We address the problem of representing the geometry and the morphology of a triangulated surface endowed with a scalar field in a combined geometric and topological multiresolution model. The model, called a Multiresolution Morse Triangulation (MMT), is composed of a multiresolution triangle mesh, and of a multiresolution Morse complex describing the morphology of the field. The MMT is built through a combined morphological and geometrical generalization, and supports queries to extract consistent geometrical and morphological representations of the field at both uniform and variable resolutions.
{"title":"Multiresolution morse triangulations","authors":"E. Danovaro, L. Floriani, P. Magillo, M. Vitali","doi":"10.1145/1839778.1839806","DOIUrl":"https://doi.org/10.1145/1839778.1839806","url":null,"abstract":"We address the problem of representing the geometry and the morphology of a triangulated surface endowed with a scalar field in a combined geometric and topological multiresolution model. The model, called a Multiresolution Morse Triangulation (MMT), is composed of a multiresolution triangle mesh, and of a multiresolution Morse complex describing the morphology of the field. The MMT is built through a combined morphological and geometrical generalization, and supports queries to extract consistent geometrical and morphological representations of the field at both uniform and variable resolutions.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123783272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Michelucci, P. Schreck, Simon E. B. Thierry, Christoph Fünfzig, J. Génevaux
This paper deals with the resolution of geometric constraint systems encountered in CAD-CAM. The main results are that the witness method can be used to detect that a constraint system is over-constrained and that the computation of the maximal rigid subsystems of a system leads to a powerful decomposition method. In a first step, we recall the theoretical framework of the witness method in geometric constraint solving and extend this method to generate a witness. We show then that it can be used to incrementally detect over-constrainedness. We give an algorithm to efficiently identify all maximal rigid parts of a geometric constraint system. We introduce the algorithm of W-decomposition to identify all rigid subsystems: it manages to decompose systems which were not decomposable by classical combinatorial methods.
{"title":"Using the witness method to detect rigid subsystems of geometric constraints in CAD","authors":"D. Michelucci, P. Schreck, Simon E. B. Thierry, Christoph Fünfzig, J. Génevaux","doi":"10.1145/1839778.1839791","DOIUrl":"https://doi.org/10.1145/1839778.1839791","url":null,"abstract":"This paper deals with the resolution of geometric constraint systems encountered in CAD-CAM. The main results are that the witness method can be used to detect that a constraint system is over-constrained and that the computation of the maximal rigid subsystems of a system leads to a powerful decomposition method.\u0000 In a first step, we recall the theoretical framework of the witness method in geometric constraint solving and extend this method to generate a witness. We show then that it can be used to incrementally detect over-constrainedness. We give an algorithm to efficiently identify all maximal rigid parts of a geometric constraint system. We introduce the algorithm of W-decomposition to identify all rigid subsystems: it manages to decompose systems which were not decomposable by classical combinatorial methods.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125369180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The centroidal Voronoi tessellation (CVT) has found versatile applications in geometric modeling, computer graphics, and visualization. In this paper, we extend the concept of the CVT from Euclidean space to hyperbolic space. A novel hyperbolic CVT energy is defined, and the relationship between minimizing this energy and the hyperbolic CVT is proved. We also show by our experimental results that the hyperbolic CVT has the similar property as its Euclidean counterpart where the sites are uniformly distributed according to given density values. Two algorithms -- Lloyd's algorithm and the L-BFGS algorithm -- are adopted to compute the hyperbolic CVT, and the convergence of Lloyd's algorithm is proved. As an example of the application, we utilize the hyperbolic CVT to compute uniform partitions and high-quality remeshing results for high-genus (genus>1) surfaces.
{"title":"Hyperbolic centroidal Voronoi tessellation","authors":"Guodong Rong, Miao Jin, X. Guo","doi":"10.1145/1839778.1839795","DOIUrl":"https://doi.org/10.1145/1839778.1839795","url":null,"abstract":"The centroidal Voronoi tessellation (CVT) has found versatile applications in geometric modeling, computer graphics, and visualization. In this paper, we extend the concept of the CVT from Euclidean space to hyperbolic space. A novel hyperbolic CVT energy is defined, and the relationship between minimizing this energy and the hyperbolic CVT is proved. We also show by our experimental results that the hyperbolic CVT has the similar property as its Euclidean counterpart where the sites are uniformly distributed according to given density values. Two algorithms -- Lloyd's algorithm and the L-BFGS algorithm -- are adopted to compute the hyperbolic CVT, and the convergence of Lloyd's algorithm is proved. As an example of the application, we utilize the hyperbolic CVT to compute uniform partitions and high-quality remeshing results for high-genus (genus>1) surfaces.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123811664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Bajaj, Radhakrishna Bettadapura, Nai Lei, Alex Mollere, Chao Peng, Alexander Rand
Whereas traditional finite element methods use meshes to define domain geometry, weighted extended B-spline finite element methods rely on a weight function. A weight function is a smooth, strictly positive function which vanishes at the domain boundary at an appropriate rate. We describe a method for generating weight functions for a general class of domains based on A-splines. We demonstrate this approach and address the relationship between weight function quality and error in the resulting finite element solutions.
{"title":"Constructing A-spline weight functions for stable WEB-spline finite element methods","authors":"C. Bajaj, Radhakrishna Bettadapura, Nai Lei, Alex Mollere, Chao Peng, Alexander Rand","doi":"10.1145/1839778.1839800","DOIUrl":"https://doi.org/10.1145/1839778.1839800","url":null,"abstract":"Whereas traditional finite element methods use meshes to define domain geometry, weighted extended B-spline finite element methods rely on a weight function. A weight function is a smooth, strictly positive function which vanishes at the domain boundary at an appropriate rate. We describe a method for generating weight functions for a general class of domains based on A-splines. We demonstrate this approach and address the relationship between weight function quality and error in the resulting finite element solutions.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133289135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Classical computational geometry algorithms handle geometric constructs whose shapes and locations are exact. However, many real-world applications require computing with geometric uncertainties, which are often coupled and mutually dependent. Existing uncertainty models cannot be used to handle dependencies among objects resulting in overestimation of the mutual errors. We have recently developed the Linear Parametric Geometric Uncertainty Model (LPGUM), a general and computationally efficient worst-case first-order linear approximation of geometric uncertainty that supports dependencies among uncertainties. In this paper, we present the properties of the uncertainty zones of a point and a line, and offer efficient algorithms to compute them. We also describe new efficient algorithms to handle relative position queries, e.g., the classification of an uncertain point with respect to an uncertain line. We show that, in all cases, the overhead of computing with dependent uncertainties is low.
{"title":"Uncertain geometry with dependencies","authors":"Yonatan Myers, Leo Joskowicz","doi":"10.1145/1839778.1839801","DOIUrl":"https://doi.org/10.1145/1839778.1839801","url":null,"abstract":"Classical computational geometry algorithms handle geometric constructs whose shapes and locations are exact. However, many real-world applications require computing with geometric uncertainties, which are often coupled and mutually dependent. Existing uncertainty models cannot be used to handle dependencies among objects resulting in overestimation of the mutual errors. We have recently developed the Linear Parametric Geometric Uncertainty Model (LPGUM), a general and computationally efficient worst-case first-order linear approximation of geometric uncertainty that supports dependencies among uncertainties. In this paper, we present the properties of the uncertainty zones of a point and a line, and offer efficient algorithms to compute them. We also describe new efficient algorithms to handle relative position queries, e.g., the classification of an uncertain point with respect to an uncertain line. We show that, in all cases, the overhead of computing with dependent uncertainties is low.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121979835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Group morphology is an extension of mathematical morphology with classical Minkowski sum and difference operations generalized respectively to Minkowski product and quotient operations over arbitrary groups. We show that group morphology is a proper setting for unifying, formulating and solving a number of important problems, including translational and rotational configuration space problems, mechanism workspace computation, and symmetry detection. The proposed computational approach is based on group convolution algebras, which extend classical convolutions and the Fourier transform to non-commutative groups. In particular, we show that all Minkowski product and quotient operations may be represented implicitly as sublevel sets of the same real-valued convolution function.
{"title":"Group morphology with convolution algebras","authors":"M. Lysenko, S. Nelaturi, V. Shapiro","doi":"10.1145/1839778.1839781","DOIUrl":"https://doi.org/10.1145/1839778.1839781","url":null,"abstract":"Group morphology is an extension of mathematical morphology with classical Minkowski sum and difference operations generalized respectively to Minkowski product and quotient operations over arbitrary groups. We show that group morphology is a proper setting for unifying, formulating and solving a number of important problems, including translational and rotational configuration space problems, mechanism workspace computation, and symmetry detection. The proposed computational approach is based on group convolution algebras, which extend classical convolutions and the Fourier transform to non-commutative groups. In particular, we show that all Minkowski product and quotient operations may be represented implicitly as sublevel sets of the same real-valued convolution function.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128712243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a new approach, called controlled linear perturbation (CLP), to the robustness problem in computational geometry and demonstrate it on Minkowski sums of polyhedra. The robustness problem is how to implement real RAM algorithms accurately and efficiently using computer arithmetic. Large errors can occur when predicates are assigned inconsistent truth values because the computation assigns incorrect signs to the associated polynomials. CLP enforces consistency by performing a small input perturbation, which it computes using differential calculus. CLP enables us to compute Minkowski sums via convex convolution, whereas prior work uses convex decomposition, which has far greater complexity. Our program is fast and accurate even on inputs with many degeneracies.
{"title":"Robust Minkowski sums of polyhedra via controlled linear perturbation","authors":"V. Milenkovic, E. Sacks, M. Kyung","doi":"10.1145/1839778.1839782","DOIUrl":"https://doi.org/10.1145/1839778.1839782","url":null,"abstract":"We present a new approach, called controlled linear perturbation (CLP), to the robustness problem in computational geometry and demonstrate it on Minkowski sums of polyhedra. The robustness problem is how to implement real RAM algorithms accurately and efficiently using computer arithmetic. Large errors can occur when predicates are assigned inconsistent truth values because the computation assigns incorrect signs to the associated polynomials. CLP enforces consistency by performing a small input perturbation, which it computes using differential calculus. CLP enables us to compute Minkowski sums via convex convolution, whereas prior work uses convex decomposition, which has far greater complexity. Our program is fast and accurate even on inputs with many degeneracies.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114368419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a volume and complexity bounded solid simplification of models represented by Binary Space Partition (BSP). Depending on the compact and robust representation of a solid model in BSP-tree, the boundary surface of a simplified model is guaranteed to be watertight and self-intersection free. Two techniques are investigated in this paper. The volume bounded convex simplification can collapse parts with small volumes on the model into a simple convex volume enclosing the volumetric cells on the input model. The selection of which region to simplify is based on a volume-difference metric, with the help of which the volume difference between the given model and the simplified one is minimized. Another technique is a plane collapse method which reduces the depth of the BSP-tree. These two techniques are integrated into our solid simplification algorithm to give satisfactory results.
提出了用二进制空间划分(Binary Space Partition, BSP)表示的模型的体积和复杂度有界实体化简方法。利用实体模型在BSP-tree中的紧凑性和鲁棒性，可以保证简化模型的边界表面是不透水和无自交的。本文研究了两种技术。体积有界的凸化简可以将模型上体积小的部件折叠成一个简单的凸体，封闭输入模型上的体积单元。选择要简化的区域是基于体积差度量，借助该度量，给定模型与简化模型之间的体积差最小。另一种技术是减少bsp树深度的平面折叠方法。将这两种技术集成到我们的实体化简算法中，得到了令人满意的结果。
{"title":"Volume and complexity bounded simplification of solid model represented by binary space partition","authors":"Pu Huang, Charlie C. L. Wang","doi":"10.1145/1839778.1839805","DOIUrl":"https://doi.org/10.1145/1839778.1839805","url":null,"abstract":"We present a volume and complexity bounded solid simplification of models represented by Binary Space Partition (BSP). Depending on the compact and robust representation of a solid model in BSP-tree, the boundary surface of a simplified model is guaranteed to be watertight and self-intersection free. Two techniques are investigated in this paper. The volume bounded convex simplification can collapse parts with small volumes on the model into a simple convex volume enclosing the volumetric cells on the input model. The selection of which region to simplify is based on a volume-difference metric, with the help of which the volume difference between the given model and the simplified one is minimized. Another technique is a plane collapse method which reduces the depth of the BSP-tree. These two techniques are integrated into our solid simplification algorithm to give satisfactory results.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"126 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128368045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mixed finite element methods solve a PDE involving two or more variables. In typical problems from electromagnetics and electrodiffusion, the degrees of freedom associated to the different variables are stored on both primal and dual domain meshes and a discrete Hodge star is used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the model and numerical stability of a finite element method. We also show how to define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods.
{"title":"A generalization for stable mixed finite elements","authors":"A. Gillette, C. Bajaj","doi":"10.1145/1839778.1839785","DOIUrl":"https://doi.org/10.1145/1839778.1839785","url":null,"abstract":"Mixed finite element methods solve a PDE involving two or more variables. In typical problems from electromagnetics and electrodiffusion, the degrees of freedom associated to the different variables are stored on both primal and dual domain meshes and a discrete Hodge star is used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the model and numerical stability of a finite element method. We also show how to define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131180096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ruosi Li, Lu Liu, Ly Phan, S. S. Abeysinghe, C. Grimm, T. Ju
Extremal surfaces are a class of implicit surfaces that have been found useful in a variety of geometry reconstruction applications. Compared to iso-surfaces, extremal surfaces are particularly challenging to construct in part due to the presence of boundaries and the lack of a consistent orientation. We present a novel, grid-based algorithm for constructing polygonal approximations of extremal surfaces that may be open or unorientable. The algorithm is simple to implement and applicable to both uniform and adaptive grid structures. More importantly, the resulting discrete surface preserves the structural property of the extremal surface in a grid-independent manner. The algorithm is applied to extract ridge surfaces from intensity volumes and reconstruct surfaces from point sets with unoriented normals.
{"title":"Polygonizing extremal surfaces with manifold guarantees","authors":"Ruosi Li, Lu Liu, Ly Phan, S. S. Abeysinghe, C. Grimm, T. Ju","doi":"10.1145/1839778.1839808","DOIUrl":"https://doi.org/10.1145/1839778.1839808","url":null,"abstract":"Extremal surfaces are a class of implicit surfaces that have been found useful in a variety of geometry reconstruction applications. Compared to iso-surfaces, extremal surfaces are particularly challenging to construct in part due to the presence of boundaries and the lack of a consistent orientation. We present a novel, grid-based algorithm for constructing polygonal approximations of extremal surfaces that may be open or unorientable. The algorithm is simple to implement and applicable to both uniform and adaptive grid structures. More importantly, the resulting discrete surface preserves the structural property of the extremal surface in a grid-independent manner. The algorithm is applied to extract ridge surfaces from intensity volumes and reconstruct surfaces from point sets with unoriented normals.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133996891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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