Discussion Algorithm for Z
12.4 Topological Simplification
In this section, we first present a case study of the five reordering algorithms described in Chapter 8 and illustrated in Figure 8.7. We then provide experi- mental evidence of the utility of the algorithms, as well as the rarity of basic and recursive conflicts. We end this chapter with visualizations of persistent complexes.12.4.1 A Case Study
In this brief picturesque study, we show the effect of the reordering algorithms in the presence of conflicts. Figure 12.8a displays the k-triangles of the data set SOD. This zeolite does not contain any basic conflicts, but it does have 26 recursive conflicts. We are interested in the tip of the region of large overlap- ping 1-triangles, shown in Figure 12.8b. The rest of the figures in c–l show how this area changes with the different reordering algorithms in Figure 8.7. Note that the differences for the pseudo-triangle algorithm cancel, as each cy- cle is given its due influence, given its persistence. Consequently, we will use this algorithm as the default method for simplification.12.4.2 Timings and Statistics
We have implemented all of the reordering algorithms for experimentation. The algorithms have the basic structure and therefore take about the same time. So, we only give the time taken for the Pseudo-triangle algorithm in Table 12.13. All timings were done on a Sun Ultra-10 with a 440 MHz Ultra- SPARC IIi processor and 256 megabyte RAM, running Solaris 8. Here, each complex is reordered with p equal to the size of the filtration. Generally, the reordering algorithms encounter the same number of conflicts, so we only list the number of basic and recursive conflicts for the pseudo-triangle algorithm in Table 12.13. The time taken for reordering correlates very well with the size of the filtration, as all algorithms make a single pass through the filter. A simplex may move multiple times during reordering, however, because of the recursive nature of the algorithms. The number of recursive conflicts is one indication of the complexity of the reordering. The table shows that the data sets with a large number of recursive conflicts, namely BOG, bearing, TAO, and bone, all have large reordering times. a SOD b Zoomed c Naive d Difference e Shift f Difference g Wormhole h Difference i Pseudo- triangle j Difference k Sudden Death l Difference Fig. 12.8. Reordering algorithms on SOD. a displays the k-triangles of SOD with the region of interest boxed and zoomed in b. c–l show the results of each reordering algorithm and the image difference between these results and b. The difference be- tween images is shown in shades of gray. I have increased the saturation by 25 for better viewing.Parts
» Spaces Topology for Computing
» New Results Topology for Computing
» Organization Topology for Computing
» Sets and Functions Topological Spaces
» Manifolds Topology for Computing
» Geometric Definition Simplicial Complexes
» Abstract Definition Simplicial Complexes
» Filtrations and Signatures Simplicial Complexes
» Manifold Sweeps Topology for Computing
» Binary Operations Groups Introduction to Groups
» Subgroups and Cosets Introduction to Groups
» Structure-Relating Maps Characterizing Groups
» Cyclic Groups Characterizing Groups
» Finitely Generated Abelian Groups
» Factor Groups Characterizing Groups
» Free Abelian Groups Advanced Structures
» Rings, Fields, Integral Domains, and Principal Ideal Domains
» Modules, Vector Spaces, and Gradings
» Structure Theorem Advanced Structures
» The Fundamental Group Justification
» Chains and Cycles Homology Groups
» Simplicial Homology Homology Groups
» Understanding Homology Homology Groups
» The Euler-Poincaré Formula Homology Groups
» Arbitrary Coefficients Topology for Computing
» Tangent Spaces Topology for Computing
» Derivatives and Morse Functions
» Critical Points Stable and Unstable Manifolds
» Morse-Smale Complex Topology for Computing
» Quasi Morse-Smale Complex Hierarchical Morse-Smale Complexes
» Piece-Wise Linear Artifacts Hierarchical Morse-Smale Complexes
» Filtration Hierarchical Morse-Smale Complexes
» Hierarchy Hierarchical Morse-Smale Complexes
» Knots and Links The Linking Number
» Marking Algorithm Topology for Computing
» Abstract Algorithm Algorithm for Z
» Cycle Search Algorithm for Z
» Canonization Algorithm for Z
» Reduction Algorithm for Fields
» Derivation Algorithm for Fields
» Algorithm Algorithm for Fields
» Algorithm for PIDs Topology for Computing
» Approach and Goals Motivation
» Persistent Betti Number Algorithm
» Lazy Migration Reordering Algorithms
» Topology Maps Topology for Computing
» Complex with Junctions The Quasi Morse-Smale Complex Algorithm
» Extending Paths The Quasi Morse-Smale Complex Algorithm
» Handle Slide Local Transformations
» Steepest Ascent Local Transformations
» Motivation Algorithm Topology for Computing
» Component Tree Enumeration Algorithm
» Methodology Topology for Computing
» Libraries and Packages Organization
» Surfaces Miscellaneous Three-Dimensional Data
» Implementation Algorithm for Fields
» Framework and Data Algorithm for Fields
» Field Coefficients Algorithm for Fields
» Higher Dimensions Algorithm for Fields
» A Case Study Timings and Statistics
» Implementation The Linking Number Algorithm
» Timings and Statistics The Linking Number Algorithm
» Topological Feature Detection Computational Structural Biology
» Knotting Computational Structural Biology
» Structure Determination Computational Structural Biology
» Hierarchical Clustering Topology for Computing
» Terrain Simplification Denoising Density Functions
» Iso-Surface Denoising Denoising Density Functions
» Time-Varying Data Denoising Density Functions
» Medial Axis Simplification Denoising Density Functions
» Surface Reconstruction Topology for Computing
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