PHAT – Persistent Homology Algorithms Toolbox

Bauer, Ulrich; Kerber, Michael; Reininghaus, Jan; Wagner, Hubert

doi:10.1007/978-3-662-44199-2_24

Cited by 94 publications

(144 citation statements)

References 8 publications

Supporting

Mentioning

142

Contrasting

Order By: Relevance

“…We compare the timings with state-of-the-art software computing persistent homology and cohomology. Specifically, we compare our implementation with the Dionysus library [17] which provides implementation for persistent homology [13,19] and persistent cohomology [10] (denoted DioCoH) with field coefficients in Z p , for any prime p. We also compare our implementation with the PHAT library (version 1.4) [1,3] which provides an implementation of the optimized algorithm for persistent homology [2,6] (using the --twist option) as well as an implementation of persistent cohomology [2,9] (using the --dualize option), with coefficients in Z 2 only. DioCoH and PHAT have been reported to be the most efficient implementation in practice [2,9].…”

Section: Methodsmentioning

confidence: 99%

“…They are constructed up to the intrinsic dimension of the space with intrinsic metric otherwise. We use a variety of both real and synthetic datasets: Cy8 is a set of points in R 24 , sampled from the space of conformations of the cyclo-octane molecule, which is the union of two intersecting surfaces; S4 is a set of points sampled from the unit 4-sphere in R 5 ; L57 and L35 are sets of points in the lens spaces L(5, 7) and L (3,5) respectively, which are non-embedded spaces; Bro is a set of 5 × 5 high-contrast patches derived from natural images, interpreted as vectors in R 25 , from the Brown database; Kl is a set of points sampled from the surface of the figure eight Klein Bottle embedded in R 5 ; Bud is a set of points sampled from the surface of the Happy Buddha (http:// graphics.stanford.edu/data/3Dscanrep/) in R 3 ; and Nep is a set of points sampled from the surface of the Neptune statue (http://shapes.aimatshape.net/). Datasets are listed in Fig.…”

Section: Methodsmentioning

confidence: 99%

See 1 more Smart Citation

The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

Boissonnat

Dey²,

Maria³

2015

Algorithmica

View full text Add to dashboard Cite

International audiencePersistent homology with coefficients in a field coincides with the samefor cohomology because of duality. We propose an implementation of a recently intro-duced algorithm for persistent cohomology that attaches annotation vectors with thesimplices. We separate the representation of the simplicial complex from the represen-tation of the cohomology groups, and introduce a new data structure for maintainingthe annotation matrix, which is more compact and reduces substantially the amountof matrix operations. In addition, we propose a heuristic to simplify further the repre-sentation of the cohomology groups and improve both time and space complexities.The paper provides a theoretical analysis, as well as a detailed experimental studyof our implementation and comparison with state-of-the-art software for persistenthomology and cohomology

show abstract

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

Boissonnat

Dey²,

Maria³

2015

Algorithmica

View full text Add to dashboard Cite

show abstract

“…. , n simultaneously, and the algorithms of persistence [4] furthermore provide the most efficient known implementation of determining these bases even when i is fixed. To compute these bases, we first create an arbitrary simplex-wise filtration of L = K 1 and augment this filtration by adding each of the simplices of K n such that simplices of K i are inserted before K j and such that the faces of any simplex σ are inserted before σ.…”

Section: A Discretization and Setupmentioning

confidence: 99%

“…Given such a simplex-wise filtration, several algorithms (see, e.g. [4]) are available to compute a basis of the persistent homology groups. We shall use the library [4] and the left-to right reduction algorithm described in [10] for this purpose.…”

Section: Persistent Relative Homologymentioning

confidence: 99%

Topological trajectory clustering with relative persistent homology

Pokorny

Goldberg

Kragić

2016

2016 IEEE International Conference on Robotics and Automation (ICRA)

View full text Add to dashboard Cite

Cloud Robotics techniques based on Learning from Demonstrations suggest promising alternatives to manual programming of robots and autonomous vehicles. One challenge is that demonstrated trajectories may vary dramatically: it can be very difficult, if not impossible, for a system to learn control policies unless the trajectories are clustered into meaningful consistent subsets. Metric clustering methods, based on a distance measure, require quadratic time to compute a pairwise distance matrix and do not naturally distinguish topologically distinct trajectories. This paper presents an algorithm for topological clustering based on relative persistent homology, which, for a fixed underlying simplicial representation and discretization of trajectories, requires only linear time in the number of trajectories. The algorithm incorporates global constraints formalized in terms of the topology of sublevel or superlevel sets of a function and can be extended to incorporate probabilistic motion models. In experiments with real automobile and ship GPS trajectories as well as pedestrian trajectories extracted from video, the algorithm clusters trajectories into meaningful consistent subsets and, as we show in an experiment with ship trajectories, results in a faster and more efficient clustering than a metric clustering by Fréchet distance.

show abstract

“…In particular the one-dimensional Betti profile presents the number of loops [β 1 (δ)] that we have as δ varies from 0 to δ max . A series of free packages are available for computing persistent homology [50][51][52]. Here, we use the R package TDA using "DIONYSUS" to perform persistent homology calculations [53].…”

Section: B Persistent Homologymentioning

confidence: 99%

Quantifying disorder in colloidal films spin-coated onto patterned substrates

et al. 2017

View full text Add to dashboard Cite

Polycrystals of thin colloidal deposits, with thickness controlled by spin-coating speed, exhibit axial symmetry with local 4-fold and 6-fold symmetric structures, termed orientationally correlated polycrystals (OCPs). While spin-coating is a very facile technique for producing large-area colloidal deposits, the axial symmetry prevents us from achieving true long-range order. To obtain true long-range order, we break this axial symmetry by introducing a patterned surface topography and thus eliminate the OCP character. We then examine symmetryindependent methods to quantify order in these disordered colloidal deposits. We find that all the information in the bond-orientational order parameters is well captured by persistent homology analysis methods that only use the centers of the particles as input data. It is expected that these methods will prove useful in characterizing other disordered structures.

show abstract

PHAT – Persistent Homology Algorithms Toolbox

Cited by 94 publications

References 8 publications

The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

Topological trajectory clustering with relative persistent homology

Quantifying disorder in colloidal films spin-coated onto patterned substrates

Contact Info

Product

Resources

About