Multi-Level Shape Representation Using Global Deformations and Locally Adaptive Finite Elements

We present a model-based method for the multi-level shape, pose estimation and abstraction of an object's surface from range data. The surface shape is estimated based on the parameters of a superquadric that is subjected to global deformations (tapering and bending) and a varying number of levels of local deformations. Local deformations are implemented using locally adaptive finite elements whose shape functions are piecewise cubic functions with C ¹ continuity. The surface pose is estimated based on the model's translational and rotational degrees of freedom. The algorithm first does a coarse fit, solving for a first approximation to the translation, rotation and global deformation parameters and then does several passes of mesh refinement, by locally subdividing triangles based on the distance between the given datapoints and the model. The adaptive finite element algorithm ensures that during subdivision the desirable finite element mesh generation properties of conformity, non-degeneracy and smoothness are maintained. Each pass of the algorithm uses physics-based modeling techniques to iteratively adjust the global and local parameters of the model in response to forces that are computed from approximation errors between the model and the data. We present results demonstrating the multi-level shape representation for both sparse and dense range data.     

Bezier曲面的适应性细分和三角形化的四叉树方法

计算机生成具有浓淡的参数曲面的方法之一是先对曲面进行适应性细分,并对所得到的曲面细分三角形化,得到曲面的三角形网表示,从而可以对每个三角形施行通常的浓淡处理算法。本文介绍了适应性细分双三次Bezier曲面的方法及曲面细分的四叉树表示,在此基础上给出了一个将曲面细分三角形化的算法。该算法防止了由于适应性细分而可能产生的曲面上的裂缝。     

Implicit Hierarchical Quad‐Dominant Meshes

We present a method for producing quad‐dominant subdivided meshes, which supports both adaptive refinement and adaptive coarsening. A hierarchical structure is stored implicitly in a standard half‐edge data structure, while allowing us to efficiently navigate through the different level of subdivision. Subdivided meshes contain a majority of quad elements and a moderate amount of triangles and pentagons in the regions of transition across different levels of detail. Topological LOD editing is controlled with local conforming operators, which support both mesh refinement and mesh coarsening. We show two possible applications of this method: we define an adaptive subdivision surface scheme that is topologically and geometrically consistent with the Catmull–Clark subdivision; and we present a remeshing method that produces semi‐regular adaptive meshes.     

Hierarchical Solutions for the Deformable Surface Problem in Visualization

In this paper we present a hierarchical approach for the deformable surface technique. This technique is a three dimensional extension of the snake segmentation method. We use it in the context of visualizing three dimensional scalar data sets. In contrast to classical indirect volume visualization methods, this reconstruction is not based on iso-values but on boundary information derived from discontinuities in the data. We propose a multilevel adaptive finite difference solver, which generates a target surface minimizing an energy functional based on an internal energy of the surface and an outer energy induced by the gradient of the volume. The method is attractive for preprocessing in numerical simulation or texture mapping. Red-green triangulation allows adaptive refinement of the mesh. Special considerations help to prevent self interpenetration of the surfaces. We will also show some techniques that introduce the hierarchical aspect into the inhomogeneity of the partial differential equation. The approach proves to be appropriate for data sets that contain a collection of objects separated by distinct boundaries. These kind of data sets often occur in medical and technical tomography, as we will demonstrate in a few examples.     

CPH: A Compact Representation for Hierarchical Meshes Generated by Primal Refinement

We present CPH (Compact Primal Hierarchy): a compact representation of the hierarchical connectivity of surface and volume manifold meshes generated through primal subdivision refinements. CPH is consistently defined in several dimensions and supports multiple kinds of tessellations and refinements, whether regular or adaptive. The basic idea is to store only the finest mesh, encoded in a classical monoresolution structure that is enriched with a minimal set of labels. These labels allow traversal of any intermediate level of the mesh concurrently without having to extract it in an additional structure. Our structure allows attributes to be stored on the cells not only on the finest level, but also on any intermediate level. We study the trade‐off between the memory cost of this compact representation and the time complexity of mesh traversals at any resolution level.     

快速收敛的四边形网格三分细分模式

提出了四边形网格的三分细分模式.对于正则和非正则四边形网格,分别采用不同的细分模板获得新的细分顶点.从双三次B样条中推导出正则四边形网格的三分细分模板,极限曲面C²连续;对细分矩阵进行傅里叶变换,推导出非正则四边形网格的三分细分模板,极限曲面C¹连续.提出的三分细分模式可以解决任意拓扑四边形网格的曲面细分问题.与其他细分模式相比,具有收敛速度快、适用范围广等优点.最后给出了四边形网格细分的实例.     

Generating a nice triangular mesh on a regular parametric surface

In this paper, we present an iterative algorithm to generate a nice triangular curvilinear mesh on a regular parametric surface. The main idea is to obtain a reparametrization of the surface that behaves approximately like a conformal map in a finite number of points. These points are the vertices of a planar triangulation T ^op in the parameter space, which we would like to lift on the surface. The image of T ^op by means of the reparametrization provides us a triangular curvilinear mesh on the surface which reflects the properties of T ^op .     

Efficient generation of discontinuity-preserving adaptive triangulations from range images

This paper presents an efficient technique for generating adaptive triangular meshes from range images. The algorithm consists of two stages. First, a user-defined number of points is adaptively sampled from the given range image. Those points are chosen by taking into account the surface shapes represented in the range image in such a way that points tend to group in areas of high curvature and to disperse in low-variation regions. This selection process is done through a noniterative, inherently parallel algorithm in order to gain efficiency. Once the image has been subsampled, the second stage applies a two and one half-dimensional Delaunay triangulation to obtain an initial triangular mesh. To favor the preservation of surface and orientation discontinuities (jump and crease edges) present in the original range image, the aforementioned triangular mesh is iteratively modified by applying an efficient edge flipping technique. Results with real range images show accurate triangular approximations of the given range images with low processing times.     

Adaptive terrain triangulation using the representation of quad trees by vertex textures and wavelet estimation of vertex significance

A method of adaptive terrain triangulation is proposed that can be implemented in hardware. The method is based on an estimate of the static error of a quad tree nodes using wavelet transforms and on the representation of the resulting quad tree by a vertex texture. The proposed method has the following characteristic features: the adjacent nodes of the generated adapted mesh can differ in any number of hierarchical levels; the triangulation process is not limited by the size of the decomposition segments, which solves the problem of joining segments without inserting additional nodes; the multiscale terrain representation used in the method makes it possible to store the levels of detail in the graphics processor memory as a multilevel vertex texture; thus, the costliest part of the algorithm can be efficiently implemented using a vertex shader. When constructing the triangulation, the algorithm takes into account both local features of the terrain and the camera location; also, it has a natural support of geomorphing.     

Optimization methods for scattered data approximation with subdivision surfaces

We present a method for scattered data approximation with subdivision surfaces which actually uses the true representation of the limit surface as a linear combination of smooth basis functions associated with the control vertices. A robust and fast algorithm for exact closest point search on Loop surfaces which combines Newton iteration and non-linear minimization is used for parameterizing the samples. Based on this we perform unconditionally convergent parameter correction to optimize the approximation with respect to the L² metric, and thus we make a well-established scattered data fitting technique which has been available before only for B-spline surfaces, applicable to subdivision surfaces. We also adapt the recently discovered local second order squared distance function approximant to the parameter correction setup. Further we exploit the fact that the control mesh of a subdivision surface can have arbitrary connectivity to reduce the L^∞ error up to a certain user-defined tolerance by adaptively restructuring the control mesh. Combining the presented algorithms we describe a complete procedure which is able to produce high-quality approximations of complex, detailed models.     

A new vector field distance transform and its application to mesh processing from 3D scanned data

In this paper we define a new 3D vector field distance transform to implicitly represent a mesh surface. We show that this new representation is more accurate than the classic scalar field distance transform by comparing both representations with an error metric evaluation. The widely used marching cube triangulation algorithm is adapted to the new vector field distance transform to correctly reconstruct the resulting explicit surface. In the reconstruction process of 3D scanned data, the useful mesh denoising operation is extended to the new vector field representation, which enables adaptive and selective filtering features. Results show that mesh processing with this new vector field representation is more accurate than with the scalar field distance transform and that it outperforms previous mesh filtering algorithms. Future work is discussed to extend this new vector field representation to other mesh useful operations and applications.     

A volumetric fusion technique for surface reconstruction from silhouettes and range data

Optical triangulation, an active reconstruction technique, is known to be an accurate method but has several shortcomings due to occlusion and laser reflectance properties of the object surface, that often lead to holes and inaccuracies on the recovered surface. Shape from silhouette, on the other hand, as a passive reconstruction technique, yields robust, hole-free reconstruction of the visual hull of the object. In this paper, a hybrid surface reconstruction method that fuses geometrical information acquired from silhouette images and optical triangulation is presented. Our motivation is to recover the geometry from silhouettes on those parts of the surface which the range data fail to capture. A volumetric octree representation is first obtained from the silhouette images and then carved by range points to amend the missing cavity information. An isolevel value on each surface cube of the carved octree structure is accumulated using local surface triangulations obtained separately from range data and silhouettes. The marching cubes algorithm is then applied for triangulation of the volumetric representation. The performance of the proposed technique is demonstrated on several real objects.     

Geometric fusion for a hand-held 3D sensor

Abstract. This article presents a geometric fusion algorithm developed for the reconstruction of 3D surface models from hand-held sensor data. Hand-held systems allow full 3D movement of the sensor to capture the shape of complex objects. Techniques previously developed for reconstruction from conventional 2.5D range image data cannot be applied to hand-held sensor data. A geometric fusion algorithm is introduced to integrate the measured 3D points from a hand-held sensor into a single continuous surface. The new geometric fusion algorithm is based on the normal-volume representation of a triangle, which enables incremental transformation of an arbitrary mesh into an implicit volumetric field function. This system is demonstrated for reconstruction of surface models from both hand-held sensor data and conventional 2.5D range images. Received: 30 August 1999 / Accepted: 21 January 2000     

Multilevel representation and transmission of real objects with progressive octree particles

We present a multilevel representation scheme adapted to storage, progressive transmission, and rendering of dense data sampled on the surface of real objects. Geometry and object attributes, such as color and normal, are encoded in terms of surface particles associated to a hierarchical space partitioning based on an octree. Appropriate ordering of surface particles results in a compact multilevel representation without increasing the size of the uniresolution model corresponding to the highest level of detail. This compact representation can progressively be decoded by the viewer and transformed by a fast direct triangulation technique into a sequence of triangle meshes with increasing levels of detail. The representation requires approximately 5 bits per particle (2.5 bits per triangle) to encode the basic geometrical structure. The vertex positions can then be refined by means of additional precision bits, resulting in 5 to 9 bits per triangle for representing a 12-bit quantized geometry. The proposed representation scheme is demonstrated with the surface data of various real objects.     

Efficient triangular surface approximations using wavelets and quadtree data structures

We present a method for adaptive surface meshing and triangulation which controls the local level of detail of the surface approximation by local spectral estimates. These estimates are determined by a wavelet representation of the surface data. The basic idea is to decompose the initial data set by means of an orthogonal or semi orthogonal tensor product wavelet transform (WT) and to analyze the resulting coefficients. In surface regions, where the partial energy of the resulting coefficients is low, the polygonal approximation of the surface can be performed with larger triangles without losing too much fine grain details. However, since the localization of the WT is bound by the Heisenberg principle, the meshing method has to be controlled by the detail signals rather than directly by the coefficients. The dyadic scaling of the WT stimulated us to build an hierarchical meshing algorithm which transforms the initially regular data grid into a quadtree representation by rejection of unimportant mesh vertices. The optimum triangulation of the resulting quadtree cells is carried out by selection from a look up table. The tree grows recursively as controlled by detail signals which are computed from a modified inverse WT. In order to control the local level of detail, we introduce a new class of wavelet space filters acting as "magnifying glasses" on the data. We show that our algorithm performs a low algorithmic complexity, so that surface meshing can be achieved at interactive rates, such as required by flight simulators, however, other applications are possible as well.     

Surface Reconstruction and Display from Range and Color Data

This paper addresses the problem of scanning both the color and geometry of real objects and displaying realistic images of the scanned objects from arbitrary viewpoints. We describe a complete system that uses a stereo camera setup with active lighting to scan the object surface geometry and color. Scans expressed in sensor coordinates are registered into a single object-centered coordinate system by aligning both the color and geometry where the scans overlap. The range data are integrated into a surface model using a robust hierarchical space carving method. The fit of the resulting approximate mesh to data is improved and the mesh structure is simplified using mesh optimization methods. In addition, a method for view-dependent texturing of the reconstructed surfaces is described. The method projects the color data from the input images onto the surface model and blends the various images depending on the location of the viewpoint and other factors such as surface orientation.     

A hierarchical structure for surface approximation

Hierarchical triangulation is a method for point selection and surface representation where the surface is approximated at successively finer levels of detail by triangular patches whose projections in the horizontal plane are nested. A tree data structure for this representation can be constructed in O(n²) worst case and O(n log n) average case time, where n is the number of data points considered. Efficient algorithms for approximation of the elevation of an arbitrary point, contour extraction, and conversion of the hierarchical structure into an ordinary triangulated irregular network, are demonstrated. The convergence and the optimality of the approximation and the relationship of the hierarchical triangulation to a structured graph representation are examined.     

Shrinkwrap: An efficient adaptive algorithm for triangulating an iso-surface

An algorithm is presented which generates a triangular mesh to approximate an iso-surface. It starts with a triangulation of a sphere and next applies a series of deformations to this triangulation to transform it into the required surface. These deformations leave the topology invariant, so the final iso-surface should be homeomorphic with a sphere. The algorithm is adaptive in the sense that the lengths of the sides of the triangles in the mesh vary with the local curvature of the underlying surface. A quantitative analysis of the accuracy of the algorithm is given along with an empirical comparison with earlier algorithms.     

Volume-Surface Trees

Many algorithms in computer graphics improve their efficiency by using Hierarchical Space Subdivision Schemes (HS³), such as octrees, kD‐trees or BSP trees. Such HS³ usually provide an axis‐aligned subdivision of the 3D space embedding a scene or an object. However, the purely volume‐based behavior of these schemes often leads to strongly imbalanced surface clustering. In this article, we introduce the VS‐Tree, an alternative HS³ providing efficient and accurate surface‐based hierarchical clustering via a combination of a global 3D decomposition at coarse subdivision levels, and a local 2D decomposition at fine levels near the surface. First, we show how to efficiently construct VS‐Trees over meshes and point‐based surfaces, and analyze the improvement it offers for cluster‐based surface simplification methods. Then we propose a new surface reconstruction algorithm based on the volume‐surface classification of the VS‐Tree. This new algorithm is faster than state‐of‐the‐art reconstruction methods and provides a final semi‐regular mesh comparable to the output of remeshing algorithms.     

一种新的自适应多分辨率细分曲面的表示法*

针对已有自适应细分方法中存在的问题,提出了一种新的自适应多分辨率细分曲面的表示方法。该方法结合拓扑细分的特点,运用二维组合映射对半边数据结构进行形式化定义,并引出超映射的概念。在超映射这个通用的理论框架中引出了半边结构在多分辨率上的扩展,并对其在自适应细分方法中的应用进行了讨论。结果表明这种结构具有普适性,支持多种细分方法,允许在网格的任何多分辨率层次上及时有效地导航,并且在自适应细分过程中避免了拓扑裂缝的产生。