Simplification Operators on a Dimension-Independent Graph-Based Representation of Morse Complexes

Čomić, Lidija; Floriani, Leila De; Iuricich, Federico

doi:10.1007/978-3-642-38294-9_2

Cited by 5 publications

(5 citation statements)

References 17 publications

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“…Several strategies have been proposed in [GBHP11], which aim at postponing a cancellation that would introduce a number of arcs greater than a predefined threshold, or nodes having more than a certain number of incident arcs. For overcoming the same problem in [ČomićDI13] a simplification algorithm based on the remove operator has been defined, which constantly reduce the number of arcs in G (see Section 7.1). In [Iur14], two simplification algorithms have been developed and compared, based on remove and cancellation, respectively.…”

Section: Discussionmentioning

confidence: 99%

“…It has been observed experimentally in [ČomićDI13] that saddle-saddle operators are likely to be performed early in the simplification process (such simplifications can be interpreted as noise removal). When the data set is small and the number of simplifications is high compared to the total number of nodes, the two algorithms behave quite similarly.…”

Section: Discussionmentioning

confidence: 99%

“…This approach provides an efficient handling for the geometry at the expense of a considerable increase in storage costs. In [ČomićDI13], the top cells and the 0-cells of the ascending and descending Morse complexes are stored in the nodes corresponding to maxima and minima. This reduces the storage cost of the geometry but affects the performances of the algorithm, since the top simplices (forming the ascending and descending top cells) have to be merged after each simplification involving extrema.…”

Section: Simplification Algorithmsmentioning

confidence: 99%

See 2 more Smart Citations

Morse complexes for shape segmentation and homological analysis: discrete models and algorithms

Floriani

Fugacci

Iuricich

et al. 2015

Computer Graphics Forum

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Morse theory offers a natural and mathematically-sound tool for shape analysis and understanding. It allows studying the behavior of a scalar function defined on a manifold. Starting from a Morse function, we can decompose the domain of the function into meaningful regions associated with the critical points of the function. Such decompositions, called Morse complexes, provide a segmentation of a shape and are extensively used in terrain modeling and in scientific visualization. Discrete Morse theory, a combinatorial counterpart of smooth Morse theory defined over cell complexes, provides an excellent basis for computing Morse complexes in a robust and efficient way. Moreover, since a discrete Morse complex computed over a given complex has the same homology as the original one, but fewer cells, discrete Morse theory is a fundamental tool for efficiently detecting holes in shapes through homology and persistent homology. In this survey, we review, classify and analyze algorithms for computing and simplifying Morse complexes in the context of such applications with an emphasis on discrete Morse theory and on algorithms based on it.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Simplification Algorithmsmentioning

confidence: 99%

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Morse complexes for shape segmentation and homological analysis: discrete models and algorithms

Floriani

Fugacci

Iuricich

et al. 2015

Computer Graphics Forum

Self Cite

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show abstract

“…To create such representation, it is useful to act on geometric and topological properties of the terrain separately. Several algorithms for modifying the topology while maintaining the triangulation of the terrain have been proposed [10,30,23,22,14,21]; our approach can be seen as acting in the other direction, maintaining topology and simplifying the triangulation. Formerly proposed methods described in [26,16] aim for a similar goal, but they focus on the restricted case of edge contractions and on how such operators affect discrete Morse gradient vector fields.…”

Section: Motivation and Further Related Workmentioning

confidence: 99%

Topology-Preserving Terrain Simplification

Fugacci¹,

Kerber²,

Manet³

2019

Preprint

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We give necessary and sufficient criteria for elementary operations in a two-dimensional terrain to preserve the persistent homology induced by the height function. These operations are edge flips and removals of interior vertices, re-triangulating the link of the removed vertex. This problem is motivated by topological terrain simplification, which means removing as many critical vertices of a terrain as possible while maintaining geometric closeness to the original surface. Existing methods manage to reduce the maximal possible number of critical vertices, but increase thereby the number of regular vertices. Our method can be used to post-process a simplified terrain, drastically reducing its size and preserving its favorable properties.

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“…To create such representation, it is useful to act on geometric and topological properties of the terrain separately. Several algorithms for modifying the topology while maintaining the triangulation of the terrain have been proposed [8,12,[19][20][21]29]; our approach can be seen as acting in the other direction, maintaining topology and simplifying the triangulation. Formerly proposed methods described in [14,24] aim for a similar goal, but they focus on the restricted case of edge contractions and on how such operators affect discrete Morse gradient vector fields.…”

mentioning

confidence: 99%