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13
Loop Surgery for Volumetric Meshes: Reeb Graphs Reduced to Contour Trees
"... Fig. 1. The Reeb graph of a pressure stress function on the volumetric mesh of a brake disk is shown at several scales of hypervolumebased simplification. At the finest resolution of this dataset (3.5 million tetrahedra), our approach computes the Reeb graph in 7.8 seconds while the fastest previou ..."
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Cited by 10 (3 self)
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Fig. 1. The Reeb graph of a pressure stress function on the volumetric mesh of a brake disk is shown at several scales of hypervolumebased simplification. At the finest resolution of this dataset (3.5 million tetrahedra), our approach computes the Reeb graph in 7.8 seconds while the fastest previous techniques [19, 12] do not produce a result. Abstract—This paper introduces an efficient algorithm for computing the Reeb graph of a scalar function f defined on a volumetric mesh M in R3. We introduce a procedure called loop surgery that transforms M into a mesh M ′ by a sequence of cuts and guarantees the Reeb graph of f (M′) to be loop free. Therefore, loop surgery reduces Reeb graph computation to the simpler problem of computing a contour tree, for which wellknown algorithms exist that are theoretically efficient (O(nlogn)) and fast in practice. Inverse cuts reconstruct the loops removed at the beginning. The time complexity of our algorithm is that of a contour tree computation plus a loop surgery overhead, which depends on the number of handles of the mesh. Our systematic experiments confirm that for reallife volumetric data, this overhead is comparable to the computation of the contour tree, demonstrating virtually linear scalability on meshes ranging from 70 thousand to 3.5 million tetrahedra. Performance numbers show that our algorithm, although restricted to volumetric data, has an average speedup factor of 6,500 over the previous fastest techniques, handling larger and more complex datasets. We demonstrate the versatility of our approach by extending fast topologically clean isosurface extraction to nonsimply connected domains. We apply this technique in the context of pressure analysis for mechanical design. In this case, our technique produces results in matter of seconds even for the largest models. For the same models, previous Reeb graph techniques do not produce a result. Index Terms—Reeb graph, scalar field topology, isosurfaces, topological simplification. 1
Algebraic relational approach for geospatial feature correlation
 In: Proceedings of International Conference on Imaging Science, Systems, and Technology (CISST’2002, June 2427, 2002), Las Vegas
, 2002
"... In Geometry in Action: Cartography and geographic Information Systems David Eppstein [6] lists several important problems of computational geometry for cartography and GIS. This paper considers two of them: (1) matching/correlating similar features from different geospatial databases (the conflation ..."
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Cited by 6 (2 self)
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In Geometry in Action: Cartography and geographic Information Systems David Eppstein [6] lists several important problems of computational geometry for cartography and GIS. This paper considers two of them: (1) matching/correlating similar features from different geospatial databases (the conflation problem), and (2) handling approximate and inconsistent data. These problems are of great practical importance for end users defense and intelligence analysts, geologists, geographers, ecologists and others. An adequate mathematical formulation and solution of these problems is still an open question due their complexity. This paper analyzes relations between these problems and topics in computational topology and geometry. This analysis concludes that a fundamentally new mathematical approach is needed. The paper develops such new approach based on the general concept of an abstract algebraic system. Such a system can uniformly express all major algebraic constructs such as groups, fields, algebras and models. We also show the benefit of developing a fundamentally new approach algebraic invariants for geospatial data analysis and correlation using this concept.
Topological quadrangulations of closed triangulated surfaces using the Reeb graph
, 2003
"... Although surfaces are more and more often represented by dense triangulations, it can be useful to convert them to Bspline surface patches, lying on quadrangles. This paper presents a method for the construction of coarse topological quadrangulations of closed triangulated surfaces, based on Morse ..."
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Cited by 6 (0 self)
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Although surfaces are more and more often represented by dense triangulations, it can be useful to convert them to Bspline surface patches, lying on quadrangles. This paper presents a method for the construction of coarse topological quadrangulations of closed triangulated surfaces, based on Morse theory. In order to construct on the surface a quadrangulation of its canonical polygonal schema, we compute first a Reeb graph then a canonical set of generators embedded on the surface. Some results are shown on different surfaces.
Geometric Objects and Cohomology Operations
 Proc. of the 5th Workshop on Computer Algebra in Scientific Computing
, 2002
"... Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes ([Mun84], [DE95,ELZ00], [DG98]), but concerning the algorith ..."
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Cited by 4 (4 self)
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Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes ([Mun84], [DE95,ELZ00], [DG98]), but concerning the algorithmic treatment of cohomology operations, very little is known. In this paper, we establish a version of the incremental algorithm for computing homology given in [ELZ00], which saves algebraic information, allowing us the computation of the cup product and the effective evaluation of the primary and secondary cohomology operations on the cohomology of a finite simplicial complex. The efficient combinatorial descriptions at cochain level of cohomology operations developed in [GR99,GR99a] are essential ingredients in our method. We study the computational complexity of these processes and a program in Mathematica for cohomology computations is presented.
Matching Image Feature Structures Using Shoulder Analysis Method, In: Algorithms and technologies for multispectral, hyperspectral and ultraspectral imagery IX. Vol
 5425, International SPIE military and aerospace symposium, AEROSENSE
"... The problems of imagery registration, conflation, fusion and search require sophisticated and robust methods. An algebraic approach is a promising new option for developing such methods. It is based on algebraic analysis of features represented as polylines. The problem of choosing points when attem ..."
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Cited by 4 (3 self)
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The problems of imagery registration, conflation, fusion and search require sophisticated and robust methods. An algebraic approach is a promising new option for developing such methods. It is based on algebraic analysis of features represented as polylines. The problem of choosing points when attempting to prepare a linear feature for comparison with other linear features is a significant challenge when orientation and scale is unknown. Previously we developed an invariant method known as Binary Structural Division (BSD). It is shown to be effective in comparing feature structure for specific cases. In cases where a bias of structure variability exists however, this method performs less well. A new method of Shoulder Analysis (SA) has been found which enhances point selection, and improves the BSD method. This paper describes the use of shoulder values, which compares the actual distance traveled along a feature to the linear distance from the start to finish of the segment. We show that shoulder values can be utilized within the BSD method, and lead to improved point selection in many cases. This improvement allows images of unknown scale and orientation to be correlated more effectively.
Cycle Bases of Graphs and Sampled Manifolds
 MAXPLANCKINSTITUT FÜR INFORMATIK, STUHLSATZENHAUSWEG 85, 66123
, 2005
"... ... are the dominant output of a multitude of 3D scanning devices. The usefulness of these devices rests on being able to extract properties of the surface from the sample. We show that, under certain sampling conditions, the minimum cycle basis of a nearest neighbor graph of the sample encodes topo ..."
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Cited by 2 (1 self)
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... are the dominant output of a multitude of 3D scanning devices. The usefulness of these devices rests on being able to extract properties of the surface from the sample. We show that, under certain sampling conditions, the minimum cycle basis of a nearest neighbor graph of the sample encodes topological information about the surface and yields bases for the trivial and nontrivial loops of the surface. We validate our results by experiments.
Linear time recognition algorithms for topological invariants in 3d
 CoRR
"... In this paper, we design linear time algorithms to recognize and determine topological invariants such as genus and homology groups in 3D. These invariants can be used to identify patterns in 3D image recognition and medical image analysis. Our method is based on cubical images with direct adjacency ..."
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Cited by 2 (0 self)
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In this paper, we design linear time algorithms to recognize and determine topological invariants such as genus and homology groups in 3D. These invariants can be used to identify patterns in 3D image recognition and medical image analysis. Our method is based on cubical images with direct adjacency, also called (6,26)connectivity images in discrete geometry. According to the fact that there are only six types of local surface points in 3D and a discrete version of the wellknown GaussBonnett Theorem in differential geometry, we first determine the genus of a closed 2Dconnected component (a closed digital surface). Then, we use the Alexander duality to obtain the homology groups of a 3D object in 3D space. This idea can be extended to general simplicial decomposed manifolds or cell complexes in 3D. 1.
An iterative algorithm for homology computation on simplicial shapes
, 2011
"... We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes, We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its subcomponents. The proposed algorithm retrieves the comple ..."
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We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes, We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its subcomponents. The proposed algorithm retrieves the complete homological information of an input shape including the Betti numbers, the torsion coefficients and the representative homology generators. To the best of our knowledge, this is the first algorithm based on the constructive MayerVietoris sequence, which relates the homology of a topological space to the homologies of its subspaces, i.e. the subcomponents of the input shape and their intersections. We demonstrate the validity of our approach through a specific shape decomposition, based only on topological properties, which minimizes the size of the intersections between the subcomponents and increases the efficiency of the algorithm.