A discrete morse-based approach to multivariate data analysis

Iuricich, Federico; Scaramuccia, Sara; Landi, Claudia; Floriani, Leila De

doi:10.1145/3002151.3002166

Cited by 10 publications

(16 citation statements)

References 25 publications

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“…Since one of our inequalities holds under the relative-perfectness assumption, as a last contribution, we prove that relative-perfectness for multi-filtrations can be actually achieved by algorithms Iuricich et al 2016), at least in the case of simplicial complex domains of dimension 2. Analogously to the one-parameter case, in Robins et al (2011) this can be seen as an optimality property among all consistent discrete gradients.…”

Section: Contributionsmentioning

confidence: 78%

“…Concerning computability, we have proven that algorithms Iuricich et al 2016) under their assumptions, i.e., for one-critical filtrations, ensure the relative-perfectness of the output discrete gradient field whenever in the case of simplicial complexes of dimension up to 2 with no restriction on the number of parameters in the filtration. A limitation of such algorithmic construction is that the gradient vector field is computed from a function which is extended from the vertices to other simplices by taking the maximum.…”

Section: Discussionmentioning

confidence: 99%

“…In this work, we address the problem of detecting the most significant piece of hidden information in multivariate data, and we try to give theoretical insights to the problem of capturing the interdependence among component values stated in Kehrer and Hauser (2013). An example of Morse-based techniques applied to visual analytics over multivariate data is provided in Iuricich et al (2016). Therein, a discrete gradient vector field simultaneously consistent with two scalar fields (temperature and pressure) is applied to segment the Hurricane Isabel dataset: the hurricane's eye is detected as the largest cluster of critical cells of the gradient vector field.…”

Section: Introductionmentioning

confidence: 99%

“…these analogies to the one-parameter case are related to the possibility of computing discrete gradient vector fields being consistent with a multifiltration, meaning by that, that the associated Morse complex preserves the MPH information. Motivated by this fact, the algorithms Iuricich et al 2016) retrieve a discrete gradient vector field consistent with a multi-filtration. The Morse-based reduction preprocessing for computing MPH invariants is shown to be effective in Scaramuccia et al (2020).…”

Section: Introductionmentioning

confidence: 99%

“…The construction of such gradient vector field is efficiently achieved, actually in linear time in Iuricich et al (2016) and whenever the worst-case size of a cell star is negligible with respect to the whole complex size. The algorithm in Iuricich et al (2016) improves that in in terms of speed but it is equivalent to it in terms of retrieved critical cells Scaramuccia et al (2020). However, even for these algorithms, the question about whether they retrieve the minimum, also known as optimal, number of critical cells necessary to get the same persistence modules was left as an open problem.…”

Section: Introductionmentioning

confidence: 99%

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Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

Landi

Scaramuccia

2021

J Comb Optim

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The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from specific coordinate systems and does so robustly to noise. Moreover, the geometric content of a discrete gradient vector field is very useful for visualization purposes. The specific case of multivariate data still demands for further investigations, on the one hand, for computational reasons, it is important to reduce the necessary amount of data to be processed. On the other hand, for analysis reasons, the multivariate case requires the detection and interpretation of the possible interdepedance among data components. To this end, in this paper we introduce and study a notion of perfectness for discrete gradient vector fields with respect to multi-parameter persistent homology, called relative-perfectness. As a natural generalization of usual perfectness in Morse theory for homology, relative-perfectness entails having the least number of critical cells relevant for multi-parameter persistence. As a first contribution, we support our definition of relative-perfectness by generalizing Morse inequalities to the filtration structure where homology groups involved are relative with respect to subsequent sublevel sets. In order to allow for an interpretation of critical cells in 2-parameter persistence, our second contribution consists of two inequalities bounding Betti tables of persistence modules from above and below, via the number of critical cells. Our last result is the proof that existing algorithms based on local homotopy expansions allow for efficient computability over simplicial complexes up to dimension 2.

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Section: Contributionsmentioning

confidence: 78%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

Landi

Scaramuccia

2021

J Comb Optim

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Algorithmic Construction of Acyclic Partial Matchings for Multidimensional Persistence

Állili

Kaczyński

Landi

et al. 2017

Discrete Geometry for Computer Imagery

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Given a simplicial complex and a vector-valued function on its vertices, we present an algorithmic construction of an acyclic partial matching on the cells of the complex. This construction is used to build a reduced filtered complex with the same multidimensional persistent homology as of the original one filtered by the sublevel sets of the function. A number of numerical experiments show a substantial rate of reduction in the number of cells achieved by the algorithm

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Local bilinear computation of Jacobi sets

et al. 2022

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We propose a novel method for the computation of Jacobi sets in 2D domains. The Jacobi set is a topological descriptor based on Morse theory that captures gradient alignments among multiple scalar fields, which is useful for multi-field visualization. Previous Jacobi set computations use piecewise linear approximations on triangulations that result in discretization artifacts like zig-zag patterns. In this paper, we utilize a local bilinear method to obtain a more precise approximation of Jacobi sets by preserving the topology and improving the geometry. Consequently, zig-zag patterns on edges are avoided, resulting in a smoother Jacobi set representation. Our experiments show a better convergence with increasing resolution compared to the piecewise linear method. We utilize this advantage with an efficient local subdivision scheme. Finally, our approach is evaluated qualitatively and quantitatively in comparison with previous methods for different mesh resolutions and across a number of synthetic and real-world examples.

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A discrete morse-based approach to multivariate data analysis

Cited by 10 publications

References 25 publications

Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

Algorithmic Construction of Acyclic Partial Matchings for Multidimensional Persistence

Local bilinear computation of Jacobi sets

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