm based on the fundamental forms to calculate the curvature on surfaces of objects discretized with respect to a regular three-dimensional grid. Thus, our new algorithm is applicable to voxel data, which are created e.g. from computed tomography (CT). Existing algorithms for binary data used the Gauss map, rather than fundamental forms. For the calculation of the fundamental forms, derivatives of a surface in tangent directions in every point of the surface have to be computed. Since the surfaces exist on grids with restricted resolution, these derivatives have to be discretized. In the presented method, this is realized by projecting the tangent plane onto the discrete object surface. The most important parameter of the proposed algorithm is the size of the chosen window for the calculation of the gradient. The size of this window has to be selected according to object size as well as with respect to distances between objects. In our experiments, an algorithm based on the Gauss map provided inconsistent values for simple test objects, whereas our method provides consistent values. We report quantitative results on various test geometries, compare our method to two algorithms working on gray value data and demonstrate the practical applicability of our novel algorithm to CT-reconstructions of Greenlandic firn.     

具有特殊效果的混合细分方法

提出了基于三角形和四边形的混合控制网格的细分曲面尖锐特征、半尖锐特征生成和控制方法,避免了已有方法仅局限于初始控制网格为单一的三角形或单一的四边形网格的缺陷.通过局部修改混合细分规则,在光滑混合曲面上产生了刺、尖、折痕、角的尖锐特征效果,并对尖锐特征处局部细分矩阵进行了详细的特征分析,讨论了极限曲面的收敛性及光滑性.同时,用特征处的离散曲率来控制特征处的尖锐程度,实现了半尖锐的特征效果,并通过自适应细分方法,把尖锐特征、半尖锐特征的生成统一起来.该方法具有多分辨率表示能力强、局部性好、简单易操作的特点.实验结果表明,该算法效果好,成功地解决了混合曲面特殊效果生成问题.     

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.     

Partitioning 3D surface meshes using watershed segmentation

This paper describes a method for partitioning 3D surface meshes into useful segments. The proposed method generalizes morphological watersheds, an image segmentation technique, to 3D surfaces. This surface segmentation uses the total curvature of the surface as an indication of region boundaries. The surface is segmented into patches, where each patch has a relatively consistent curvature throughout, and is bounded by areas of higher, or drastically different, curvature. This algorithm has applications for a variety of important problems in visualization and geometrical modeling including 3D feature extraction, mesh reduction, texture mapping 3D surfaces, and computer aided design     

Improved,feature-centric EMD for 3D surface modeling and processing

Since late 1990s, Empirical Mode Decomposition (EMD) starts to emerge as a powerful tool for processing non-linear and non-stationary signals. Nonetheless, the research on exploring EMD-relevant techniques in the domain of geometric modeling and processing is extremely rare. Directly applying EMD to coordinate functions of 3D shape geometry will not take advantage of the attractive EMD properties. To ameliorate, in this paper we articulate a novel 3D surface modeling and processing framework founded upon improved, feature-centric EMD, with a goal of realizing the full potential of EMD. Our strategy starts with a measure of mean curvature as a surface signal for EMD. Our newly-formulated measure of mean curvature is computed via the inner product of Laplacian vector and vertex normal. Such measure is both rotation-invariant and translation-invariant, facilitates the computation of different scale features for original surfaces, and avoids boundary shrinkage when processing open surfaces. Moreover, we modify the original EMD formulation by devising a feature-preserving multiscale decomposition algorithm for surface analysis and synthesis. The key idea is to explicitly formulate details as oscillation between local minima and maxima. Within our novel framework, we could accommodate many modeling and processing operations, such as filter design, detail transfer, and feature-preserving smoothing and denoising. Comprehensive experiments and quantitative evaluations/comparisons on popular models have demonstrated that our new surface processing methodology and algorithm based on the improved, feature-centric EMD are of great value in digital geometry processing, analysis, and synthesis.     

Surface processing methods for point sets using finite elements

We present a framework for processing point-based surfaces via partial differential equations (PDEs). Our framework efficiently and effectively brings well-known PDE-based processing techniques to the field of point-based surfaces. At the core of our method is a finite element discretization of PDEs on point surfaces. This discretization is based on the local assembly of PDE-specific mass and stiffness matrices, using a local point coupling computation. Point couplings are computed using a local tangent plane construction and a local Delaunay triangulation of point neighborhoods. The definition of tangent planes relies on moment-based computation with proven scaling and stability properties. Once local stiffness matrices are obtained, we are able to easily assemble global matrices and efficiently solve the corresponding linear systems by standard iterative solvers. We demonstrate our framework by several types of PDE-based surface processing applications, such as segmentation, texture synthesis, bump mapping, and geometric fairing.     

Shape differentiation of freeform surfaces using a similarity measure based on an integral of Gaussian curvature

Freeform surfaces are popularly used to design and model complex 3D objects. These 3D models are stored as computerized models in databases. To facilitate data retrieval and shape matching, a major challenge lies in defining and computing the level of similarity between two or more freeform surfaces. In order to explore the useful 3D information associated with the surfaces, an integrated approach based on the integral of Gaussian curvature is proposed to develop the measures of similarity of freeform surfaces. Specifically, the integral of Gaussian curvature is mapped into the 2D space, and a shape-based measure is developed using statistical methods to compute the level of similarity. For smooth surfaces, a fast approximation algorithm is developed to calculate the curvature of individual subregions. In cases where the target surface has a complex topological structure or a smooth surface is not available, the integral of Gaussian curvature for the discrete surface is first calculated at each vertex, followed by mapping onto a 2D spherical coordinate. The distance measure focuses on the local geometry, which is critical to investigate models with a certain level of resemblance such as products in a family. This proposed approach can be applied to surfaces under various transformations, as well as 3D data from various sources.     

三维网格曲面的可展性优化算法

可展曲面是每点高斯曲率均为0的曲面,具有许多良好的性质,因此在工业中具有很多应用.将一般的曲面用可展曲面来逼近表示具有现实意义.以此为目的,文中设计了一个有效的算法来处理一般的曲面,使得处理后的曲面尽可能满足可展的性质,同时与初始的曲面尽量地接近.首先利用最小范数方法来对网格曲面进行处理,得到初始预测的网格曲面.初始预测曲面具有良好的可展性,但是不能较好地保持网格的局部结构.然后利用尽可能刚性(As-rigid-as-possible)的方法,在初始预测曲面的基础上进行修正得到新的网格曲面.为了保持局部结构,作者的方法可以是基于顶点邻域的,也可以是基于三角形的.这两个过程可以迭代进行,直至得到满足要求的结果.与以往的算法相比,文中算法能保证结果收敛,迭代次数更少,且能得到更好的结果.     

基于多分辨率的厄米高斯矩的LBP纹理分类

针对LBP（局部二值模式)纹理描述子局限于在单一分辨率下捕获纹理图像的纹理信息的问题,提出一种基于多分辨率的厄米高斯矩的LBP纹理分类方法。首先结合图像纹理的多分辨率特性,采用厄米高斯矩对图像进行多分辨率重构,然后利用LBP纹理描述子对重构图像进行特征提取,最后采用K近邻特征空间距离的分类方法进行纹理分类。选取KTH-TIPS纹理数据库的纹理图像进行测试实验,实验结果表明,与传统LBP纹理分类方法相比,使用多分辨率的厄米高斯矩的LBP纹理分类方法进行纹理分类,可以更加全面地描述图像的纹理信息,使纹理分类准确率更高。     

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.     

An adaptive machine learning algorithm for color image analysis and processing

In this paper, a new adaptive machine learning algorithm for analyzing and processing color images of natural scenes is presented. The eventual goal of this research is to obtain a mathematical training algorithm to guide the operation of an unsupervised pattern recognition and classification technique for detecting and extracting the image modes or clusters in a selected or constructed feature space. For this purpose, the peak modality of one-dimensional (1-D) image histograms is selected as the mathematical training criterion. Area, mode dispersion, approximated curvature and steepness are some of the measured quantities for a modality test. Linear discriminant function is then used to extract the detected image clusters in the feature or measurement space.     

Digital Differential Geometry Processing

The theory and methods of digital geometry processing has been a hot research area in computer graphics, as geometric models serves as the core data for 3D graphics applications. The purpose of this paper is to introduce some recent advances in digital geometry processing, particularly mesh fairing, surface parameterization and mesh editing, that heavily use differential geometry quantities. Some related concepts from differential geometry, such as normal, curvature, gradient, Laplacian and their counterparts on digital geometry are also reviewed for understanding the strength and weakness of various digital geometry processing methods.     

用主动轮廓模型优化网格曲面上的特征线

对主动轮廓模型在三维网格曲面上的表示进行研究．首先提出一种根据输入的点快速确定初始特征线的追踪投影法；然后计算出特征线的主动轮廓模型能量，其中特征能用平均曲率来表示；最后，特征线经多次迭代后移动到能量极小处，实现优化．实例表明，优化后的特征线既光滑又逼近特征．     

一种基于深度图像的自遮挡检测方法

针对视觉目标存在的自遮挡现象,并为更好地界定、规避自遮挡现象提供依据,提出一种完全基于目标深度图像信息、仅需通过分析深度图像平均曲率变化特征并结合使用二次阈值法进行自遮挡检测的方法.为避免平均曲率计算的复杂性,该方法首先采用改进的离散正交多项式局部曲面拟合法计算深度图像的平均曲率;然后,通过分析图像各点的平均曲率并结合曲率阈值提取与其八邻域点存在曲率异号的点组成自遮挡候选点集;最后,依据候选点与以其为中心的窗口内其它点存在深度值不连续的现象,再次使用阈值法,实现对自遮挡的检测.实验结果表明该方法能够有效地检测出自遮挡现象并获得自遮挡边界.