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35
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 61 (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 ...
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 27 (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...
Applications of computational geometry to geographic information systems
 In Handbook of computational geometry
, 2000
"... ..."
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 21 (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.
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 17 (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...
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.
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 12 (6 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
IsoContour Queries and Gradient Descent with Guaranteed Delivery in Sensor Networks
"... Abstract—We study the problem of datadriven routing and navigation in a distributed sensor network over a continuous scalar field. Specifically, we address the problem of searching for the collection of sensors with readings within a specified range. This is named the isocontour query problem. We ..."
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Cited by 10 (3 self)
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Abstract—We study the problem of datadriven routing and navigation in a distributed sensor network over a continuous scalar field. Specifically, we address the problem of searching for the collection of sensors with readings within a specified range. This is named the isocontour query problem. We develop a gradient based routing scheme such that from any query node, the query message follows the signal field gradient or derived quantities and successfully discovers all isocontours of interest. Due to the existence of local maxima and minima, the guaranteed delivery requires preprocessing of the signal field and the construction of a contour tree in a distributed fashion. Our approach has the following properties: (i) the gradient routing uses only local node information and its message complexity is close to optimal, as shown by simulations; (ii) the preprocessing message complexity is linear in the number of nodes and the storage requirement for each node is a small constant. The same preprocessing also facilitates route computation between any pair of nodes where the the route lies within any user supplied range of values. I.
Topology based selection and curation of level sets
 In TopoInVis 2007, Accepted
"... Summary. The selection of appropriate level sets for the quantitative visualization of three dimensional imaging or simulation data is a problem that is both fundamental and essential. The selected level set needs to satisfy several topological and geometric constraints to be useful for subsequent ..."
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Cited by 6 (3 self)
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Summary. The selection of appropriate level sets for the quantitative visualization of three dimensional imaging or simulation data is a problem that is both fundamental and essential. The selected level set needs to satisfy several topological and geometric constraints to be useful for subsequent quantitative processing and visualization. For an initial selection of an isosurface, guided by contour tree data structures, we detect the topological features by computing stable and unstable manifolds of the critical points of the distance function induced by the isosurface. We further enhance the description of these features by associating geometric attributes with them. We then rank the attributed features and provide a handle to them for curation of the topological anomalies. 1