Efficient Computation of Simplicial Homology through Acyclic Matching

Fugacci, Ulderico; Iuricich, Federico; Floriani, Leila De

doi:10.1109/synasc.2014.84

Cited by 10 publications

(11 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An efficient implementation of [24], focused on regular grids, is discussed in [25] while, for simplicial complexes, the same algorithm as been extended to triangle [26] and tetrahedral meshes [27]. The first dimension independent implementation for simplicial complexes is presented in [28].…”

Section: Computing Persistent Homologymentioning

confidence: 99%

Computing multiparameter persistent homology through a discrete Morse-based approach

Scaramuccia

Iuricich

Floriani

et al. 2020

Computational Geometry

Self Cite

View full text Add to dashboard Cite

Persistent Homology (PH) allows tracking homology features like loops, holes and their higher-dimensional analogs, along with a single-parameter family of nested spaces. Currently, computing descriptors for complex data characterized by multiple functions is becoming a major challenging task in several applications, including physics, chemistry, medicine, geography, etc. Multiparameter Persistent Homology (MPH) generalizes persistent homology opening to the exploration and analysis of shapes endowed with multiple filtering functions. Still, computational constraints prevent MPH to be feasible over realsized data. In this paper, we consider discrete Morse Theory [1] as a tool to simplify the computation of MPH on a multiparameter dataset. We propose a new algorithm, well suited for parallel and distributed implementations and we provide the first evaluation of the impact on MPH computations of a preprocessing approach.

show abstract

Section: Computing Persistent Homologymentioning

confidence: 99%

Computing multiparameter persistent homology through a discrete Morse-based approach

Scaramuccia

Iuricich

Floriani

et al. 2020

Computational Geometry

Self Cite

View full text Add to dashboard Cite

show abstract

“…An encoding for the Forman gradient on such data structures associates the gradient pairs to the top simplices, and is called a compact gradient. The use of such data structures with the compact gradient makes the computation of the Forman gradient and of the Morse and Morse-Smale complexes feasible on simplicial complexes on large size [WIFD13,FID14,FIDW14].…”

Section: Forman Gradient Encoding On Simplicial Complexesmentioning

confidence: 99%

“…Comparison The two approaches described above are dual to each other. The equivalence in the use of reductions or coreductions has been proven in [FID14] in the context of simplicial complexes. Any Forman gradient obtained through a sequence of reductions and removals of top cells can be obtained through a suitable sequence of coreductions and removals of free cells, and vice versa.…”

Section: Algorithms Based On Reduction and Coreductionmentioning

confidence: 99%

“…Any Forman gradient obtained through a sequence of reductions and removals of top cells can be obtained through a suitable sequence of coreductions and removals of free cells, and vice versa. In [FID14] an algorithm has been proposed to build a Forman gradient vector field by executing reduction and coreduction pairs in an interleaved way. It has been shown that any interleaved approach still produces a Forman gradient and that such a gradient can be obtained through reduction or coreduction pairings.…”

Section: Algorithms Based On Reduction and Coreductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Floriani

Fugacci

Iuricich

et al. 2015

Computer Graphics Forum

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…We describe a third method initially formulated in [16], obtained by interleaving reductions and coreductions, and we prove the equivalence of all three techniques. This equivalence will provide us the freedom to implement the method that best fits any given data structure.…”

Section: Introductionmentioning

confidence: 98%

Computing discrete Morse complexes from simplicial complexes

2019

Self Cite

View full text Add to dashboard Cite

We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial complexes, and we define an efficient encoding for the discrete Morse gradient on the most compact of such representations. We theoretically compare methods based on reductions and coreductions for computing a discrete Morse gradient, proving that the combination of reductions and coreductions produces new mutually equivalent approaches. We design and implement a new algorithm for computing a discrete Morse complex on simplicial complexes. We show that our approach scales very well with the size and the dimension of the simplicial complex also through comparisons with the only existing public-domain algorithm for discrete Morse complex computation. We discuss applications to the computation of multi-parameter persistent homology and of extrema graphs for visualization of time-varying 3D scalar fields.

show abstract

Efficient Computation of Simplicial Homology through Acyclic Matching

Cited by 10 publications

References 26 publications

Computing multiparameter persistent homology through a discrete Morse-based approach

Computing multiparameter persistent homology through a discrete Morse-based approach

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

Computing discrete Morse complexes from simplicial complexes

Contact Info

Product

Resources

About