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Mesh Denoising based on Differential Coordinates Zhi-Xun Su1, Hui Wang1, Jun-Jie Cao1 1 Department of Applied Mathematica, Dalian University of Technology, Dalian 116023, P.R. China Keywords: mesh denoising; differential coordinates; mean filter Abstract: In this paper, we propose a novel triangle mesh denoising method based on the differential coordinates. The proposed approach consists of the application of the mean filter to differential coordinates of the mesh and the reconstruction of mesh vertices’ Cartesian coordinates to make them fit to the modified differential coordinates. The presented method is simple, stable and able to effectively remove large noise. Experimental results demonstrate that the proposed Mesh Mean Filter does not cause surface shrinkage and shape distortion during the denoising process, and preserves geometric detail features to a certain extent. (2)
An effective third-order local fitting patch and its application Zhong Li1,2,3, Brian Barsky3, Xiaogang Jin2 1 Department of Mathematics and Science, Zhejiang Sci-Tech University, Hangzhou 310018, P.R. China 2 State Key Lab of CAD&CG, Zhejiang University, Hangzhou 310058, P.R. China 3 Computer Science Division, University of California, Berkeley, CA 94720, USA E-mail:
[email protected] Keywords: bicubic Bézier surface; normal vector; curvature; torsion Abstract: In this paper, we extend Razdan and Bae’s second-order local fitting method [11] to construct an effective third-order fitting patch. Compared to other estimation algorithms, this weighted bicubic Bézier patch more accurately obtains the normal vector and curvature estimation of a triangular mesh model. Furthermore, we define the principal geodesic torsion of each vertex on the mesh model and estimate it through this local fitting patch. In the end of this paper, we apply the third-order fitting patch for the mesh smoothing and hole-filling which can get the satisfactory results. . (3)
Image based modeling via plane sweep based surface Growing Bo Shu 1,2, Xianjie Qiu 1, Zhaoqi Wang1 1 Institute of Computing Technology, Chinese Academy of Sciences, 100190 Beijing China 2 Graduate School of the Chinese Academy of Sciences, 100190 Beijing, China
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{bshu, qxj, zqwang}@ict.ac.cn Keywords: multi-view stereo; plane based sweep; surface growing; quasi dense Abstract: In this paper, we present a multi-view stereo based shaped modeling method. Using images captured from different viewpoints, our approach can provide objects’ 3d models with high fidelity details automatically and efficiently. We firstly use a strict plane based sweep stereo method via GPU to compute quasi-dense depth maps which usually have many holes. Then, a simplified patch based surface growing method is used to compute dense depth maps and the corresponding 3d geometry model. Different from other multi-view stereo methods, we do not optimize object normal during expansion process, but gradually compute the normal information from reconstructed quasi dense neighbors. This makes our method works well on much more difficult scenarios, such as textureless, wide baseline, varying illumination than plane based sweep methods and more efficient than surface growing methods. Experiments show that our method can generate high fidelity 3D object shape model quite efficiently. (4)
Automatic Generation of Coarse Bounding Cages from Dense Meshes Chuhua Xian1, Hongwei Lin1,*, Shuming Gao1 1 State Key Lab. of CAD & CG, Zhejiang University, Hangzhou 310058, P.R. China Keywords: bounding box, coarse bounding cage, automatic generation, deformation, subdivision surface fittingAbstract: The coarse bounding cage of a dense mesh plays important roles in computer graphics, computer vision, and geometric design. Specifically, in volume-based deformation, a coarse bounding cage is required to manipulate the dense mesh model it enclosed; in subdivision surface fitting, the fitting starts from a coarse cage bounding the fitted dense mesh or point set; and so on. However, the generation of a coarse bounding cage is mainly by interactive ways, which are very tedious and time-consuming. In this paper, we develop a fully automatic method to generate a coarse cage bounding a dense mesh model. The automatically generated coarse bounding cage can keep the topological structure and major geometric features of the original mesh model, which is validated by theoretical analysis and experimental data presented in this paper. Further more, we employ the automatically generated coarse bounding cage in some applications, such as deformation, and subdivision fitting, producing good results. (5)
Shape Isophotic Error Metric Controllable Re-Sampling for Point-sampled Surfaces Yongwei Miao1;2;3, Pablo Diaz-Gutierrez4, Renato Pajarola3, M. Gopi4, Jieqing Feng1
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State Key Laboratory of CAD & CG, Zhejiang University, Zhejiang 310027, China College of Science, Zhejiang University of Technology, Zhejiang 310032, China 3 Department of Informatics, University of Z¨urich, Zurich 8050, Switzerland 4 Department of Computer Science, University of California, Irvine, CA 92697-3425, USA 2
Keywords: point-sampled surfaces; isophotic error metric; Gaussian sphere; error metric controllable; re-samplingAbstract: Shape simpli_cation and re-sampling of underlying point-sampled surfaces under userde _ned error bounds is an important and challenging issue. Based on the regular triangulation of the Gaussian sphere and the surface normals mapping onto the Gaussian sphere, a Gaussian sphere based re-sampling scheme is presented that generates a non-uniformly curvature-aware simpli_cation of the given point-sampled model. Owing to the theoretical analysis of shape isophotic error metric for did that Gaussian sphere based sampling, the proposed simpli_cation scheme provides a convenient way to control the re-sampling results under a user-speci_ed error metric bound. The novel algorithm has been implemented and demonstrated on several examples. (6)
A framework for the objective evaluation of segmentation algorithms using a ground-truth of human segmented 3D-models H. Benhabiles1, J-P. Vandeborre1,2, G. Lavou´e3, M. Daoudi1,2 1 LIFL (UMR USTL/CNRS 8022), University of Lille, France 2 Institut TELECOM ; TELECOM Lille 1, France 3 University of Lyon, CNRS, INSA-Lyon, LIRIS, UMR5205, F-69621, France Keywords: 3D-mesh; segmentation; evaluation; ground-truth Abstract: In this paper, we present an evaluation method of 3D-mesh segmentation algorithms based on a ground-truth corpus. This corpus is composed of a set of 3D-models grouped in different classes (animals, furnitures, etc.) associated with several manual segmentations produced by human observers. We define a measure that quantifies the consistency between two segmentations of a 3D-model, whatever their granularity. Finally, we propose an objective quality score for the automatic evaluation of 3D-mesh segmentation algorithms based on these measures and on the ground-truth corpus. Thus the quality of segmentations obtained by automatic algorithms is evaluated in a quantitative way thanks to the quality score, and on an objective basis thanks to the groundtruth corpus. Our approach is illustrated through the evaluation of two recent 3D-mesh segmentation methods. (7)
Approximate Shape Matching And Symmetry Detection for 3D 3
Shapes With Guaranteed Error Bounds Shankar Krishnan1 and Suresh Venkatasubramanian2 1 180 Park Avenue, AT&T Shannon Laboratory, Florham Park, NJ 07932, USA 2 School of Computing, Univ. of Utah, Salt Lake City, UT 84112, USA Keywords: shape matching; symmetry detection; pattern matching; geometric algorithms Abstract: In this paper, we describe a system for approximate shape matching and symmetry (rotation and reflection) detection of geometric shapes represented as point clouds. Rather than using the leastsquares distance as a measure of similarity between shapes, we use the Hausdorff distance between point sets as the underlying shape metric. This allows us to exploit methods from geometric pattern matching to return symmetries and rigid transformation matches with guaranteed error bounds on the quality of our solution. The approximation is determined by intuitive user-specified input precision and distance threshold parameters. Another important feature of our method is that it leverages FFT-based techniques for string matching to compute all approximate symmetries simultaneously. Our algorithm is simple to implement and is efficient; we present a detailed experimental study. (8)
Interpolation over arbitrary topology meshes using Doo-Sabin surfaces Chongyang Deng1, Xunnian Yang2 1 Institute of Applied Mathematics and Engineering Computations, Hangzhou Dianzi University, Hangzhou 310018, P.R. China 2 Department of Mathematics, Zhejiang University, Hangzhou 310027, P.R. China Keywords: Subdivision surfaces; Interpolation; Doo-Sabin subdivision Abstract: Interpolating an arbitrary topology mesh by a smooth surface plays an important role in geometric modeling and computer graphics. In this paper we present an efficient new algorithm for constructing a Doo-Sabin subdivision surface that interpolates a given mesh. By introducing additional degrees of freedom, the control vertices of the Doo-Sabin subdivision surface can be obtained directly with no need to solve any initial or intermediate large systems. The control points are computed by modifying the geometric rules of the first step of Doo-Sabin subdivision scheme and the resulting surface interpolates given vertices and optionally normal vectors at the vertices. The method has several merits for surface modeling purposes: (1) Efficiency: we obtain a generalized quadratic B-spline surface to interpolate a given mesh in a robust and simple manner. (2) Simplicity: we use only simple geometric rules to construct a smooth surface interpolating given data. (3) Locality: the perturbation of a given vertex only influences the surface shape near this vertex. (4) Freedom: for each vertex, there is one degree of freedom to adjust the shape of the interpolation surface. These features make surface interpolation using Doo-Sabin surface very
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simple and thus make the method itself suitable for interactive free-form shape design. (9)
Detail-preserving axial deformation using curve pairs Wen-Bing Ge1;2, Gang Xu1;3;4, Kin-Chuen Hui1, Guo-Ping Wang2 1 CAD Laboratory, Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong 2 The key lab on machine perception and intelligence of MOE, Peking University, Beijing, 100871, China 3 College of Computer, Hangzhou Dianzi University, Hangzhou 310018, P.R.China 4 Galaad, INRIA Sophia-Antipolis, France Keywords: axial deformation; curve pair; detailpreserving; Laplacian coordinate; free form deformation Abstract: Traditional axial deformation is simple and intuitive for users to modify the shape of objects. However, unexpected twist of the object may be obtained. The use of a curve-pair allows the local coordinate frame to be controlled intuitively. However, some important geometric details may be lost and changed in the deformation process. In this paper, we present a detail-preserving axial deformation algorithm based on Laplacian coordinates. Instead of embedding the absolute coordinates into deformation space in traditional axial deformation, we transform the Laplacian coordinates at each vertex according to the transformation of local frames at the closest points on the axial curve. Then the deformed mesh is reconstructed by solving a linear system that describes the reconstruction of the local details in least squares sense. By associating a complex 3D object to a curve-pair, the object can be stretched, bend, twisted intuitively through manipulating the curve-pair, and can also be edited by means of view-dependent sketching. This method combines the advantages of axial deformation and Laplacian mesh editing. Experimental results are presented to show the effectiveness of the proposed method. (10)
Filling Holes in Triangular Meshes by Curve Unfolding Alan Brunton1, Stefanie Wuhrer1,2, Chang Shu1, Prosenjit Bose2, Erik D. Demaine3 1 National Research Council of Canada, Ottawa, Canada 2 Carleton University, Ottawa, Canada 3 Massachusetts Institute of Technology, Massachusetts, USA Keywords: Computational Geometry; Object Modeling; Hole Filling; Curve Unfolding Abstract: We propose a novel approach to automatically fill holes in triangulated models. Each hole is filled using a minimum energy surface that is obtained in three steps. First, we unfold the hole boundary onto a plane using energy minimization. Second, we triangulate the unfolded hole using a
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constrained Delaunay triangulation. Third, we embed the triangular mesh as a minimum energy surface in R3. The running time of the method depends primarily on the size of the hole boundary and not on the size of the model, thereby making the method applicable to large models. Our experiments demonstrate the applicability of the algorithm to the problem of filling holes bounded by highly curved boundaries in large models. (11)
Contributing vertices-based Minkowski sum of a non-convex polyhedron without fold and a convex polyhedron Hichem Barki, Florence Denis, Florent Dupont Universit´e de Lyon, CNRS - Universit´e Lyon 1, LIRIS, UMR5205 - 43 Bd. du 11 novembre 1918, F-69622 Villeurbanne, France Keywords: Minkowski sum; contributing vertices; 3D envelope computation Abstract: We present an original approach for the computation of the Minkowski sum of a non-convex polyhedron without fold and a convex polyhedron, without decomposition and union steps—that constitute the bottleneck of convex decomposition-based algorithms. A non-convex polyhedron without fold is a polyhedron whose boundary is completely recoverable from three orthographic projections defined by three orthogonal basis vectors in R3. First, we generate a superset of the Minkowski sum facets using the concept of contributing vertices we accommodate for a non-convex–convex pair of polyhedra. The generated superset guarantees that its envelope is the boundary of the Minkowski sum polyhedron. Secondly, we extract the Minkowski sum facets and handle the intersections among the superset facets by using 3D envelope computation. Our approach is limited to non-convex polyhedra without fold because of the use of 3D envelope computation to recover the Minkowski sum boundary. Models with holes are not handled by our method. The implementation of our algorithm uses exact number types, produces exact results, and is based on CGAL, the Computational Geometry Algorithms Library. (12)
Volumetric texture synthesis of bone micro-structure as a base for scaffold design Y. Holdstein, A. Fischer, L. Podshivalov, P. Z. Bar-Yoseph Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel Keywords: 3D texture synthesis; bone micro structure; tissue engineering; finite element analysis Abstract: Bones consist of hierarchical biocomposite materials arranged in multi-scale structural geometry exhibiting complex behavior. This structure is vulnerable to various damaging factors that may cause its degradation, such as accidents, medical operations and diseases. Current technology cannot precisely reconstruct damaged bone tissue and can only roughly approximate
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such damaged structures. The aim of this research is to develop a method to identify the damaged regions and provide a best fitting scaffold to imitate the original structure, thus offering better rehabilitation. New imaging techniques at the micro-scale level are emerging. Imaging can already provide highly detailed micro-features of a bone sample or even a complete volumetric micro-structure of a bone. A three-dimensional model of the bone can then be reconstructed and analyzed. This study proposes a new method for applying volumetric texture synthesis that can adapt according to location, size and shape. Such 3D olumetric texture may be irregular, but can still imitate the textural behavior of its surroundings. The method has the ability to create a smooth and continuous structure according to topological and geometrical characteristics. Moreover, the texture captures the stochastic and porous nature of the bone micro-structure. In addition, the resulting texture is tested by applying mechanical analysis to the new synthesized structure, thus controlling the mechanical properties of the reconstructed bone. We believe our method will contribute to understanding bone structure and behavior and make it possible to customize the design and fabrication of scaffolds for bone micro-structures. Moreover, such scaffolds can facilitate the process of rehabilitating damaged bone. (13)
Parts-based 2D shape decomposition by convex hull Lili Wan Institute of Information Science, Beijing Jiaotong University, Beijing 100044, P.R. China Keywords: Shape decomposition; Convex hull; Concavity measurement Abstract: Parts-based 2D shape decomposition is important to shape analysis and recognition. Much research in psychology has shown that the human visual system tends to segment complex objects at regions of deep concavities, so concavity measurement is very important to shape decompositions, but it still has not a well accepted definition. In this paper, we propose a method for measuring concavities and segmenting a 2D shape without holes by 2D convex hulls. The primary motivation for using 2D convex hull in our SLAconcavity (straight line and angle concavity) is to grasp global variation trends of the polygon boundary, and furthermore, determine concave vertexes before computing interior angles for representing local attribute. SLA-concavity is invariant despite the presence of arbitrary translations, rotations and scales after normalizing the polygon by its area. For dealing with oversegmentation, we introduce a decomposition method in order of decreasing concavities, avoiding connection of two vertexes in the same pocket generated by a convex hull. Experimental results show that our approach has good performance. (14)
On minimal orthographic view covers for polyhedra Min Liu1, Karthik Ramani2 1 Institute of Manufacturing Engineering, Tsinghua University, Beijing, 100084, China 2 School of Mechanical Engineering, Purdue University, West Lafayette, IN, 47907, USA
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Keywords: minimal view cover; global visibility map; orthographic view Abstract: In this paper, we consider the external visibility coverage for polyhedra under the orthographic viewing model. The problem is to compute whether the whole boundary of a polyhedron is visible from a finite set of view directions, and if so, how to compute a minimal set of such view directions. A global visibility map based method is developed to calculate an optimal or near-optimal solution using object space segmentation and viewpoint space sampling. Our method subdivides the concave regions of a polyhedron into a ground set containing two types of segments: concave regions with nonempty global visibility maps and convex polygons in the concave regions whose global visibility maps are empty. The viewpoint space is sampled using a generate-as-required heuristic. The corresponding visibility matrix is computed based on global visibility calculation. Our problem is then modeled as an instance of the classical set-cover problem and solved using a minimal visible set based branch-and-bound algorithm. (15)
Shape watermarking based on minimizing the quadric error metric Ming Luo and Adrian G. Bors Department of Computer Science, University of York, YO10 5DD, UK E-mail: { ming , adrian.bors }@cs.york.ac.uk Keywords: Digital Watermarking; Quadric Error Metric; Shape Distortion Analysis Abstract: Blind and robust watermarking of 3D object aims to embed codes into a 3D object such that the object is not visually distorted from the original shape. An essential condition is that the message should be securely extracted even after the graphical object was processed. In this paper, we propose a novel blind and robust mesh watermarking method based on the quadric error metric. The vertices are firstly grouped into bins using a secret key according to their distances to the object center. The statistics of the distances in each bin is modified when embedding the message. A novel quadric selective vertex placement scheme is proposed for finding the best location of each vertex, following watermark embedding, such that the resulting shape distortion is minimal. Experimental results show that the proposed method reduces the distortion to a minimum in the 3D shape. (16)
Distance Field Transform with an Adaptive Iteration Method Fan Chen , Ye Zhao Department of Computer Science, Kent State University Kent, Ohio, USA, Email: [fchen, zhao]@cs.kent.edu
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Keywords: Distance Transform, GPU, Iteration, Wavefront Propagation, Narrow Band, FCC, Multiple-segment Abstract: We propose a novel distance field transform method based on an iterative method adaptively performed on an evolving active band. Our method utilizes a narrow band to store active grid points being computed. Unlike the conventional fast marching method, we do not maintain a priority queue, and instead, perform iterative computing inside the band. This new algorithm alleviates the programming complexity and the data-structure (e.g. a heap) maintenance overhead, and leads to a parallel amenable computational process. During the active band propagating from a starting boundary layer, each grid point stays in the band for a lifespan time, which is determined by analyzing the particular geometric property of the grid structure. In this way, we find the Face-Centered Cubic (FCC) grid is a good 3D structure for distance transform.We further develop a multiple-segment method for the band propagation, achieving the computational complexity of O(m . N) with a segment-related constant m. (17)
Curvature normal vector driven interpolatory subdivision Huanxi Zhao *, Xia Qiu, Luming Liang *, Chuan Sun, Beiji Zou School of Information Science and Engineering, Central South University Changsha 410083, People's Republic of China Keywords: Nonlinear subdivision; Curvature normal vector; Convexity preservingAbstract: We present an intrinsically nonlinear interpolatory subdivision scheme with geometric information and some free parameters via discrete curvatures normal vector. Our scheme can produce fair G1-continuous curves, which can avoid the potential pitfalls and unacceptable cases appeared in the four-point subdivision scheme. Furthermore, with the proper parameter choice, the proposed scheme is convexity-preserving, and reproduces the conic curve. Finally, the experimental results show our scheme is effective. (18)
Interpolation to C1 Boundary Conditions by Polynomial of Degree Six* Caiming Zhang1;2, Feng Li1, Dongmei Niu1, Xingqiang Yang1 1 School of Computer Science and Technology, Shandong University, Jinan, China, 250061 2 Department of Computer Science and Technology, Shandong Economic University, Jinan, China, 250014 Keywords: infinite interpolation; triangular patch; polynomial
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Abstract: A new method for constructing triangular patches to pass the C1 interpolation conditions (boundary curves and cross-boundary slopes), on the boundary of triangles is presented. The triangular patch is constructed by a basic triangular operator and an error triangular operator. The basic operator is a polynomial of degree six, which approximates the interpolation conditions with a higher approximation precision, while the error operator is constructed by the side-vertex method, which passes the C1 error boundary conditions. The C1 error boundary conditions are formed by the C1 interpolation conditions minus the boundary curves and cross-boundary slopes taken from the basic operator. The basic operator and the error operator are put together to form the triangular patch. Comparison results of the new method with other two methods are included. (19)
An Exact Representation of Polygonal Objects by C1-continuous Scalar Fields Based on Binary Space Partitioning Oleg Fryazinov, Alexander Pasko, Valery Adzhiev The National Centre for Computer Animation, Bournemouth University, UK Keywords: Implicit surfaces, Boundary representation, Function representation, Binary Space Partitioning, BSP-field, Exact Conversion Abstract: The problem considered in this work is to find a dimension independent algorithm for the generation of signed scalar fields exactly representing polygonal objects and satisfying the following requirements: the defining real function takes zero value exactly at the polygonal object boundary; no extra zero-value isosurfaces should be generated; C1 continuity of the function in the entire domain. The proposed algorithms are based on the binary space partitioning (BSP) of the object by the planes passing through the polygonal faces and are independent of the object genus, the number of disjoint components, and holes in the initial polygonal mesh. Several extensions of the basic algorithm are proposed to satisfy the selected optimization criteria. The generated BSP-fields allow for applying techniques of the function-based modelling to already existing legacy objects from CAD and computer animation areas, which is illustrated by several examples. (20)
Parameterised free-form feature templates J-P. Pernot1, F. Giannini2, B. Falcidieno2, J-C. L´eon3 1 LSIS, UMR CNRS 6168, Aix-en-Provence, France 2 IMATI-CNR, Genova, Italy 3 G-SCOP Laboratory, FRE CNRS 3028, Grenoble, France Keywords: free-form surfaces; NURBS deformation; feature templates; aesthetic and engineering design. Abstract:
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Even if today’s CAD systems can easily represent free-form shapes by means of NURBS surfaces, their definition and modification still require a deep knowledge and a great skill in the manipulation of the underlying mathematical models. This paper presents an attempt to bring the feature concepts, well-known in the classical mechanical domain, to the free-form domain. The paper extends our previous work on fully free-form features to include parameterised feature templates. The free-form shapes are obtained by deformation according to specific constraint lines taking part to the feature templates definition. The feature template is adapted to the user-specified parameter values by our deformation engine, which can applied either to surfaces and curves. The method is illustrated with examples obtained with our prototype software. (21)
Feature Sensitive Bas Relief Generation Jens Kerber1, Art Tevs1, Alexander Belyaev2, Rhaleb Zayer3, Hans-Peter Seidel1 1 MPI Informatik, Saarbr¨ucken, Germany 2 Heriot-Watt University, Edinburgh, Scotland, UK 3 INRIA, Nancy, France Keywords: shape deformation, computer art, sculpture, tone mapping Abstract: Among all forms of sculpture, bas-relief is arguably the closest to painting. Although inherently a two dimensional sculpture, a bas-relief suggests a visual spatial extension of the scene in depth through the combination of composition, perspective, and shading. Most recently, there have been significant results on digital bas-relief generation but many of the existing techniques may wash out high level surface detail during the compression process. The primary goal of this work is to address the problem of fine features by tailoring a filtering technique that achieves good compression without compromising the quality of surface details. As a secondary application we explore the generation of artistic relief which mimic cubism in painting and we show how it could be used for generating Picasso like portraits. (22)
GridMesh: Fast and High Quality 2D Mesh Generation for Interactive 3D Shape Modeling Andrew Nealen1, Justus Pett2, Marc Alexa2,3, Takeo Igarashi4 1 Rutgers University, 2TU Berlin, 3Walt Disney Animation Studios, 4The University of Tokyo / PRESTOJST Keywords: Meshing; Mesh Generation; Interactive Shape Modeling; Sketch Based Modeling; Fair Surface Design Abstract: In this paper we present an algorithm for watertight meshing of closed, sketched curves. The
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sketch is resampled as a piecewise linear (PWL) curve and placed onto a triangular grid. A small boundary (seed) that describes a closed path along grid points is placed inside the sketch and grown until it resembles the sketch. Vertices of the evolved grid boundary are projected onto the stroke to establish a bijective, ordered mapping. Finally, valences along the boundary are optimized while retaining the previously established mapping. The resulting mesh patch can be duplicated, stitched and inflated to generate a new shape, or used to fill a hole in an existing shape. We have implemented our algorithm in FiberMesh [1], an interactive sketch based interface for designing freeform surfaces, where it is used for the all mesh generation processes. The triangulation generated with our algorithm improves the quality of the model by reducing the number of irregular vertices, while running at real time rates. (23)
Haptic Rendering Using C1 Continuous Reconstructed Distance Fields Wei Li1, Youngung Shon1, Sara McMains1 1 Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA Keywords: haptic rendering; distance fields; Coons interpolation; gradients; collision detection Abstract: Existing haptic rendering algorithms are limited in the complexity of the models that can be simulated realistically without artifacts such as extraneous force discontinuities or pop-through. High haptic update rates for realistic feedback require fast collision detection methods, such as those achievable using distance fields. We introduce an efficient algorithm for reconstructing a C1 distance field from discretely sampled distances and gradients. Our Coons-patch based interpolation method supports 50 kHz 3-DOF haptic rendering of complex models with high accuracy, while eliminating undesirable force discontinuities and pop-through. The approximation error of our representation on a 256×256×256 grid is similar to that of polygonal model reconstructions containing hundreds of thousands of polygons, an accuracy previously unobtainable with haptic force rendering. (24)
Canonical Homotopy Class Representative Using Hyperbolic Structure Wei Zeng1, Miao Jin2, Feng Luo3, Xianfeng David Gu1 1 Department of Computer Science, Stony Brook University, Stony Brook, NY 11794, USA 2 Center for Advanced Computer Studies, University of Louisiana at Lafayette, Lafayette, LA 70504, USA 3 Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA Keywords: homotopy class, hyperbolic structure, hyperbolic Yamabe _ow
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Abstract: Homotopy group plays a role in computational topology with a fundamental importance. Each homotopy equivalence class contains an in_nite number of loops. Finding a canonical representative within a homotopy class will simplify many computational tasks in computational topology, such as loop homotopy detection, pants decomposition. Furthermore, the canonical representative can be used as the shape descriptor. This work introduces a rigorous and practical method to compute a unique representative for each homotopy class. The main strategy is to use hyperbolic structure, such that each homotopy class has a unique closed geodesic, which is the representative. The following is the algorithm pipeline: for a given surface with negative Euler number, we apply hyperbolic Yamabe curvature _ow to compute the unique Riemannian metric, which has constant negative one curvature everywhere and is conformal to the original metric. Then we compute the Fuchsian group generators of the surface on the hyperbolic space. For a given loop on the surface, we lift it to the universal covering space, to obtain the Fuchsian transformation corresponding to the homotopy class of the loop. The unique closed geodesic inside the homotopy class is the axis of the Fuchsian transformation, which is the canonical representative. Theories and algorithms are explained thoroughly in details. Experimental results are reported to show the ef_ciency and ef_cacy of the algorithm. The unique homotopy class representative can be applied for homotopy detection and shape comparison. (25)
Classification of non-manifold singularities from transformations of 2-manifolds J-C. L´eon1, L. De Floriani2, F. H´etroy3 1 Grenoble University, CNRS, Laboratoire G-SCOP, 46 av. F´elix Viallet, 38000 Grenoble, France 2 University of Genova, Department of Computer Science, Via Dodecaneso, 35, 16129 Genova, Italy 3 Grenoble University, CNRS and INRIA, Laboratoire Jean Kuntzmann, 655 av. de l’Europe, 38334 Montbonnot, France Keywords: non-manifold models; shape features; topology; simplicial complexes Abstract: Non-manifold models are frequently encountered in engineering simulations and design as well as in computer graphics. However, these models lack shape characterization for modelling and searching purposes. Topological properties act as a kernel for deriving key features of objects. Here we propose a classification for the non-manifold singularities of non-manifold objects through continuous shape transformations of 2-manifolds without boundary up to the creation of non-manifold singularities. As a result, the non-manifold objects thus created can be categorized and contribute to the definition of a general purpose taxonomy for non-manifold shapes. (26)
On the Computation of the Minimal Ellipse Enclosing a Set of Planar Curves 13
Dan Albocher, Gershon Elber Department of Computer Science, Technion—Israel Institute of Technology, Technion City, Haifa 32000, Israel Keywords: bounding areas, minimal spanning circles, collision detection, linear programming Abstract: A method of finding the precise ellipse of minimal area, enclosing a finite set of regular planar curves (and points), is presented. We start with a direct approach of prescribing the problem using a set of algebraic constraints and solving them. This approach turns out to be intractable using contemporary memory support and computing power and several improvements are presented to alleviate these difficulties: the number of degrees of freedom and constraints is limited and the search domain is restricted. As a result, an alternative set of algebraic constraints is created whose solution is found in a reasonable amount of memory size and/or computing time. (27)
Computing Fenchel-Nielsen Coordinates in Teichm¨uller Shape Space Miao Jin1, Wei Zeng2, Ning Ding1, Xianfeng Gu2 1 The Center for Advanced Computer Studies , University of Louisiana at Lafayette, Lafayette 70506, USA 2 Department of Computer Science, Stony Brook University, Stony Brook 11794, USA Keywords: conformal geometry, Teichm¨ uller space, shape space, shape analysis, shape classification. Abstract: Teichm¨ uller shape space is a finite dimensional Riemannian manifold, where each point represents a class of surfaces, which are conformally equivalent, and a path represents a deformation process from one shape to the other. Two surfaces in the real world correspond to the same point in the Teichm¨ uller space, only if they can be conformally mapped to each other. Teichm¨ uller shape space can be used for surface classification purpose in shape modeling. This work focuses on the computation of the coordinates of high genus surfaces in the Teichm¨ uller space. The coordinates are called as Fenchel-Nielsen coordinates. The main idea is to decompose the surface to pairs of hyperbolic pants. Each pair of pants is a genus zero surface with three boundaries, equipped with hyperbolic metric. Furthermore, all the boundaries are geodesics. Each pair of hyperbolic pants can be uniquely described by the lengths of its boundaries. The way of gluing different pairs of pants can be represented by the twisting angles between two adjacent pairs of pants which share a common boundary. The algorithms are based on Teichm¨ uller space theory in conformal geometry, and they utilize the discrete surface Ricci flow. Most computations are carried out using hyperbolic geometry. The method is automatic, rigorous and efficient. The Teichm¨ uller shape space coordinates can be used for surface classification and indexing. Experimental results on surfaces acquired from real
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world showed the potential value of the method for geometric database indexing, shape comparison and classification. (28)
An Edge Matching Technique for Non-Planar Face Intersections in Geometric Parametric Models M. Baba-ali1,2, D. Marcheix2, X. Skapin1 1 XLIM-SIC, University of Poitiers, UMR CNRS 6172, Bat. SP2MI, Teleport 2, Bvd Marie et Pierre Curie, BP 30179 86962 Futuroscope Chasseneuil Cedex, France 2 Laboratory of Applied Computer Science (LISI), ENSMA, T´el´eport 2 - 1 avenue Cl´ement Ader BP 40109 86961 Futuroscope Chasseneuil cedex, France Keywords: Persistent naming; edge matching; geometric transformation; non-planar intersection Abstract: In current feature-based parametric design systems, the reusability principle is not as fully supported as it should be. Both unpredictability and ambiguity of models often happen during design modification within one system. This reference deficiency significantly reduces the power of feature-based parametric modeling, where geometry reevaluation generates some unexpected topology, especially when considering edges.We propose in this paper a new edge matching method resulting from the intersection of non-planar faces. This method considers the matching between edges as a geometric transformation which overlays edges of the initial model on edges of the reevaluated model. It finds, among all possible edge mappings, the one which represents the design intent as close as possible. In order to find the right mapping, it quantifies the relevance of each mapping. Moreover, in addition to the edges resulting from the intersection of the same pair of faces, this method uses ”virtual edges” for solving some ambiguous cases. (195)
A modified advancing layers mesh generation for thin three-dimensional objects with variable thickness Young Choia, Ki-Youn Kwonb, Soo-Won Chaec,*, Dong-Min Kimc a Department of Mechanical Engineering, Chung-Ang University, Seoul 156-756, Republic of Korea b Samsung Heavy Industries Co., Daejeon 305-714, Republic of Korea c Department of Mechanical Engineering, Korea University, Anam-Dong, Seongbuk-Gu, Seoul 136-713, Republic of Korea Keywords: Modified advancing layers mesh; Prismatic element; Tetrahedral element; Adaptive offsetting; Node relocation Abstract: A method of generating modified advancing layers mesh is proposed. In this paper the mesh
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generation process of semi-unstructured prismatic/tetrahedral mesh is presented for relatively thin three- dimensional geometries with variable thickness, as in the case of injection molding analysis. Prismatic meshes are generated by offsetting initial surface triangular meshes. During the mesh generation process, mesh quality is efficiently improved with the use of a new node relocation method. Finally, tetrahedral meshes are automatically generated in the rest of the domain. The mesh generating capability of the proposed algorithm is demonstrated with the several practical test cases. (204)
Assembling curvature continuous surfaces from triangular patches Kestutis Karciauskas a , Jorg Peters b,* a Vilnius University, Lithuania b Department of CISE, CSE Bldg, University of Florida, Gainesville, FL 32611-6120, USA Keywords: C2 surface construction; Shape; Guide surface; Triangularpatches Abstract: We assemble triangular patches of total degree at most eight to form a curvature continuous surface. The construction illustrates how separation of local shape from representation and formal continuity yields an effective construction paradigm in partly underconstrained scenarios. The approach localizes the technical challenges and applies the spline approach, i.e. keeping the degree fixed but increasing the number of pieces, to deal with increased complexity when many patches join at a central point. (211)
Estimating body shape of dressed humans Nils Hasler a, * , Carsten Stoll a, Bodo Rosenhahn b, Thorsten Thormahlen a, Hans-Peter Seidel a a MPI Informatik, Germany b HannoverUniversity, Germany Keywords: Human modelling; Shape estimation; Surface fitting; Statistical modelling Abstract: The paper presents a method to estimate the detailed 3D body shape of a person even if heavy or loose clothing is worn. The approach is based on a space of human shapes, learned from a large database of registered body scans. Together with this database we use as input a 3D scan or model of the person wearing clothes and apply a fitting method, based on ICP (iterated closest point) registration and Laplacian mesh deformation. The statistical model of human body shapes enforces that the model stays within the space of human shapes. The method therefore allows us to compute the most likely shape and pose of the subject, even if it is heavily occluded or body parts are not visible. Several experiments demonstrate the applicability and accuracy of our approach to recover occluded or missing body parts from 3D laser scans.
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(217)
Smooth reverse subdivision Javad Sadeghi *, Faramarz F.Samavati Department of Computer Science, University of Calgary, Alberta, Canada Keywords: Multiresolution; Reverse subdivision; Energy minimization; Least squares; Wavelets Abstract: In this paper we present a new multiresolution framework that takes into consideration reducing the coarse points’ energy during decomposition. We start from initial biorthogonal filters to include energy minimization in multiresolution. Decomposition and reconstruction are main operations for any multiresolution representation. We formulate decomposition as smooth reverse subdivision, based on a least squares problem. Both approximation of overall shape and energy are taken into account in the least squares formulation through different weights. Using this method, significant smoothness in decomposition of curves and tensor product surfaces can be achieved; while their overall shape is preserved. Having smooth coarse points yields details with maximum characteristics. Our method works well with synthesizing applications in which re-using high-energy details is important. We use our method for finding the smooth reverse of three common subdivision schemes. We also provide examples of our method in curve synthesis and terrain synthesis applications. (226)
Variational Bayesian noise estimation of point sets Mincheol Yoon a, Ioannis Ivrissimtzis b, Seungyong Lee a, * a Department of Computer Science and Engineering, POSTECH, San31, Hyoja-dong, Pohang 790-784, Republic of Korea b Department of Computer Science, Durham University, South Road, Durham DH1 3LE, UK Keywords: Noise estimation; Variational Bayesian method Abstract: Scanning devices acquire geometric information from the surface of an object in the form of a 3D point set. Such point sets, as any data obtained by means of physical measurement, contain some noise. To create an accurate model of the scanned object, this noise should be resolved before or during the process of surface reconstruction. In this paper, we develop a statistical technique to estimate the noise in a scanned point set. The noise is represented as normal distributions with zero mean and their variances determine the amount of the noise. These distributions are estimated with a variational Bayesian method, which is known to provide more robust estimations than point estimate methods, such as maximum likelihood and maximum a posteriori. Validation experiments and further tests with real scan data show that the proposed technique can accurately estimate the noise in a 3D point set.
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Efficient triangle–triangle intersection test for OBB-based collision detection Jung-Woo Chang, Myung-Soo Kim* School of Computer Science and Engineering, Seoul National University, Seoul 151-744, Republic of Korea Keywords: Triangle–triangle intersection; OBB; Collision detection; Coordinate representation Abstract: We present an efficient algorithm for triangle–triangle intersection test in oriented bounding box (OBB)-based collision detection. In testing two OBB leaf nodes (i.e., rectangles), many intermediate computation results can be reused for the intersection test of two triangles they contain. It is considerably easier to detect redundant operations when we work in the local coordinate of the bounding rectangle rather than in the global coordinate of the object. The performance improvement of our algorithm is based on this observation that eliminates redundant computations. Compared with conventional algorithms, we have observed 15–79% improvement in computing time. We demonstrate the effectiveness of our approach using several experimental results. (241)
Mesh simplification by stochastic sampling and topological clustering TamyBoubekeur a, *, MarcAlexa b a TELECOM ParisTech, CNRS LTCI, France b TU Berlin, Germany Keywords: Surface simplification;
Mesh subsampling; Clustering; Stochastic geometry processing
Abstract: We introduce TOPSTOC, a fast mesh simplification algorithm. The two main components are stochastic vertex selection and re-indexing of triangles. The probability for vertex selection depends on a local feature estimator, which prefers areas of high curvatures but still ensures sufficient sampling in flat parts. Re-indexing the triangles is done by breadth-first traversal starting from the selected vertices and then identifying triangles incident upon three regions. Both steps are linear in the number of triangles, require minimal data, and ar every fast, while still preserving geometrical and topological features. Additional optional processing steps improve sampling properties and/or guarantee homotopy equivalence with the input. These properties provide an alternative to vertex clustering especially for CAD/CAM models in the areas of previewing or network graphics. (250)
Quality encoding for tetrahedral mesh optimization 18
Kai Xu a,b, Zhi-Quan Cheng a, Yanzhen Wang a, Yueshan Xiong a, HaoZhang b, * a School of Computer Science, National University of Defense Technology, China b School of Computing Science, Simon Fraser University, Canada Keywords: Tetrahedral mesh optimization; Tetrahedral element quality; Algebraic metrics; Quality differential coordinates Abstract: We define quality differential coordinates (QDC) for per-vertex encoding of the quality of a tetrahedral mesh. QDC measures the deviation of a mesh vertex from a position which maximizes the combined quality of the set of tetrahedra incident at that vertex. Our formulation allows the incorporation of different choices of element quality metrics into QDC construction to penalize badly shaped and inverted tetrahedra. We develop an algorithm for tetrahedral mesh optimization through energy minimization driven by QDC. The variational problem is solved efficiently and robustly using gradient flow based on a stable semi-implicit integration scheme. To ensure quality boundary of the resulting tetrahedral mesh, we propose a harmonic-guided optimization scheme which leads to consistent handling of both the interior and boundary tetrahedra. (262)
Consistent segmentation of 3D models Aleksey Golovinskiy*, Thomas Funkhouser Department of Computer Science, Princeton University, 35 Olden Street, Princeton NJ08540, USA Keywords: Mesh segmentation; Mesh analysis Abstract: This paper proposes a method to segment a set of models consistently. The method simultaneously segments models and creates correspondences between segments. First, a graph is constructed whose nodes represent the faces of every mesh, and whose edges connect adjacent faces within a mesh and corresponding faces in different meshes. Second, a consistent segmentation is created by clustering this graph, allowing for outlier segments that are not present in every mesh. The method is demonstrated for several classes of objects and used for two applications: symmetric segmentation and segmentation transfer. (270)
Extracting a polyhedron from a single-view sketch: Topological construction of a wireframe sketch with minimal hidden elements Sofia Kyratzi *, Nickolas Sapidis Department of Product and System Design Engineering, University of the Aegean, Hermoupolis, Syros 84100, Greece
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Keywords: 3D reconstruction; Graph-based sketch model; Solid modeling; Computer-aided design; Object modeling Abstract: An essential prerequisite to construct a manifold trihedral polyhedron from a given natural (or partial-view) sketch is solution of the ‘‘wire frame sketch from a single natural sketch(WSS)’’ problem, which is the subject of this paper. Published solutions view WSS as an ‘‘image-processing’’ / ‘‘computervision’’ problem where emphasis is placed on analyzing the given input (natural sketch) using various heuristics. This paper proposes a new WSS method based on robust tools from graph theory, solid modeling and Euclidean geometry. Focus is placed on producing a minimal wireframe sketch that corresponds to a topologically correct polyhedron. (280)
Semantics-driven best view of 3D shapes Michela Mortara* , Michela Spagnuolo CNR IMATI Ge, via de Marini 6, 16149 Genova, Italy Keywords: Viewpoint selection; Semantics; Segmentation; 3D shapes Abstract: The problem of automatically selecting the pose of a 3D object that corresponds to the most informative and intuitive view of the shape is known as he best view problem. In this paper we address the selection of the best view driven by the meaningful features of the shape, in order to maximize the visibility of salient components from the context or from the application point of view. Meaningful features can be automatically detected by means of semantic-oriented segmentations: we tested several approaches with very pleasant results in the automatic generation of thumbnails for large 3D model databases. (291)
Optimal rotation alignment of 3D objects using a GPU-based similarity function Michael Martinek*, Roberto Grosso University of Erlangen, Department of Computer Science, Erlangen, Germany Keywords: Rotation alignment; Similarity measure; Rotational symmetry; 3D objects; Optimization Abstract: In this paper, we address the challenging task of finding the best alignment between two 3D objects by solving a global optimization problem in the space of rotations SO(3). The objective function to be optimized is a newly developed rotation-variant similarity measure, which is obtained directly from the object’s geometry and is entirely implemented on the GPU. By exploiting the modern GPU’s parallel architecture, we can process considerably greater amounts
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of data than a CPU implementation can do in the same amount of time. This allows us to create a similarity measure which combines speed and accuracy. The actual problem of rotation alignment is then solved by finding the global maximum of this similarity function in the space of rotations. A special rotation representation allows for an efficient local optimization on the manifold SO(3). Furthermore, unwanted local maxima can be avoided by a heuristic global optimization procedure which exploits rotational symmetry. Due to this common sense heuristics, the global search can be gradually reduced to a lower-dimensional problem up to a 1D line search to handle objects with high rotational symmetry. We show that our method is superior to existing normalization techniques such as PCA and provides a high degree of precision despite remarkably short runtimes. (299)
Fourier method for large-scale surface modeling and registration Li Shen*, Sungeun Kim, Andrew J. Saykin Center for Neuroimaging, Division of Imaging Sciences, Department of Radiology, Center for Computational Biology and Bioinformatics, Indiana University School of Medicine, 950 W Walnut St, R2 E124, Indianapolis, IN 46074, USA Keywords: Spherical harmonics; Spherical parameterization; SPHARM expansion; Surface registration; Spherical thin plate spline Abstract: Spherical harmonic (SPHARM) description is a powerful Fourier shape modeling method for processing arbitrarily shaped but simply connected 3D objects. As a highly promising method, SPHARM has been widely used in several domains including medical imaging. However, its primary use has been focused on modeling small or moderately sized surfaces that are relatively smooth, due to challenges related to its applicability, robustness and scalability. This paper presents an enhanced SPHARM framework that addresses these issues and show that the use of SPHARM can expand into broader areas. In particular, we present a simple and efficient Fourier expansion method on the sphere that enables large-scale modeling, and propose a new SPHARM registration method that aims to preserve the important homological properties between 3D models. Although SPHARM is a global descriptor, our experimental results show that the proposed SPHARM framework can accurately describe complicated graphics models and highly convoluted 3D surfaces and the proposed registration method allows for effective alignment and registration of these 3D models for further processing or analysis. These methods greatly enable the potential of applying SPHARM to broader areas such as computer graphics, medical imaging, CAD/CAM, bioinformatics, and other related geometric modeling and processing fields. (312)
Variational implicit surface meshing Arnaud Gelas a,c, *, Sebastien Valette b, Remy Prost b, Wieslaw L.Nowinski a a Biomedical Imaging Laboratory, Singapore BioImaging Consortium, Biopolis, Singapore b CREATIS-LRMN, University of Lyon, INSA-Lyon, CNRS UMR5220, Inserm U630, France
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c
Department of Systems Biology, Harvard Medical School, Boston, MA, USA
Keywords: Implicit surface; Meshing; Variational approach; Restricted Delaunay Abstract: In this paper, we propose a new algorithm to mesh implicit surfaces which produces meshes both with a good triangle aspect ratio as well as a good approximation quality. The number of vertices of the output mesh is defined by the end-user. For this goal, we perform a two-stage processing: an initialization step followed by an iterative optimization step. The initialization step consists in capturing the surface topology and allocating the vertex budget. The optimization algorithm is based on a variational vertices relaxation and triangulation update. In addition a gradation parameter can be defined to adapt the mesh sampling to the curvature of the implicit surface. We demonstrate the efficiency of the approach on synthetic models as well as real-world acquired data, and provide comparisons with previous approaches. (321)
Modeling and 3D object reconstruction by implicitly defined surfaces with sharp features Xinghua Song a,b, Bert Juttler b, * a University of Science and Technology of China, China b Johannes Kepler University Linz, Austria Keywords: Sharp features; Circular spline; Edge descriptor; Surface modeling; Surface fitting Abstract: We propose a new method for describing sharp features(i.e.,edges and vertices) of implicitly defined surfaces. We consider an initial implicitly defined surface, which is represented as the zero set of a C1 smooth scalar field with non-vanishing gradients. Inorder to represent sharp edges and vertices, this surface is augmented by adding new types of implicit representations, called edge descriptors and vertex descriptors. They are defined with the help of the distance field of edge curves. In our implementation, we use circular splines to describe these edge curves, since they support a fast and non-iterative closest point computation. After adding the edge and vertex descriptors to the initial scalar field, the zero set of the augmented function contains the sharp features. We apply the new representation to surface modeling by implicitly defined surfaces with sharp features and to object reconstruction. In the latter case we describe an algorithm for detecting the sharp curves and vertices of a shape which is given by an unorganized point cloud, which are then approximated by circular splines, in order to define the edge and vertex descriptors. (331)
A new construction of smooth surfaces from triangle meshes using parametric pseudo-manifolds 22
Marcelo Siqueiraa,1, Dianna Xub, *, Jean Gallierc, Luis Gustavo Nonatod,e,2, Dimas Martınez Moreraf, Luiz Velhog a UFMS, Campo Grande(MS), Brazil b Department of Computer Science, Bryn Mawr College, Bryn Mawr, PA 19067, USA c University of Pennsylvania, Philadelphia(PA), USA d ICMC-USP, Sao Carlos(SP), Brazil e Scientific Computing and Imaging Institute, University of Utah, SaltLake City(UT),USA f UFAL, Maceio(AL), Brazil g IMPA, Riode Janeiro(RJ), Brazil Keywords: Geometric modeling; Manifolds; Triangle meshes Abstract: We introduce a new manifold-based construction for fitting a smooth surface to a triangle mesh of arbitrary topology. Our construction combines in novel ways most of the best features of previous constructions and, thus, it fills the gap left by them. We also introduce a theoretical framework that provides a sound justification for the correctness of our construction. Finally, we demonstrate the effectiveness of our manifold-based construction with a few concrete examples. (341)
Surface reconstruction configurations
using
bivariate
simplex
splines
on
Delaunay
Juan Cao *, Xin Li b, *, Guozhao Wang a, Hong Qin c a Institute of Computer Graphics and Image Processing, Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China b Department of Electrical and Computer Engineering and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA c Department of Computer Science, State University of New York at Stony Brook, Stony Brook, NY 11794-4400, USA Keywords: B-splines; Simplex splines; Surface fitting; Delaunay configuration; Triangular B-splines; Conformal mapping; Computational geometry; Object modeling; Geometric algorithms Abstract: Recently, a new bivariate simplex spline scheme based on Delaunay configuration has been introduced into the geometric computing community, and it defines a complete spline space that retains many attractive theoretic and computational properties. In this paper, we develop a novel shape modeling framework to reconstruct a closed surface of arbitrary topology based on this new spline scheme. Our framework takes a triangulated set of points, and by solving a linear least-square problem and iteratively refining parameter domains with newly added knots, we can finally obtain a continuous spline surface satisfying the requirement of a user-specified error tolerance. Unlike existing surface reconstruction methods based on triangular B-splines (or DMS
23
splines), in which auxiliary knots must be explicitly added in advance to form a knot sequence for construction of each basis function, our new algorithm completely avoids this less-intuitive and labor-intensive knot generating procedure. We demonstrate the efficacy and effectiveness of our algorithm on real-world, scattered datasets for shape representation and computing. (351)
Persistence-based handle and tunnel loops computation revisited for speed up Tamal K.Dey*, Kuiyu Li Department of Computer Science and Engineering, The Ohio State University, 2015 Neil Avenue, Columbus, OH 43210, USA Keywords: Shape analysis; Topology; Loops in surfaces; Homology; Persistent homology; Topological persistence Abstract: Loops in surfaces associated with topological features such as handles and tunnels are important entities in many applications including surface parameterization, feature identification, and topological simplification. Recently, a persistent homology based algorithm has been proposed to compute them. The algorithm has several advantages including its simplicity, combinatorial nature and independence from computing other extra structures. In this paper, we propose changes to this loop computation algorithm based on some novel observations. These changes reduce the computation time of the algorithm dramatically. In particular, our experimental results show that the suggested changes achieve considerable speed up for large data sets without sacrificing loop qualities. (359)
Geometry-aware domain decomposition for T-spline-based manifold modeling Hongyu Wang a, *, Ying He b, Xin Li c, Xianfeng Gua, Hong Qin a a Computer Science Department, Stony Brook University, Stony Brook, NY 11794-4400, USA b Nanyang Technological University,S ingapore c Louisiana State University, USA Keywords: Manifold splines; T-Splines; Tensor-product B-splines; Shape modeling; Solid modeling; Shape computing; Object segmentation Abstract: This paper presents a new and effective method to construct manifold T-splines of complicated topology/geometry. The fundamental idea of our novel approach is the geometry-aware object segmentation, by which an arbitrarily complicated surface model can be decomposed into a group of disjoint components that comprise branches, handles, and base patches. Such a domain decomposition simplifies objects of arbitrary topological type into a family of genus-zero/one open surfaces, each of which can be conformally parameterized into a set of rectangles. In contrast
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to the conventional decomposition approaches, our method can guarantee that the cutting locus are consistent on the parametric domain. As a result, the resultant T-splines of decomposed components are automatically glued and have high-order continuity everywhere except at the extraordinary points. We show that the number of extraordinary points of the domain manifold is bounded by the number of segmented components. Furthermore, the entire mesh-to-spline data conversion pipeline can be implemented with full automation, and thus, has potential in shape modeling and reverse engineering applications of complicated real-world objects. (369)
A divide-and-conquer approach for automatic polycube map construction Ying He a,*, Hongyu Wangb, Chi-Wing Fua, Hong Qinb a School of Computer Engineering, Nanyang Technological University, 50 Nanyang Avenue, Blk N4, Singapore 639798,Singapore b Computer Science Department, Stony Brook University, NY, USA Keywords: Shape modeling; Polycube map; Discrete Ricci flow; Uniform flat metric; Computational geometry; Object modeling; Geometric algorithms Abstract: Polycube map is a global cross-surface parameterization technique, where the polycube shape can roughly approximate the geometry of modeled objects while retaining the same topology. The large variation of shape geometry and its complex topological type in real-world applications make it difficult to effectively construct a high-quality polycube that can serve as a good global parametric domain for a given object. In practice, existing polycube map construction algorithms typically require a large amount of user interaction for either pre-constructing the polycubes with great care or interactively specifying the geometric constraints to arrive at the user-satisfied maps. Hence, it is tedious and labor intensive to construct polycube maps for surfaces of complicated geometry and topology. This paper aims to develop an effective method to construct polycube maps for surfaces with complicated topology and geometry. Using our method, users can simply specify how close the target polycube mimics a given shape in a quantitative way. Our algorithm can both construct a similar polycube of high geometric fidelity and compute a high-quality polycube map in an automatic fashion. In addition, our method is theoretically guaranteed to output a one-to-one map. To demonstrate the efficacy of our method, we apply the automatically-constructed polycube maps in a number of computer graphics applications, such as seamless texture tiling, T-spline construction, and quadrilateral mesh generation. (381)
Discrete Laplace–Beltrami operators for shape analysis and segmentation Martin Reuter a,b, Silvia Biasotti c, *, Daniela Giorgi c, Giuseppe Patane` c, Michela Spagnuolo c a Massachusetts Institute of Technology, Cambridge, MA, USA b A.A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Boston, MA, USA c Istituto di Matematica Applicatae Tecnologie Informatiche – Consiglio Nazionale delle Ricerche,
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Genova, Italy Keywords: Laplace–Beltrami operator; Eigenfunctions; Nodal sets; Nodal domains; Shape analysis; Shape segmentation Abstract: Shape analysis plays a pivotal role in a large number of applications, ranging from traditional geometry processing to more recent 3D content management. In this scenario, spectral methods are extremely promising as they provide a natural library of tools for shape analysis, intrinsically defined by the shape itself. In particular, the eigenfunctions of the Laplace–Beltrami operator yield a set of real-valued functions that provide interesting insights in the structure and morphology of the shape. In this paper, we first analyze different discretizations of the Laplace–Beltrami operator (geometric Laplacians, linear and cubic FEM operators) in terms of the correctness of their eigenfunctions with respect to the continuous case. We then present the family of segmentations induced by the nodal sets of the eigenfunctions, discussing its meaningfulness for shape understanding. (391)
Dynamic harmonic fields for surface processing Kai Xu a,b, Hao Zhang b, *, Daniel Cohen-Or c, Yueshan Xiong a a School of Computer Science, National University of Defense Technology, China b School of Computing Science, SimonFraserUniversity, Canada c School of Computer Science ,Tel-AvivUniversity, Israel Keywords: Harmonic fields; Dynamic update; Boundary constraints; Penalty method; Multi-rank updating Abstract: Harmonic fields have been shown to provide effective guidance for a number of geometry processing problems. In this paper, we propose a method for fast updating of harmonic fields defined on polygonal meshes, enabling real-time insertion and deletion of constraints. Our approach utilizes the penalty method to enforce constraints in harmonic field computation. It maintains the symmetry of the Laplacian system and takes advantage of fast multi-rank updating and downdating of Cholesky factorization, achieving both speed and numerical stability. We demonstrate how the interactivity induced by fast harmonic field update can be utilized in several applications, including harmonic- guided quadrilateral remeshing, vector field design, interactive geometric detail modeling, and handle- driven shape editing and animation transfer with a dynamic handle set. (399)
Computing smooth approximations of scalar functions with constraints Giuseppe Patane*, BiancaFalcidieno Istituto di Matematica Applicata e Tecnologie Informatiche - Consiglio Nazionale delle Ricerche,
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Genova, Italy Keywords: Signal and function smoothing; Critical points; Laplacian matrix; Shape analysis; Morse complex; Level sets Abstract: In engineering, geographical applications, scientific visualization, and bio-informatics, a variety of phenomena is described by a large set of datamodeled a sthe values of a scalar functio f : µ → R defined on a surface µ . Alow quality of the discrete representations of the input data, unstable computations, numerical approximations, and noise might produce functions with a high number of critical points. In this context, we propose an algorithmic framework for smoothing an arbitrary scalar function, while simplifying its redundant critical points and preserving those that are mandatory for its description. From our perspective, the critical points of f are an atural choice to guide the approximation scheme; infact they usually represent relevant information about the behavior of f or the shape itself. To address the aforementioned aims, we compute a smooth approximation f : µ → R of f whose set of critical points contains those that have been preserved by the simplification process. The idea behind the proposed approach is to combine smoothing techniques, criticalpoints, and spectral properties of the Laplacianmatrix. Inserting constraints in the smoothing of f allows us to overcome the traditional error-driven approximation of f , which does not provide constraints on the preserved topological features. Finally, the computational cost of the proposed approach is O ( n log n ) , where n is the number of vertices of µ . (414)
2D shape morphing via automatic feature matching and hierarchical interpolation Wenwu Yang,Jieqing Feng* State Key Laboratory of CAD&CG, Zhejiang University, Hangzhou 310027, PR China Keywords: Shape morphing; Animation; Feature matching; Path interpolation Abstract: The paper presents a new method to interpolate a pair of 2D shapes that are represented by piecewise linear curves. The method addresses two key problems in 2D shape morphing process: feature correspondence and path interpolation. First, a robust feature metric is defined to measure the similarity of a pair of 2D shapes in terms of visual appearance, orientation and relative size. Based on the metric, an optimal problem is defined and solved to associate the features on the source shape with the corresponding ones on the target shape. Then, a two-level hierarchical approach is proposed to solve the corresponding features interpolation trajectory problem. The algorithm decomposes the input shapes into a pair of corresponding coarse polygons and several pairs of corresponding features. Then the corresponding coarse polygons are interpolated in an as-rigid-as-possible plausible way; meanwhile the corresponding features are interpolated using
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the intrinsic method. Thus interior distortions of the intermediate shapes could be avoided and the feature details on the input shapes could be well preserved. Experimental results show that the method can generate smooth, natural and visually pleasing 2D shape morphing effects. (424)
Gradient field based inhomogeneous volumetric mesh deformation for maxillofacial surgery simulation Sheng-hui Liao a, *, Ruo-feng Tong b, Jin-xiang Dong a, Fu-dong Zhu b a State Key Laboratory of CAD and CG, Department of Computer Science and Engineering, ZhejiangUniversity, Hangzhou, China b The Affiliated of Stomatological Hospital, College of Medicine, Zhejiang University, Hangzhou, China Keywords: Volumetric mesh deformation; Volumetric gradient field; Local transformation; Inhomogeneous material; Maxillofacial surgery; Soft tissue simulation Abstract: This paper presents a novel inhomogeneous volumetric mesh deformation approach by gradient field manipulation, and uses it for maxillofacial surgery simulation. The study is inspired by the state-of-the- art surface deformation techniques based on differential representations. Working in the volumetric domain instead of on only the surface can preserve the volumetric details much better, avoid local self- intersections, and achieve better deformation propagation because of the internal mesh connections. By integrating the mesh cell material stiffness parameter into our new discrete volumetric Laplacian operator, it is very convenient to incorporate inhomogeneous materials into the deformation framework. In addition, the system matrix for solving the volumetric harmonic field to handle the local transformation problem is the same used for Poisson reconstruction equation, thus it requires solving essentially only one global linear system. The system is easy to use, and can accept explicit rotational constraints, or only translational constraints to drive the deformation. One typical maxillofacial surgery case was simulated by the new methodology with inhomogeneous material estimated directly from CT data, and compared to the commonly used finite element method (FEM) approach. The results demonstrated that the deformation methodology achieved good accuracy, as well as interactive performance. Therefore, the usage of our volumetric mesh deformation approach is relevant and suitable for daily clinical practice. (433)
Content-aware model resizing based on surface deformation Kun-PengWang a, Cai-MingZhang a,b, * a School of Computer Science and Technology, Shandong University, Jinan 250061,China b School of Computer Science and Technology, University of Shandong Economics, Jinan250014, China Keywords:
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3D meshes; Scaling; Resizing; Feature preserving; Deformation Abstract: Model resizing is a common operation when we adapt available models to suit different surrounding scenes. Uniform scaling is not ideal in such scenario, since it will result in loss of features and unwanted distortions. In this work, we propose a novel method for content-aware mesh resizing. Unlike uniform scaling, significant features can be well preserved after scaling. Compared with the seminal work by Kraevoy et al., our method does not need an auxiliary regular grid, and directly deforms the mesh models according to local sensitivity to geometric scaling. The method is efficient, easy-to-implement and produces reasonable scaling results.
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