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27
Hierarchical Morse Complexes for Piecewise Linear 2Manifolds
, 2001
"... We present algorithms for constructing a hierarchy of increasingly coarse Morse complexes that decompose a piecewise linear 2manifold. While Morse complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simu ..."
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Cited by 47 (5 self)
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We present algorithms for constructing a hierarchy of increasingly coarse Morse complexes that decompose a piecewise linear 2manifold. While Morse complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simulating differentiability. We then simplify Morse complexes by cancelling pairs of critical points in order of increasing persistence. Keywords Computational topology, PL manifolds, Morse theory, topological persistence, hierarchy, algorithms, implementation, terrains 1. INTRODUCTION In this paper, we define the Morse complex decomposing a piecewise linear 2manifold and present algorithms for constructing and simplifying this complex. 1.1 Motivation Physical simulation problems often start with a space and measurements over this space. If the measurements are scalar values, we talk about a height function of that space. We use this name throughout the paper, although the functions can ...
Applications of Computational Geometry to Geographic Information Systems
"... Contents 1 Introduction 2 2 Map Data Modeling 4 2.1 TwoDimensional Spatial Entities and Relationships . . . . . . . . . . . . . . . . . . . . . 4 2.2 Raster and Vector Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Subdivisions as Cell Complexes . . . . . . . ..."
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Cited by 22 (1 self)
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Contents 1 Introduction 2 2 Map Data Modeling 4 2.1 TwoDimensional Spatial Entities and Relationships . . . . . . . . . . . . . . . . . . . . . 4 2.2 Raster and Vector Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Subdivisions as Cell Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Topological Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Multiresolution Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Map data processing 8 3.1 Spatial Queries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Map Overlay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Geometric Problems in Map Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Map Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parallel Computation of the Topology of Level Sets
 Algorithmica
, 2003
"... This paper introduces two efficient algorithms that compute the Contour Tree of a 3D scalar field F and its augmented version with the Betti numbers of each isosurface. The Contour Tree is a fundamental data structure in scientific visualization that is used to preprocess the domain mesh to allow op ..."
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Cited by 21 (7 self)
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This paper introduces two efficient algorithms that compute the Contour Tree of a 3D scalar field F and its augmented version with the Betti numbers of each isosurface. The Contour Tree is a fundamental data structure in scientific visualization that is used to preprocess the domain mesh to allow optimal computation of isosurfaces with minimal overhead storage. The Contour Tree can also be used to build user interfaces reporting the complete topological characterization of a scalar field, as shown in Figure 1. Data exploration time is reduced since the user understands the evolution of level set components with changing isovalue. The Augmented Contour Tree provides even more accurate information segmenting the range space of the scalar field in portion of invariant topology. The exploration time for a single isosurface is also improved since its genus is known in advance. Our first new algorithm...
Simple and Optimal OutputSensitive Construction of Contour Trees Using Monotone Paths
, 2004
"... Contour trees are used when highdimensional data are preprocessed for efficient extraction of isocontours for the purpose of visualization. So far, efficient algorithms for contour trees are based on processing the data in sorted order. We present a new algorithm that avoids sorting of the whole da ..."
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Cited by 19 (1 self)
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Contour trees are used when highdimensional data are preprocessed for efficient extraction of isocontours for the purpose of visualization. So far, efficient algorithms for contour trees are based on processing the data in sorted order. We present a new algorithm that avoids sorting of the whole dataset, but sorts only a subset of socalled componentcritical points. They form only a small fraction of the vertices in the dataset, for typical data that arise in practice. The algorithm is simple, achieves the optimal outputsensitive bound in running time, and works in any dimension. Our experiments show that the algorithm compares favorably with the previous best algorithm.
Efficient Methods for Isoline Extraction from a Digital Elevation Model based on Triangulated Irregular Networks
 University of Utrecht, the Netherlands
, 1994
"... A data structure is presented to store a triangulated irregular network digital elevation model, from which isolines (contour lines) can be extracted very efficiently. If the network is based on n points, then for any elevation, the isolines can be obtained in O(log n + k) query time, where k is the ..."
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Cited by 16 (0 self)
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A data structure is presented to store a triangulated irregular network digital elevation model, from which isolines (contour lines) can be extracted very efficiently. If the network is based on n points, then for any elevation, the isolines can be obtained in O(log n + k) query time, where k is the number of line segments that form the isolines. This compares favorably with O(n) time by straightforward computation. When a structured representation of the isolines is needed, the same query time applies. For a fully topological representation (with adjacency), the query requires additional O(c log c) or O(c log log n) time, where c is the number of connected components of isolines. In all three cases, the required data structure has only linear size.
VARIANT: A System for Terrain Modeling at Variable Resolution
 Geoinformatica
, 2000
"... . We describe VARIANT (VAriable Resolution Interactive ANalysis of Terrain), an extensible system for processing and visualizing terrain represented through Triangulated Irregular Networks (TINs), featuring the accuracy of the representation, possibly variable over the terrain domain, as a further p ..."
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Cited by 14 (3 self)
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. We describe VARIANT (VAriable Resolution Interactive ANalysis of Terrain), an extensible system for processing and visualizing terrain represented through Triangulated Irregular Networks (TINs), featuring the accuracy of the representation, possibly variable over the terrain domain, as a further parameter in computation. VARIANT is based on a multiresolution terrain model, which we developed in our earlier research. Its architecture is made of a kernel, which provides primitive operations for building and querying the multiresolution model; and of application programs, which access a terrain model based on the primitives in the kernel. VARIANT directly supports basic queries (e.g., windowing, buering, computation of elevation at a given point, or along a given line) as well as highlevel operations (e.g., yover visualization, contour map extraction, viewshed analysis). However, the true power of VARIANT lies in the possibility of extending it with new applications that can explo...
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
Simple and Optimal OutputSensitive Computation of Contour Trees
, 2003
"... Isosurface extraction is one of the most powerful techniques in the investigation of volume datasets in scientific visualization. The contour tree is a fundamental data structure for fast isosurface extraction, and has also been used to build user interfaces to report the complete topological charac ..."
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Cited by 6 (1 self)
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Isosurface extraction is one of the most powerful techniques in the investigation of volume datasets in scientific visualization. The contour tree is a fundamental data structure for fast isosurface extraction, and has also been used to build user interfaces to report the complete topological characterization of the isosurfaces embedded in the volume data, as well as to simplify the volume data to build a multiresolution hierarchy while preserving the isosurface topologies. In this paper, we present a new outputsensitive algorithm for computing the contour tree. Our algorithm is simple, and achieves the optimal bound of ¢¤£¦¥¨§�©�������©� � in running time for both structuredand unstructuredgrid volume datasets, where ¥ is the number of cells of the input volume, and © is the number of critical points in the input volume, which bounds the size of the contour tree. Our algorithm improves the previous best running time of ��£¦¥���£�����§����������� � given in [5] for unstructured grids (where � is the number of vertices of the input volume), and as the algorithm of [5], works in all dimensions as well. The experiments show that typically © is less than 5 % of the overall number � of the input vertices, and that our algorithm is 2 to 3 times as fast as the previous best algorithm [5].
Contour Trees and Small Seed Sets for Isosurface Traversal
 In Proceedings of the 13th Annual ACM Symposium on Computational Geometry
, 1998
"... For 2D or 3D meshes that represent the domain of continuous function to the reals, the contoursor isosurfacesof a specified value are an important way to visualize the function. To find such contours, a seed set can be used for the starting points from which the traversal of the contours can b ..."
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Cited by 5 (0 self)
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For 2D or 3D meshes that represent the domain of continuous function to the reals, the contoursor isosurfacesof a specified value are an important way to visualize the function. To find such contours, a seed set can be used for the starting points from which the traversal of the contours can begin. This paper gives the first methods to obtain seed sets that are provably small in size. They are based on a variant of the contour tree (or topographic change tree). We give a new, simple algorithm to compute such a tree in regular and irregular meshes that requires O(n log n) time in 2D for meshes with n elements, and in O(n 2 ) time in higher dimensions. The additional storage overhead is proportial to the maximum size of any contour (linear in the worst case, but typically less). Given the contour tree, a minimum size seed set can be computed in roughly quadratic time. Since in practice this can be excessive, we develop a simple approximation algorithm giving a seed set of size a...