An analytical representation of conformal mapping for genus-zero implicit surfaces and its application to surface shape similarity assessment

This paper develops an analytical representation of conformal mapping for genus-zero implicit surfaces based on algebraic polynomial functions, and its application to surface shape similarity assessment. Generally, the conformal mapping often works as a tool of planar or spherical parameterization for triangle mesh surfaces. It is further exploited for implicit surface matching in this study. The method begins with discretizing one implicit surface by triangle mesh, where a discrete harmonic energy model related to both the mesh and the other implicit surface is established based on a polynomial-function mapping. Then both the zero-center constraint and the landmark constraints are added to the model to ensure the uniqueness of mapping result with the Möbius transformation. By searching optimal polynomial coefficients with the Lagrange–Newton method, the analytical representation of conformal mapping is obtained, which reveals all global and continuous one-to-one correspondent point pairs between two implicit surfaces. Finally, a shape similarity assessment index for (two) implicit surfaces is proposed through calculating the differences of all the shape index values among those corresponding points. The proposed analytical representation method of conformal mapping and the shape assessment index are both verified by the simulation cases for the closed genus-zero implicit surfaces. Experimental results show that the method is effective for genus-zero implicit surfaces, which will offer a new way for object retrieval and manufactured surface inspection.     

A methodology for shape matching of non-rigid structures based on integrated graphical information

Shape matching is a long-studied problem and lies at the core of many applications in statistical shape analysis, virtual reality and human–computer interaction. This paper presents an automatic dense correspondence method to match the mesh vertices of two 3D shapes under near-isometric and non-rigid deformations. The goal is achieved by combining three types of graphic structure information. The method includes three major steps: first, we describe the vertices based on three types of graphical information, Euclidean structure information, Riemannian structure information, and conformal structure information; second, the match between two shapes is formulated as an optimization problem and a novel objective function is proposed; third, we resolve the optimal solution by using the projected descent optimization procedure to solve the objective function. The method is tested on various shape pairs with different poses, surface details, and topological noises. We demonstrate the performance of our approach through an extensive quantitative and qualitative evaluation on several challenging 3D shape matching datasets where we achieve superior performance to existing methods.     

Conformal self-organizing map for a genus-zero manifold

This paper presents the implementation of a surface mesh on a genus-zero manifold with 3D scattered data of sculpture surfaces using the conformal self-organizing map (CSM). It starts with a regular mesh on a sphere and gradually shapes the regular mesh to match its object’s surface by using the CSM. It can drape a uniform mesh on an object with a high degree of conformality. It accomplishes the surface reconstruction and also defines a conformal mapping from a sphere to the object’s manifold.     

Robust Spherical Parameterization of Triangular Meshes

Parameterization of 3D mesh data is important for many graphics and mesh processing applications, in particular for texture mapping, remeshing and morphing. Closed, manifold, genus-0 meshes are topologically equivalent to a sphere, hence this is the natural parameter domain for them. Parameterizing a 3D triangle mesh onto the 3D sphere means assigning a 3D position on the unit sphere to each of the mesh vertices, such that the spherical triangles induced by the mesh connectivity do not overlap. This is called a spherical triangulation. In this paper we formulate a set of necessary and sufficient conditions on the spherical angles of the spherical triangles for them to form a spherical triangulation. We formulate and solve an optimization procedure to produce spherical triangulations which reflect the geometric properties of a given 3D mesh in various ways.     

限制失真的网格参数化方法

针对当前网格参数化效率较低、映射失真较严重的问题，提出一种限制失真的网格参数化方法。首先，预处理原始网格模型。输入原3D网格模型，采用Half-Edge数据结构来重新组织网格并切割网格模型产生相应的切缝；构建Tutte映射把3D网格映射到一个2D凸多边形域，即构建2D网格模型。然后，进行限制失真的网格参数化计算。将Tutte映射后的2D网格模型作为限制失真计算的初始数据，建立相对于原3D模型网格的失真度量函数；求得该度量函数的最小值点，即为映射后的网格坐标集合；将映射后的网格作为限制失真映射的输入网格，设定迭代终止条件，循环迭代直至迭代结束，得到收敛的最优网格坐标；在计算映射失真度时，针对等距映射失真采用Dirichlet能量函数度量，针对共形映射失真采用尽可能等距（MIPS）能量函数度量；在求解映射失真度量函数的最小值点时采用代理函数法结合组合牛顿法的最优解方法。最终，实现了该方法并开发了一个原型系统。在原型系统中，分别设计了限制等距失真和限制共形失真的网格参数化实验，对程序执行时间和失真能量下降情况进行了统计和对比，提供了相应的纹理映射效果展示。实验数据表明，所提出的方法执行效率高、映射失真能量下降快，最优值收敛质量稳定；纹理映射时纹理着色均匀、布局紧致、线条均匀，符合实际应用的标准。     

Enhancing 3D mesh topological skeletons with discrete contour constrictions

This paper describes a unified and fully automatic algorithm for Reeb graph construction and simplification as well as constriction approximation on triangulated surfaces. The key idea of the algorithm is that discrete contours – curves carried by the edges of the mesh and approximating the continuous contours of a mapping function – encode both topological and geometrical shape characteristics. Therefore, a new concise shape representation, enhanced topological skeletons, is proposed, encoding the contours’ topological and geometrical evolution. First, mesh feature points are computed. Then they are used as geodesic origins for the computation of an invariant mapping function that reveals the shape most significant features. Next, for each vertex in the mesh, its discrete contour is computed. As the set of discrete contours recovers the whole surface, each of them can be analyzed, both to detect topological changes and constrictions. Constriction approximations enable Reeb graphs refinement into more visually meaningful skeletons, which we refer to as enhanced topological skeletons. Extensive experiments showed that, without any preprocessing stage, proposed algorithms are fast in practice, affine-invariant and robust to a variety of surface degradations (surface noise, mesh sampling and model pose variations). These properties make enhanced topological skeletons interesting shape abstractions for many computer graphics applications.     

一种三维模型形状检索描述符

为了解决基于关键字的搜索方式不能准确查找三维模型的问题,对三角网格形式的三维模型提出一种基于内容的模型搜索算法.首先将模型中的三角网格再次划分,增加形状函数的维度,定义了一种新的三维模型形状函数,以提高形状函数的准确性;然后确定三维模型的极轴,将三维模型绕任意轴的旋转转换为绕极轴旋转,增强了球面调和方法提取的形状描述符旋转不变性;最后根据形状描述符之间的欧氏距离来度量三维模型的相似程度.实验结果表明,该算法适用于基于内容的三维模型搜索,并且在准确性和时效性方面都令人满意.     

Harmonic Functions for Rotational Symmetry Vector Fields

Representing rotational symmetry vector as a set of vectors is not suitable for design due to lacking of a consistent ordering for measurement. In this paper we introduce a spectral method to find rotation invariant harmonic functions for symmetry vector field design. This method is developed for 3D vector fields, but it is applicable in 2D. Given the finite symmetry group G of a symmetry vector field v (x) on a 3D domain Ω, we formulate the harmonic function h(s) as a stationary point of group G. Using the real spherical harmonic (SH) bases, we showed the coefficients of the harmonic functions are an eigenvector of the SH rotation matrices corresponding to group G. Instead of solving eigen problems to obtain the eigenvector, we developed a forward constructive method based on orthogonal group theory. The harmonic function found by our method is not only invariant under G, but also expressive and can distinguish different rotations with respect to G. At last, we demonstrate some vector field design results with tetrahedron‐symmetry, cube‐symmetry and dodecahedron‐symmetry groups.     

A pipeline for computer aided polyp detection

We present a novel pipeline for computer-aided detection (CAD) of colonic polyps by integrating texture and shape analysis with volume rendering and conformal colon flattening. Using our automatic method, the 3D polyp detection problem is converted into a 2D pattern recognition problem. The colon surface is first segmented and extracted from the CT data set of the patient's abdomen, which is then mapped to a 2D rectangle using conformal mapping. This flattened image is rendered using a direct volume rendering technique with a translucent electronic biopsy transfer function. The polyps are detected by a 2D clustering method on the flattened image. The false positives are further reduced by analyzing the volumetric shape and texture features. Compared with shape based methods, our method is much more efficient without the need of computing curvature and other shape parameters for the whole colon surface. The final detection results are stored in the 2D image, which can be easily incorporated into a virtual colonoscopy (VC) system to highlight the polyp locations. The extracted colon surface mesh can be used to accelerate the volumetric ray casting algorithm used to generate the VC endoscopic view. The proposed automatic CAD pipeline is incorporated into an interactive VC system, with a goal of helping radiologists detect polyps faster and with higher accuracy.     

3D anatomical shape atlas construction using mesh quality preserved deformable models

3D anatomical shape atlas construction has been extensively studied in medical image analysis research, owing to its importance in model-based image segmentation, longitudinal studies and populational statistical analysis, etc. Among multiple steps of 3D shape atlas construction, establishing anatomical correspondences across subjects, i.e., surface registration, is probably the most critical but challenging one. Adaptive focus deformable model (AFDM) [1] was proposed to tackle this problem by exploiting cross-scale geometry characteristics of 3D anatomy surfaces. Although the effectiveness of AFDM has been proved in various studies, its performance is highly dependent on the quality of 3D surface meshes, which often degrades along with the iterations of deformable surface registration (the process of correspondence matching). In this paper, we propose a new framework for 3D anatomical shape atlas construction. Our method aims to robustly establish correspondences across different subjects and simultaneously generate high-quality surface meshes without removing shape details. Mathematically, a new energy term is embedded into the original energy function of AFDM to preserve surface mesh qualities during deformable surface matching. More specifically, we employ the Laplacian representation to encode shape details and smoothness constraints. An expectation–maximization style algorithm is designed to optimize multiple energy terms alternatively until convergence. We demonstrate the performance of our method via a set of diverse applications, including a population of sparse cardiac MRI slices with 2D labels, 3D high resolution CT cardiac images and rodent brain MRIs with multiple structures. The constructed shape atlases exhibit good mesh qualities and preserve fine shape details. The constructed shape atlases can further benefit other research topics such as segmentation and statistical analysis.     

不规则三维网格纹理映射的均匀化

采用单一纹理对不规则三维网格进行纹理映射,常常使网格密集处映射不足;而采用纹理地图集方法又使映射过程变得复杂。因此文章提出一种新的方法:通过离散保角变换方法,先将三维网格无遮挡地映射到平面正方形中,随后通过一种离散均匀面积变换方法,使其平面网格较均匀地分布,然后求得其纹理坐标,从而有效实现不规则三维网格的纹理映射。该方法算法快速、稳定,纹理映射均匀化效果良好。     

质心坐标变换及其在纹理映射均匀化中的应用

在现有质心坐标变换方法基础上,提出一种改进方法——均匀面积质心变换方法：在某一顶点邻域中,采用相应点所对应的边高比之和作为质心坐标进行分析推导,并将其应用到复杂三维形体的纹理映射均匀化中．首先通过面积权重质心坐标变换将复杂三维网格映射到平面上;在此基础上进行均匀面积质心坐标变换,就可使平面网格较均匀地分布．求解其纹理坐标可实现采用单幅图像的纹理映射均匀化．通过典型三维模型的实验和比较可以看到：采用文中方法所获得的纹理映射均匀化效果较现有的保角变换、保面积变换方法有显著改善,而且算法简单、稳定、快速．     

同心球面网格最远距离特征的三维模型检索方法

距离是三维模型特征提取中最有效、直观,也最容易让人理解的特征之一。目前基于距离信息类特征提取算法因为需要大量计算射线与模型面片的交点,时间耗费很大。针对这一问题,提出了同心球面网格最远距离特征描述子,细分三维模型,统计落人各球面网格上面片中心的最远距离,构造层次深度球面上的距离函数,然后对构造球面函数进行球面调和分析,截取球面调和系数作为特征向量。实验结果表明其检索性能优于其他同类基于距离的特征描述算法,计算时间花费也最少,计算复杂度仅为O（n）。     

A spherical representation for recognition of free-form surfaces

Introduces a new surface representation for recognizing curved objects. The authors approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object. The mesh is computed from the data by deforming a standard shaped mesh, for example, an ellipsoid, until it fits the surface of the object. The authors define local regularity constraints that the mesh must satisfy. The authors then define a canonical mapping between the mesh describing the object and a standard spherical mesh. A surface curvature index that is pose-invariant is stored at every node of the mesh. The authors use this object representation for recognition by comparing the spherical model of a reference object with the model extracted from a new observed scene. The authors show how the similarity between reference model and observed data can be evaluated and they show how the pose of the reference object in the observed scene can be easily computed using this representation. The authors present results on real range images which show that this approach to modelling and recognizing 3D objects has three main advantages: (1) it is applicable to complex curved surfaces that cannot be handled by conventional techniques; (2) it reduces the recognition problem to the computation of similarity between spherical distributions; in particular, the recognition algorithm does not require any combinatorial search; and (3) even though it is based on a spherical mapping, the approach can handle occlusions and partial views     

三角网格曲面共形参数化研究综述

曲面参数化是计算机图形学和计算机辅助设计中的一个重要问题,有着广泛的应用背景.曲面参数化旨在谋求从三维曲面到某参数区域变换过程中某些内蕴几何量(例如边长、角度、面积)的变形最小化.保角参数化前后保持曲面局部形状,又称作共形参数化.本文针对三角网格曲面,介绍了共形参数化领域的研究现状.根据目标参数域的不同,主要讨论了平面...     

System for reconstruction of three-dimensional micro objects from multiple photographic images

We introduce a system to reconstruct a three-dimensiojnal (3D) polygonal model of 3D micro objects with outer dimensions ranging from several hundred microns to several millimeters from multiple two-dimensional (2D) images of an object taken from different views. The data acquisition system consists of a digital microscope that captures still images at a resolution of 1600 × 1200 pixels and a computer-controlled turntable. We employ the shape-from-silhouette (SFS) method to construct a voxel-based 3D model from silhouette images. The concave shapes are further carved by using the space carving technique. In order to make the resulting model compatible with a commercial CAD/CAM system, the voxel model is converted into a triangular mesh using the marching cubes algorithm. Because the mesh generated from the voxel model by using the marching cubes algorithm inherits the staircase effect, the mesh is adjusted to recover the object precisely by using silhouette images. Finally, we evaluate the accuracy of the proposed method. The reconstructed models of complex micro objects indicate the effectiveness of the 3D shape reconstruction system for micro objects.