Efficient Interpolation of Articulated Shapes Using Mixed Shape Spaces

Interpolation between compatible triangle meshes that represent different poses of some object is a fundamental operation in geometry processing. A common approach is to consider the static input shapes as points in a suitable shape space and then use simple linear interpolation in this space to find an interpolated shape. In this paper, we present a new interpolation technique that is particularly tailored for meshes that represent articulated shapes. It is up to an order of magnitude faster than state‐of‐the‐art methods and gives very similar results. To achieve this, our approach introduces a novel shape space that takes advantage of the underlying structure of articulated shapes and distinguishes between rigid parts and non‐rigid joints. This allows us to use fast vertex interpolation on the rigid parts and resort to comparatively slow edge‐based interpolation only for the joints.     

Transitive mesh space of a progressive mesh

The paper investigates the set of all selectively refined meshes that can be obtained from a progressive mesh. We call the set the transitive mesh space of a progressive mesh and present a theoretical analysis of the space. We define selective edge collapse and vertex split transformations, which we use to traverse all selectively refined meshes in the transitive mesh space. We propose a complete selective refinement scheme for a progressive mesh based on the transformations and compare the scheme with previous selective refinement schemes in both theoretical and experimental ways. In our comparison, we show that the complete scheme always generates selectively refined meshes with smaller numbers of vertices and faces than previous schemes for a given refinement criterion. The concept of dual pieces of the vertices in the vertex hierarchy plays a central role in the analysis of the transitive mesh space and the design of selective edge collapse and vertex split transformations.     

Combinatorial mesh optimization

A new mesh optimization framework for 3D triangular surface meshes is presented, which formulates the task as an energy minimization problem in the same spirit as in Hoppe et al. (SIGGRAPH’93: Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, 1993). The desired mesh properties are controlled through a global energy function including data attached terms measuring the fidelity to the original mesh, shape potentials favoring high quality triangles, and connectivity as well as budget terms controlling the sampling density. The optimization algorithm modifies mesh connectivity as well as the vertex positions. Solutions for the vertex repositioning step are obtained by a discrete graph cut algorithm examining global combinations of local candidates.     

基于几何特征的自适应三维模型数字水印算法

针对三维三角网格模型提出一种稳健的数字水印算法.首先将三维三角网格模型进行仿射变换,以获得模型的旋转不变性和缩放不变性;然后将各顶点邻域内顶点位置的平均差值作为掩蔽因子确定水印嵌入的强度,使得嵌入的水印具有不可见性.实验结果表明:该算法简单,对几何变换、简化、随机噪声和剪切攻击具有较好的鲁棒性.     

A framework for quad/triangle subdivision surface fitting: Application to mechanical objects

In this paper we present a new framework for subdivision surface approximation of three‐dimensional models represented by polygonal meshes. Our approach, particularly suited for mechanical or Computer Aided Design (CAD) parts, produces a mixed quadrangle‐triangle control mesh, optimized in terms of face and vertex numbers while remaining independent of the connectivity of the input mesh. Our algorithm begins with a decomposition of the object into surface patches. The main idea is to approximate the region boundaries first and then the interior data. Thus, for each patch, a first step approximates the boundaries with subdivision curves (associated with control polygons) and creates an initial subdivision surface by linking the boundary control points with respect to the lines of curvature of the target surface. Then, a second step optimizes the initial subdivision surface by iteratively moving control points and enriching regions according to the error distribution. The final control mesh defining the whole model is then created assembling every local subdivision control meshes. This control polyhedron is much more compact than the original mesh and visually represents the same shape after several subdivision steps, hence it is particularly suitable for compression and visualization tasks. Experiments conducted on several mechanical models have proven the coherency and the efficiency of our algorithm, compared with existing methods.     

A Constrained Resampling Strategy for Mesh Improvement

In many geometry processing applications, it is required to improve an initial mesh in terms of multiple quality objectives. Despite the availability of several mesh generation algorithms with provable guarantees, such generated meshes may only satisfy a subset of the objectives. The conflicting nature of such objectives makes it challenging to establish similar guarantees for each combination, e.g., angle bounds and vertex count. In this paper, we describe a versatile strategy for mesh improvement by interpreting quality objectives as spatial constraints on resampling and develop a toolbox of local operators to improve the mesh while preserving desirable properties. Our strategy judiciously combines smoothing and transformation techniques allowing increased flexibility to practically achieve multiple objectives simultaneously. We apply our strategy to both planar and surface meshes demonstrating how to simplify Delaunay meshes while preserving element quality, eliminate all obtuse angles in a complex mesh, and maximize the shortest edge length in a Voronoi tessellation far better than the state‐of‐the‐art.     

Structure‐Aware Mesh Decimation

We present a novel approach for the decimation of triangle surface meshes. Our algorithm takes as input a triangle surface mesh and a set of planar proxies detected in a pre‐processing analysis step, and structured via an adjacency graph. It then performs greedy mesh decimation through a series of edge collapse, designed to approximate the local mesh geometry as well as the geometry and structure of proxies. Such structure‐preserving approach is well suited to planar abstraction, i.e. extreme decimation approximating well the planar parts while filtering out the others. Our experiments on a variety of inputs illustrate the potential of our approach in terms of improved accuracy and preservation of structure.     

边收缩池化的网格变分自编码器

目的 3D形状分析是计算机视觉和图形学的一个重要研究课题。虽然现有方法使用基于图的卷积将基于图像的深度学习推广到3维网格,但缺乏有效的池化操作限制了其网络的学习能力。针对具有相同连通性,但几何形状不同的网格模型数据集,本文利用网格简化的边收缩操作建立网格层次结构,提出了一种新的网格池化操作。方法 本文改进了传统的网格简化方法,以避免生成高度不规则的三角形,利用改进的网格简化方法定义了新的网格池化操作。网格简化的边收缩操作建立的网格层次结构之间存在对应关系,有利于网格池化的定义。新定义的池化操作有效地编码了层次结构中较粗糙和较稠密网格之间的对应关系。最后提出了一种带有边收缩池化和图卷积的变分自编码器（variational auto-encoder,VAE）结构,以探索3D形状的隐空间并用于3D形状的生成。结果 由于引入了新定义的池化操作和图卷积操作,提出的网络结构比原始MeshVAE需要的参数更少,因此可以处理更稠密的网格模型。结论 实验表明提出的方法具有更好的泛化能力,并且在各种应用中更可靠,包括形状生成、形状插值和形状嵌入。     

Skeleton‐driven Adaptive Hexahedral Meshing of Tubular Shapes

We propose a novel method for the automatic generation of structured hexahedral meshes of articulated 3D shapes. We recast the complex problem of generating the connectivity of a hexahedral mesh of a general shape into the simpler problem of generating the connectivity of a tubular structure derived from its curve‐skeleton. We also provide volumetric subdivision schemes to nicely adapt the topology of the mesh to the local thickness of tubes, while regularizing per‐element size. Our method is fast, one‐click, easy to reproduce, and it generates structured meshes that better align to the branching structure of the input shape if compared to previous methods for hexa mesh generation.     

A Colour Interpolation Scheme for Topologically Unrestricted Gradient Meshes

Gradient meshes are a 2D vector graphics primitive where colour is interpolated between mesh vertices. The current implementations of gradient meshes are restricted to rectangular mesh topology. Our new interpolation method relaxes this restriction by supporting arbitrary manifold topology of the input gradient mesh. Our method is based on the Catmull‐Clark subdivision scheme, which is well‐known to support arbitrary mesh topology in 3D. We adapt this scheme to support gradient mesh colour interpolation, adding extensions to handle interpolation of colours of the control points, interpolation only inside the given colour space and emulation of gradient constraints seen in related closed‐form solutions. These extensions make subdivision a viable option for interpolating arbitrary‐topology gradient meshes for 2D vector graphics.     

Surface interpolation of meshes by geometric subdivision

Subdivision surfaces are generated by repeated approximation or interpolation from initial control meshes. In this paper, two new non-linear subdivision schemes, face based subdivision scheme and normal based subdivision scheme, are introduced for surface interpolation of triangular meshes. With a given coarse mesh more and more details will be added to the surface when the triangles have been split and refined. Because every intermediate mesh is a piecewise linear approximation to the final surface, the first type of subdivision scheme computes each new vertex as the solution to a least square fitting problem of selected old vertices and their neighboring triangles. Consequently, sharp features as well as smooth regions are generated automatically. For the second type of subdivision, the displacement for every new vertex is computed as a combination of normals at old vertices. By computing the vertex normals adaptively, the limit surface is G¹ smooth. The fairness of the interpolating surface can be improved further by using the neighboring faces. Because the new vertices by either of these two schemes depend on the local geometry, but not the vertex valences, the interpolating surface inherits the shape of the initial control mesh more fairly and naturally. Several examples are also presented to show the efficiency of the new algorithms.     

细分曲面拟合的局部渐进插值方法

逼近型细分方法生成的细分曲面其品质要优于插值型细分方法生成的细分曲面.然而,逼近型细分方法生成的细分曲面不能插值于初始控制网格顶点.为使逼近型细分曲面具有插值能力,一般通过求解全局线性方程组,使其插值于网格顶点.当网格顶点较多时,求解线性方程组的计算量很大,因此,难以处理稠密网格.与此不同,在不直接求解线性方程组的情况下,渐进插值方法通过迭代调整控制网格顶点,最终达到插值的效果.渐进插值方法可以处理稠密的任意拓扑网格,生成插值于初始网格顶点的光滑细分曲面.并且经证明,逼近型细分曲面渐进插值具有局部性质,也就是迭代调整初始网格的若干控制顶点,且保持剩余顶点不变,最终生成的极限细分曲面仍插值于初始网格中被调整的那些顶点.这种局部渐进插值性质给形状控制带来了更多的灵活性,并且使得自适应拟合成为可能.实验结果验证了局部渐进插值的形状控制以及自适应拟合能力.     

基于Hermite插值的网格拼接和融合-CIDE2013

网格模型的拼接和融合是3维形状编辑和造型中的一个重要方面。基于Hermite插值技术,提出了一种适用于具有一般边界点空间分布的三角网格模型之间无缝光滑拼接和融合方法。首先查找网格模型待拼接区域的边缘点集,并利用二次B样条曲线插值边缘点集分别得到边缘曲线;然后对边缘曲线进行Hermite插值得到拼接区域连续曲面;最后对拼接曲面分别进行三角网格化和Laplacian光顺平滑处理以实现网格模型的光滑拼接和融合。由于利用B样条曲线插值待拼接模型边界,本文方法适用于具有各种不同边界情形的网格模型拼接和融合,它不仅仅可以处理平面边界曲线情形也可以处理空间边界曲线情形。结合Hermite曲面插值拼接过渡区域,使得产生的拼接网格能光滑地衔接待拼接模型。实验结果表明,本文方法能够有效地实现三角网格模型的光滑拼接、模型修复和模型融合。     

基于Hermite插值的网格拼接和融合

网格模型的拼接和融合是3维形状编辑和造型中的一个重要方面。基于Hermite插值技术,提出一种适用于具有一般边界点空间分布的三角网格模型之间无缝光滑拼接和融合方法。首先查找网格模型待拼接区域的边缘点集,并利用二次B样条曲线插值边缘点集分别得到边缘曲线;然后对边缘曲线进行Hermite插值得到拼接区域连续曲面;最后对拼接曲面分别进行三角网格化和Laplacian光顺平滑处理以实现网格模型的光滑拼接和融合。由于利用B样条曲线插值待拼接模型边界,本文方法适用于具有各种不同边界情形的网格模型拼接和融合,它不仅仅可以处理平面边界曲线情形也可以处理空间边界曲线情形。结合Hermite曲面插值拼接过渡区域,使得产生的拼接网格能光滑地衔接待拼接模型。实验结果表明,本文方法能够有效地实现三角网格模型的光滑拼接、模型修复和模型融合。     

Modeling Polyhedral Meshes with Affine Maps

We offer a framework for editing and modeling of planar meshes, focusing on planar quad, and hexagonal‐dominant meshes, which are held in high demand in the field of architectural design. Our framework manipulates these meshes by affine maps that are assigned per‐face, and which naturally ensure the planarity of these faces throughout the process, resulting in a linear subspace of compatible planar deformations for any given mesh. Our modeling metaphors include classical handle‐based editing, mesh interpolation, and shape‐space exploration, all of which allow for an intuitive way to produce new polyhedral and near‐polyhedral meshes by editing.     

Curvilinear Mesh Adaptation Using Radial Basis Function Interpolation and Smoothing

We present a new iterative technique based on radial basis function (RBF) interpolation and smoothing for the generation and smoothing of curvilinear meshes from straight-sided or other curvilinear meshes. Our technique approximates the coordinate deformation maps in both the interior and boundary of the curvilinear output mesh by using only scattered nodes on the boundary of the input mesh as data sites in an interpolation problem. Our technique produces high-quality meshes in the deformed domain even when the deformation maps are singular due to a new iterative algorithm based on modification of the RBF shape parameter. Due to the use of RBF interpolation, our technique is applicable to both 2D and 3D curvilinear mesh generation without significant modification.     

Animation‐Aware Quadrangulation

Geometric meshes that model animated characters must be designed while taking into account the deformations that the shape will undergo during animation. We analyze an input sequence of meshes with point‐to‐point correspondence, and we automatically produce a quadrangular mesh that fits well the input animation. We first analyze the local deformation that the surface undergoes at each point, and we initialize a cross field that remains as aligned as possible to the principal directions of deformation throughout the sequence. We then smooth this cross field based on an energy that uses a weighted combination of the initial field and the local amount of stretch. Finally, we compute a field‐aligned quadrangulation with an off‐the‐shelf method. Our technique is fast and very simple to implement, and it significantly improves the quality of the output quad mesh and its suitability for character animation, compared to creating the quad mesh based on a single pose. We present experimental results and comparisons with a state‐of‐the‐art quadrangulation method, on both sequences from 3D scanning and synthetic sequences obtained by a rough animation of a triangulated model.     

渐进插值的LOOP曲面细分*

目前很多细分方法都存在不能用同一种方法处理封闭网格和开放网格的问题。对此,一种新的基于插值技术的LOOP曲面细分方法,其主要思想就是给定一个初始三角网格M,反复生成新的顶点,新顶点是通过其相邻顶点的约束求解得到的,从而构造一个新的控制网格M,在取极限的情况下,可以证明插值过程是收敛的;因为生成新顶点使用的是与其相连顶点的约束求解得到的,本质上是一种局部方法,所以,该方法很容易定义。它在本地方法和全局方法中都有优势,能处理任意顶点数量和任意拓扑结构的网格,从而产生一个光滑的曲面并忠实于给定曲面的形状,其控制