Delfinado C., Edelsbrunner H.: An Incremental Algorithm for Betti Numbers of Simplicial Complexes on the 3--Sphere. Comput. Aided Geom. Design 12 (1995) 771--784  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the family of triangulations that yield the topological structure of the molecular surfaces (solvent accessible or solvent contact surfaces) while the solvent radius grows. The determination of the topological structure of such molecular surfaces is an important problem addressed by several papers [16]. The family of shapes obtained from a weighted ff shape [20,24] is based on quadratic growth of the radii of the balls and is therefore not directly related to the family based on the growth of the solvent ball radius. In fact the fundamental property on which the ff shape construction is based ....

C. J. A. Delfinado and H. Edelsbrunner. An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere. Computer Aided Geometric Design, 12(7):771--784, 1995.

....is much too complex. For dimension 4 and above, this cannot be computed even with an ideal computer [Mar58] Even from dimension 2 on, we will encounter in this work some NP hard problems (cf section 3.4. 3) However, there exist other computable tools that can describe topological properties [Del93, Dey01] and that can prove, in some cases, that two spaces are not homeomorphic. For example, if there is no homotopy (i.e. continuous deformation) between two spaces, they cannot be topologically equivalent. Morse theory [Mil63] unifies many of those tools. 22 2.2.2 Cell complexes A cell complex is, ....

C. J. A. Delfinado and H. Edelsbrunner. An Incremental Algorithm for Betti Numbers of Simplicial Complexes. In Proceedings of 9th Annual Symposium on Computer Geometry, pages 232--239, 1993.

....Seville, Spain rogodi, real us.es Abstract. 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 ....

....in some sense, counts the number of holes of the object. We can cite two relevant algorithms for computing homology groups H K of a simplicial complex K in R : 1) the classical algorithm based on reducing certain matrices to their Smith normal form [Mun84] 2) the incremental algorithm [DE95,ELZ00,EZ01], avoiding the severe computational costs of the reduction to Smith normal form and consisting of assembling the complex simplex by simplex and at each step updates the Betti numbers of the current complex. Starting with the boundary of a negative simplex, this persistence process finds the cycle ....

[Article contains additional citation context not shown here]

C.J.A. Delfinado, H. Edelsbrunner. An Incremental Algorithm for Betti Numbers of Simplicial Complexes on the 3--Sphere. Comput. Aided Geom. Design, v. 12 (1995), 771--784.

.... groups, for example, the reduction algorithm consisting in reducing the matrices corresponding to the di#erential in each dimension to the Smith normal form, from which one can read o# (co)homology groups of the complex [Mun84] or the incremental algorithm for computing Betti numbers [DE93]. However, there is a gap in the literature concerning general methods for computing cohomology operations. For a given finite simplicial complex K, we sketch a procedure including the computation of some primary and secondary cohomology operations and the A# algebra structure on the ....

C.J.A. Delfinado, H. Edelsbrunner. An Incremental Algorithm for Betti Numbers of Simplicial Complexes. Proc. 9th Ann. Symp. Comput. Geom. (1993) 232--239.

....and empty entries in the edge and tetra hedra rank tables are marked with rank = F rank = 0. 4 Master list. In general, end points of more than one s interval will be mapped to the same s threshold in the spectrum. For many application, such as computing Betti numbers of a complexes [15], or computing the volume, surface, and other metric features of diagrams or space filling diagrams [25] it is desirable to be able to retrieve, given a rank r , all faces whose intervals have . as an end point. In other words, for each s threshold . in the spectrum, we need to store the list of ....

....;qndependent roughly means that one tunnel cannot be obtained as a sum of other tunnels (addition respects orientation and oppositely oriented simplices erase each other) finally, b2 is the number of voids of the complex or shape. Further details are omitted; refer to Delfinado and Edelsbrunner [15] who give an efficient algorithm computing b0, b, and b2 for simplicial cell complexes. An implementation for a shapes, based on the master list, is available. Here, the term old is used for a bounded connected component of the complement of the shape. 72 The author believes that the ....

C Delfinado and H Edelsbrunner. An incremental algorithm for Betti numbers of simplicial complexes. In Proceedings of the Nineth Annual 'ymposium on Computational Geometry, pages 232 239, 1993. 97

....work in that we are computing a natural Morse function given only the triangulated domain as input, our aim being to provide a tool for structural analysis of the domain. As opposed to a global invariant such as the Betti numbers (for which there exists efficient algorithms in dimension 3 and less [4]) critical points of a Morse function give local structural information at each vertex. The output of the computation can then be represented with a visual display or an auditory display, the latter being particularly useful in higher dimensions. 3 Main Results. A general function f on a ....

C. J. A. Delfinado and H. Edelsbrunner, An incremental algorithm for betti numbers of simplicial complexes on the 3-sphere, Computationally Aided Geometric Design 12 (1995), 771--784.

.... 2 are equal compared to the original volume: i) One single connected component ( 0 ) is obtained. 19 ii) There exist as many cavities, tunnels ( 1 ) and voids ( 2 ) as in the original dataset. These two conditions are calculated in a direct manner from the alpha complex (Delfinado and Edelsbrunner, 1995). If topology is not preserved a new point set with a larger number of pseudoatoms is created. 2) Shape preservation: As for the shape similarity, a number of shape measures are calculated between the original data set and the voxel based model obtained from the alpha complex, requesting that ....

Delfinado, C.J.A., H. Edelsbrunner. 1995. An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere, Grid generation, finite elements, and geometric design. Comput. Aided Geom. Design 12;771-784.

....the help of the Smith normal form algorithm is explored by GonzalezDiez and Real [8] in the context of cohomology computation. Among algorithms of computing homology in low dimensions (simplicial complexes in R 3 and S 3 ) probably the most e#cient one is given by Delfinado and Edelsbrunner [5]. Finally, we deal in Section 4 with the most complex project of computing homology of continuous maps. This project is especially motivated by the previously mentioned applications to dynamical systems but there are also many problems in nonlinear analysis where homology of a map provides a ....

....cubical set in the plane R 2 . Then the homology of C(X) is be computed in linear time. Moreover this also holds true for computation in integer coe#cients. The fact that Betti numbers of polyhedra in R 2 can be computed in linear time has already been observed by Delfinado and Edelsbrunner [5]. We refer the reader to [11, 13, 14] for further discussion of cases when the complexity of the reduction algorithm can be improved. 3.3 Computing Homology of a Chain Map Given a triple (C, D, #) where C, D are finite chain complexes with coe# cients in a field F and # : C # D is a chain ....

C. J. A. Delfinado and H. Edelsbrunner, An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere, Computer Aided Geometric Design 12 (1995), 771-784.

.... the expected running time of the classical algorithm is only O(n 2 ) This probabilistic analysis was recently done by Chang and Donald ( Cha Don] The worst case complexity is significantly improved, in the case of a finite simplicial complex embedded in R 3 , by Delfinado and Edelsbrunner ([D E]) who compute Betti numbers in that case in an almost linear time. Their method, however, does not apply to complexes in R n with n 4. In this paper we propose an algorithm for computing homology of a finitely generated chain complex based on reducing the size of the complex to a minimum while ....

C.J.A.Delfinado and H.Edelsbrunner, An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere, Computer Aided Geometric Design, 12 (1995), 771--784.

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C. J. A. DELFINADO AND H. EDELSBRUNNER. An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere. Comput. Aided Geom. Design 12 (1995), 771--784.

....homology groups and Betti numbers [7] These sequences form a framework within which our result on persistent Betti numbers may be placed. The algorithm we develop for computing persistence of non bounding cycles is based on the incremental Betti number algorithm of Delfinado and Edelsbrunner [2]. Three dimensional alpha shapes and complexes may be found in Edelsbrunner and Mucke [4] The problem of topological simplification was also approached by El Sana and Varshney [5] using alpha shape inspired ideas of geometric growth. There is a large body of parallel work on iso surfaces or ....

....before higherdimensional ones, breaking remaining ties arbitrarily. We call the resulting sequence the age filter of the Delaunay triangulation. Incremental algorithm. The ordering of simplices in a filter permits a simple algorithm for computing Betti numbers of all complexes in a filtration [2]. We review the essential steps of the algorithm here. Suppose the sequence , for 0 i m, is a filter and the sequence of = f j 0 j ig, for 0 i m, is the corresponding filtration. Before running the algorithm, the Betti number variables are set to the Betti numbers of the empty ....

[Article contains additional citation context not shown here]

C. J. A. Delfinado and H. Edelsbrunner. An incremental algorithm for betti numbers of simplicial complexes on the 3-sphere. Comput. Aided Geom. Design, 12:771--784, 1995.

....number for each complex in a filtration of m complexes. Let [m] denote the set f1; 2; mg. Then, the linking number may be viewed as a signature function : m] Z that maps each index i 2 [m] to an integer (i) 2 Z. For other signature functions for filtrations of alpha complexes, see [5, 7]. 4 Basis and Surfaces To compute the linking numbers for an alpha complex, we need to recognize cycles, establish a basis for the set of cycles, and find spanning surfaces for the basis cycles. We do so by extending an algorithm we developed for computing Figure 6: Six complexes in the ....

....for each homology class. We use these representatives to compute linking numbers for the complex. A simplex of dimension d in a filtration either creates a d cycle or destroys a (d 1) cycle by turning it into a boundary. We mark simplices as positive or negative, according to this action [5]. In particular, edges in a filtration which connect components are marked as negative. The set of all negative edges gives us a spanning tree of the complex, as shown in Figure 8. We use this spanning tree to define our canonical basis. Every time a positive edge i is added to the complex, it ....

DELFINADO, C. J. A., AND EDELSBRUNNER, H. An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere. Comput. Aided Geom. Design 12 (1995), 771--784.

....homology groups and Betti numbers [6] These sequences form a framework within which our result on persistent Betti numbers may be placed. The algorithm we develop for computing persistence of non bounding cycles is based on the incremental Betti number algorithm of Delfinado and Edelsbrunner [2]. Three dimensional alpha shapes and complexes may be found in Edelsbrunner and Mucke [3] The problem of topological simplification was also ap1 proached by El Sana and Varshney [4] using alpha shape inspired ideas of geometric growth. There is a large body of parallel work on iso surfaces or ....

....before higher dimensional ones, breaking remaining ties ar3 bitrarily. We call the resulting sequence the age filter of the Delaunay triangulation. Incremental algorithm. The ordering of simplices in a filter permits a simple algorithm for computing Betti numbers of all complexes in a filtration [2]. We review the essential steps of the algorithm here. Suppose the sequence of oe i , for 0 i m, is a filter and the sequence of K i = foe j j 0 j ig, for 0 i m, is the corresponding filtration. Before running the algorithm, the Betti number variables are set to the Betti numbers of ....

[Article contains additional citation context not shown here]

C. J. A. Delfinado and H. Edelsbrunner. An incremental algorithm for betti numbers of simplicial complexes on the 3sphere. Comput. Aided Geom. Design, 12:771--784, 1995.

....number for each complex in a filtration of m complexes. Let [m] denote the set 1, 2, m . Then, the linking number may be viewed as a signature function # : m] # Z that maps each index i # [m] to an integer #(i) # Z. For other signature functions for filtrations of alpha complexes, see [5, 7]. 4 Basis and Surfaces To compute the linking numbers for an alpha complex, we need to recognize cycles, establish a basis for the set of cycles, and find spanning surfaces for the basis cycles. We do so by extending an algorithm we developed for computing persistent homology [6] We dispense ....

....for each homology class. We use these representatives to compute linking numbers for the complex. A simplex of dimension d in a filtration either creates a d cycle or destroys a (d 1) cycle by turning it into a boundary. We mark simplices as positive or negative, according to this action [5]. In particular, edges in a filtration which connect components are marked as negative. The set of all negative edges gives us a spanning tree of the complex, as shown in Figure 6. We use this spanning s i Fig. 6. Solid negative edges combine to form a spanning tree. The dashed positive edge # i ....

Delfinado, C. J. A., and Edelsbrunner, H. An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere. Comput. Aided Geom. Design 12 (1995), 771--784.

....possibly non trivial Betti numbers of K j . For example # 0 = # n 0 = 1, # 1 = # n 1 , and # 2 = # n 2 = 1 are the Betti numbers of K = K n . The Betti numbers of K j 1 can be computed from those of K j merely by looking at the type of u j 1 and how its lower star connects to K j [6]. It is convenient to adopt reduced homology groups, but we will freely talk about components and holes when we mean reduced homology classes of non bounding 0 and 1 cycles. We start with # 0 1 = 1 and # 0 0 = # 0 1 = # 0 2 = 0. Regular vertices u j 1 can be skipped because they do not ....

DELFINADO, C. J. A., AND EDELSBRUNNER, H. An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere. Comput. Aided Geom. Design 12 (1995), 771--784.

....non trivial Betti numbers of K j . For example fi 0 = fi n 0 = 1, fi 1 = fi n 1 , and fi 2 = fi n 2 = 1 are the Betti numbers of K = K n . The Betti numbers of K j 1 can be computed from those of K j merely by looking at the type of u j 1 and how its lower star connects to K j [6]. It is convenient to adopt reduced homology groups, but we will freely talk about components and holes when we mean reduced homology classes of non bounding 0and 1 cycles. We start with fi 0 Gamma1 = 1 and fi 0 0 = fi 0 1 = fi 0 2 = 0. Regular vertices u j 1 can be skipped because they ....

Delfinado, C. J. A., and Edelsbrunner, H. An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere. Comput. Aided Geom. Design 12 (1995), 771--784.

....homology groups and Betti numbers [6] These sequences form a framework within which our result on persistent Betti numbers may be placed. The algorithm we develop for computing persistence of nonbounding cycles is based on the incremental Betti number algorithm of Delfinado and Edelsbrunner [3]. Threedimensional alpha shapes and complexes may be found in Edelsbrunner and Mucke [4] The problem of topological simplification was also approached by El Sana and Varshney [5] using alpha shape inspired ideas of geometric growth. There is a large body of parallel work on iso surfaces or level ....

....before higher dimensional ones, breaking remaining ties arbitrarily. We call the resulting sequence the age filter of the Delaunay triangulation. Incremental algorithm. The ordering of simplices in a filter permits a simple algorithm for computing Betti numbers of all complexes in a filtration [3]. We review the essential steps of the algorithm here. Suppose the sequence of oe i , for 0 i m, is a filter and the sequence of K i = foe j j 0 j ig, for 0 i m, is the corresponding filtration. Before running the algorithm, the Betti number variables are set to the Betti numbers of ....

[Article contains additional citation context not shown here]

C. J. A. Delfinado and H. Edelsbrunner. An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere. Comput. Aided Geom. Design 12 (1995), 771--784.

....homology groups and Betti numbers [5] These sequences form a framework within which our result on persistent Betti numbers may be placed. The algorithm we develop for computing persistence of non bounding cycles is based on the incremental Betti number algorithm of Delfinado and Edelsbrunner [2]. Three dimensional alpha shapes and complexes may be found in Edelsbrunner and Mucke [3] The problem of topological simplification was also approached by El Sana and Varshney [4] using alpha shape inspired ideas of geometric growth. There is a large body of parallel work on iso surfaces or ....

....before higher dimensional ones, breaking remaining ties ar bitrarily. We call the resulting sequence the age filter of the Delaunay triangulation. Incremental algorithm. The ordering of simplices in a filter permits a simple algorithm for computing Betti numbers of all complexes in a filtration [2]. We review the essential steps of the algorithm here. Suppose the sequence of oe i , for 0 i m, is a filter and the sequence of K i = foe j j 0 j ig, for 0 i m, is the corresponding filtration. Before running the algorithm, the Betti number variables are set to the Betti numbers of ....

[Article contains additional citation context not shown here]

C. J. A. Delfinado and H. Edelsbrunner. An incremental algorithm for betti numbers of simplicial complexes on the 3sphere. Comput. Aided Geom. Design, 12:771--784, 1995.

....in the ordering. The result is a sequence of collections of Delaunay simplices that captures the evolution of the complex. Note that every prefix of the sequence is itself a complex. Because of this property, we also have a fast algorithm for deciding how and when the homotopy type of K changes [7]. The underlying space of K(t) and the body bounded by the skin at time t are homotopy equivalent [9] It follows that the metamorphoses for the two structures happen at exactly the same moments in time, and these moments can be computed from the sequence of simplices. Assuming general position, ....

....Where do we go from here The first job on the agenda is the rigorous implementation of the dynamic skin triangulation algorithm. The first author of this paper has already taken steps in that direction, partially reusing prior software on Alpha Shapes [10] and on computing Betti numbers [7]. It will be interesting to study the algorithm experimentally and measure the influence of design decisions on its performance. For example, the bounds on the constants C; Q controlling the curvature adaptation algorithm derived in this paper are all conservative. Perhaps it is possible to relax ....

C. J. A. DELFINADO AND H. EDELSBRUNNER. An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere. Comput. Aided Geom. Design 12 (1995), 771--784.

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Delfinado C., Edelsbrunner H.: An Incremental Algorithm for Betti Numbers of Simplicial Complexes on the 3--Sphere. Comput. Aided Geom. Design 12 (1995) 771--784

No context found.

C.J.A. Delfinado, H. Edelsbrunner. An Incremental Algorithm for Betti Numbers of Simplicial Complexes. Proc. 9th Ann. Symp. Comput. Geom. (1993) 232--239.

No context found.

Delfinado C.J.A., Edelsbrunner H.: An Incremental Algorithm for Betti Numbers of Simplicial Complexes on the 3--Sphere. Comput. Aided Geom. Design 12 (1995) 771--784

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C. Delfinado, H. Edelsbrunner, An incremental algorithm for Betti numbers of simplicial complexes, in: Proceedings of the Ninth Annual Symposium on Computational Geometry, 1993, pp. 232--239.

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C.J.A. Delfinado, H. Edelsbrunner, An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere, Computer Aided Geometric Design 12 (1995) 771--784.

No context found.

C. J. A. Delfinado and H. Edelsbrunner. An incremental algorithm for betti numbers of simplicial complexes. Technical Report UIUCDCS-R-93-1787, Computer Science Department, University of Illinois at Urbana-Champaign, 1992.

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