Optimal discrete Morse functions for 2-manifolds

Lewiner, Thomas; Lopes, Hélio; Tavares, Geovan

doi:10.1016/s0925-7721(03)00014-2

Cited by 67 publications

(81 citation statements)

References 6 publications

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“…In [13] it is shown that computing optimal Morse matchings in the setting of simplicial complexes is NP-hard. In [15], a linear algorithm to define optimal discrete Morse functions on discrete 2-manifolds is introduced. In [1] the authors establish conditions under which a 2-dim.…”

Section: Relations Between Minimal Barcodes and Optimal Discrete Morsmentioning

confidence: 99%

Towards Minimal Barcodes

González-Dı́az

Jiménez

Krim³

2013

Graph-Based Representations in Pattern Recognition

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Abstract. In the setting of persistent homology computation, a useful tool is the persistence barcode representation in which pairs of birth and death times of homology classes are encoded in the form of intervals. Starting from a polyhedral complex K (an object subdivided into cells which are polytopes) and an initial order of the set of vertices, we are concerned with the general problem of searching for filters (an order of the rest of the cells) that provide a minimal barcode representation in the sense of having minimal number of "k-significant" intervals, which correspond to homology classes with life-times longer than a fixed number k. As a first step, in this paper we provide an algorithm for computing such a filter for k = 1 on the Hasse diagram of the poset of faces of K.

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Section: Relations Between Minimal Barcodes and Optimal Discrete Morsmentioning

confidence: 99%

Towards Minimal Barcodes

González-Dı́az

Jiménez

Krim³

2013

Graph-Based Representations in Pattern Recognition

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“…Optimal discrete Morse functions has been widely studied in the literature [6,7]. However, this question is not usually considered as an optimization problem in terms of obtaining discrete Morse functions with as less critical simplices as possible.…”

Section: Introductionmentioning

confidence: 99%

“…It can be explained by two main reasons: either a wrong selection of the field of coefficients of the Betti numbers (so it can be sorted out by choosing a suitable field) or the considered complex is homologically trivial (acyclic) but not homotopically trivial (non-contractible). In this sense, since we consider the problem on general 2-complexes, not just on surfaces as in [7], our results are useful to find examples of 2-complexes on which is not possible to define perfect discrete Morse functions. These kind of 2-complexes are interesting since the notions of perfect and optimal discrete Morse function are not equivalent on them.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, we establish conditions under which a 2-complex admits a perfect discrete Morse function and conversely, we get topological properties of a 2-complex admitting such kind of functions. This approach is more general than the known results in the literature [7], since our study is not restricted to surfaces. These results can be considered as a first step in the study of perfect discrete Morse functions on 3-manifolds.…”

mentioning

confidence: 99%

“…First, it can be applied to discrete objects more general than manifolds. Second, it is more suitable in the digital context on topics like pattern recognition, shape classification and recognition, thinning 2D-objects where usually discretized functions are used.Optimal discrete Morse functions has been widely studied in the literature [6,7]. However, this question is not usually considered as an optimization problem in terms of obtaining discrete Morse functions with as less critical simplices as possible.…”

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Perfect discrete Morse functions on 2-complexes

Fernández-Ternero

Vilches

2012

Pattern Recognition Letters

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A bst r ac t This paper is focused on the study of perfect discrete Morse functions on a 2-simplicial complex. These are those discrete Morse functions such that the number of critical i-simplices coincides with the i-th Betti number of the complex. In particular, we establish conditions under which a 2-complex admits a perfect discrete Morse function and conversely, we get topological properties of a 2-complex admitting such kind of functions. This approach is more general than the known results in the literature [7], since our study is not restricted to surfaces. These results can be considered as a first step in the study of perfect discrete Morse functions on 3-manifolds.K e y wor ds perfect discrete Morse function, Betti number, simplicial complex. I nt roductionSince it was introduced, Morse theory has been a powerful tool in the study of smooth manifolds by means of differential geometry techniques. Basically, it allows us to describe the topology of a manifold in terms of the cellular decomposition generated by the critical points of a scalar smooth map defined on it.At the end of the last century, R. Forman [4] developed a discrete version of Morse theory that turned out to be a fruitful and efficient method for the study of the topology of discrete objects, such as simplicial and cellular complexes, which play a central role in many different fields of pure and applied mathematics.Essentially, a discrete Morse function on a simplicial complex is a way to assign a real number to each simplex of a complex, without any continuity, in such a way that for each simplex the natural order given by the dimension simplices is respected, except at most in one (co)face of the given simplex. As in the smooth setting, changes in the topology of the level subcomplexes are deeply related to the presence of critical simplices of the function, and the analysis of the evolution of the homology of these complexes can be a very useful tool in computer vision to deal with shape recognition problems by means of topological shape descriptors. In our opinion, there are many advantages of using Forman's theory. First, it can be applied to discrete objects more general than manifolds. Second, it is more suitable in the digital context on topics like pattern recognition, shape classification and recognition, thinning 2D-objects where usually discretized functions are used.Optimal discrete Morse functions has been widely studied in the literature [6,7]. However, this question is not usually considered as an optimization problem in terms of obtaining discrete Morse functions with as less critical simplices as possible. On the contrary, this problem is mainly settled as a problem of perfect discrete Morse functions, that is, those functions satisfyingwhere m i ( f ) is indicating the number of i-critical simplices of f on K and b i ( K ; F ) is the i-th Betti number of M . In this setting is interesting to point out that there are complexes which do not admit perfect discrete Morse functions. It can be explained by two main reasons:...

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