Reeb graphs for shape analysis and applications

Biasotti, Silvia; Giorgi, Daniela; Spagnuolo, Michela; Falcidieno, Bianca

doi:10.1016/j.tcs.2007.10.018

Cited by 247 publications

(176 citation statements)

References 61 publications

(95 reference statements)

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“…We demonstrate how these methods can be extended to design a transfer function that explicitly highlights similar regions. Identifying repeating patterns in scalar fields can also be used to enhance applications that use the contour tree for segmentation [7,21,42,49] and shape matching [3,5,18,43].…”

Section: Contour Tree For Data Exploration and Visualizationmentioning

confidence: 99%

Symmetry in Scalar Field Topology

Thomas

Natarajan

2011

IEEE Trans. Visual. Comput. Graphics

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Abstract-Study of symmetric or repeating patterns in scalar fields is important in scientific data analysis because it gives deep insights into the properties of the underlying phenomenon. Though geometric symmetry has been well studied within areas like shape processing, identifying symmetry in scalar fields has remained largely unexplored due to the high computational cost of the associated algorithms. We propose a computationally efficient algorithm for detecting symmetric patterns in a scalar field distribution by analysing the topology of level sets of the scalar field. Our algorithm computes the contour tree of a given scalar field and identifies subtrees that are similar. We define a robust similarity measure for comparing subtrees of the contour tree and use it to group similar subtrees together. Regions of the domain corresponding to subtrees that belong to a common group are extracted and reported to be symmetric. Identifying symmetry in scalar fields finds applications in visualization, data exploration, and feature detection. We describe two applications in detail: symmetry-aware transfer function design and symmetry-aware isosurface extraction.

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Section: Contour Tree For Data Exploration and Visualizationmentioning

confidence: 99%

Symmetry in Scalar Field Topology

Thomas

Natarajan

2011

IEEE Trans. Visual. Comput. Graphics

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“…Hence, the Reeb graph provides a simple yet meaningful abstraction of the input scalar field. It has been used in a range of applications in computer graphics and visualization; see, for example, the survey [2] and references therein on applications of Reeb graph.…”

Section: Introductionmentioning

confidence: 99%

Reeb Graphs: Approximation and Persistence

Dey

Wang

2012

Discrete Comput Geom

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Given a continuous function f : X → IR on a topological space X, its level set f −1 (a) changes continuously as the real value a changes. Consequently, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of f summarizes this information into a graph structure. Previous work on Reeb graph mainly focused on its efficient computation. In this paper, we initiate the study of two important aspects of the Reeb graph which can facilitate its broader applications in shape and data analysis.The first one is the approximation of the Reeb graph of a function on a smooth compact manifold M without boundary. The approximation is computed from a set of points P sampled from M. By leveraging a relation between the Reeb graph and the so-called vertical homology group, as well as between cycles in M and in a Rips complex constructed from P , we compute the H 1 -homology of the Reeb graph from P . It takes O(n log n) expected time, where n is the size of the 2-skeleton of the Rips complex. As a by-product, when M is an orientable 2-manifold, we also obtain an efficient near-linear time (expected) algorithm for computing the rank of H 1 (M) from point data. The best known previous algorithm for this problem takes O(n 3 ) time for point data.The second aspect concerns the definition and computation of the persistent Reeb graph homology for a sequence of Reeb graphs defined on a filtered space. For a piecewise-linear function defined on a filtration of a simplicial complex K, our algorithm computes all persistent H 1 -homology for the Reeb graphs in O(nn 3 e ) time, where n is the size of the 2-skeleton and n e is the number of edges in K.

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“…The important critical points defined here correspond to component-critical points (as used in [13,7]) of the object's exterior, seen as 3-manifold with boundary. We distinguish important and unimportant critical points with respect to the height function using only the surface of the object (see Section 5).…”

Section: Important Critical Pointsmentioning

confidence: 99%

“…In Morse theory, the topology of a manifold is analyzed using critical points of a differentiable function f over the surface, the Morse function [24,25,4]. A Reeb graph's vertices represent these critical points, its edges correspond to parts of the manifold of which all points with the same f value are connected (see [7] for an introduction). Applications of Reeb graphs include shape abstraction [2,6,22] and recognition [8,36].…”

Section: Introductionmentioning

confidence: 99%

Horizontal decomposition of triangulated solids for the simulation of dip-coating processes

Strodthoff

Schifko

Jüttler

2011

Computer-Aided Design

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In dip-coating processes a three-dimensional object, e.g. an entire car body, is dipped into a liquid bath. In order to simulate such processes, the space surrounding the object is decomposed into the so-called flow volumes, for which each intersection with a horizontal plane is connected. At any time the liquid's surface then has a unique level within such a flow volume, which greatly simplifies the simulation of the liquid. The decomposition into flow volumes corresponds to the Reeb graph of the object's exterior (considered as 3-manifold with boundary) with respect to the height function. This article presents an algorithm which computes this decomposition for an object represented as oriented triangular boundary mesh. First critical vertices of the surface are identified, which include the upper and lower ends of flow volumes. Using local information about horizontal intersection planes near the critical points, a sweep plane algorithm then constructs the volume decomposition in a second step. It is shown that the method can deal with realistic data.

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Reeb graphs for shape analysis and applications

Cited by 247 publications

References 61 publications

Symmetry in Scalar Field Topology

Symmetry in Scalar Field Topology

Reeb Graphs: Approximation and Persistence

Horizontal decomposition of triangulated solids for the simulation of dip-coating processes

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