Reconstructing embedded graphs from persistence diagrams

Belton, Robin; Fasy, Brittany Terese; Mertz, Rostik; Micka, Samuel; Millman, David L.; Salinas, Daniel; Schenfisch, Anna; Schupbach, Jordan; Williams, Lucia

doi:10.1016/j.comgeo.2020.101658

Cited by 13 publications

(25 citation statements)

References 8 publications

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“…To understand this question Turner, Mukherjee and Boyer introduced the persistent homology transform (PHT) [33], and proved that indeed under certain circumstances PHT is injective on shapes embedded in R 3 , see also [14,19] for a generalisation to higher dimensions. It has even been possible to find algorithmically a left inverse for PHT for some specific classes of sets [4,5,18,24].…”

Section: Related Previous Workmentioning

confidence: 99%

The Fiber of Persistent Homology for simplicial complexes

Leygonie¹,

Tillmann²

2021

Preprint

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We study the inverse problem for persistent homology: For a fixed simplicial complex K, we analyse the fiber of the continuous map PH on the space of filters that assigns to a filter f : K Ñ R the total barcode of its associated sublevel set filtration of K. We find that PH is best understood as a map of stratified spaces. Over each stratum of the barcode space the map PH restricts to a (trivial) fiber bundle with fiber a polyhedral complex. Amongst other we derive a bound for the dimension of the fiber depending on the number of distinct endpoints in the barcode. Furthermore, taking the inverse image PH ´1 can be extended to a monodromy functor on the (entrance path) category of barcodes. We demonstrate our theory on the example of the simplicial triangle giving a complete description of all fibers and monodromy maps. This example is rich enough to have a Möbius band as one of its fibers.

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Section: Related Previous Workmentioning

confidence: 99%

The Fiber of Persistent Homology for simplicial complexes

Leygonie¹,

Tillmann²

2021

Preprint

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“…Moreover, this ECT is a sufficient statistic that effectively summarizes all information regarding shape. Although there are infinite possible directional filters, there is ongoing research into defining a sufficient finite number of directions such that we can effectively reconstruct shapes based solely on their finite ECT (Belton et al, 2020; Betthauser, 2018; Curry et al, 2018; Fasy et al, 2019). Nonetheless, a computationally efficient reconstruction procedure for large 3D images remains elusive.…”

Section: Introductionmentioning

confidence: 99%

Measuring hidden phenotype: Quantifying the shape of barley seeds using the Euler Characteristic Transform

Amézquita

Quigley

Ophelders

et al. 2021

Preprint

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Shape plays a fundamental role in biology. Traditional phenotypic analysis methods measure some features but fail to measure the information embedded in shape comprehensively. To extract, compare, and analyze this information embedded in a robust and concise way, we turn to Topological Data Analysis (TDA), specifically the Euler Characteristic Transform (ECT). TDA measures shape comprehensively using mathematical terms based on algebraic topology features. To study its use, we compute both traditional and topological shape descriptors to quantify the morphology of 3121 barley seeds scanned with X-ray Computed Tomography (CT) technology at 127 micron resolution. The ECT measures shape by analyzing topological features of an object at thresholds across a number of directional axes. We optimize the number of directions and thresholds for classification to 158 and 8 respectively, creating vectors of length 1264 that are topological signatures for each barley seed. Using these vectors, we successfully train a support vector machine to classify 28 different accessions of barley based on the 3D shape of their grains. We observe that combining both traditional and topological descriptors classifies barley seeds to their correct accession better than using just traditional descriptors alone. This improvement suggests that TDA is thus a powerful complement to traditional morphometrics to describe comprehensively a multitude of shape nuances which are otherwise not picked up. Using TDA we can quantify aspects of phenotype that have remained "hidden" without its use, and the ECT opens the possibility of accurately reconstructing objects from their topological signatures.

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“…The PH aims to capture topological evolution of data set by varying scale (parameter), and extracts topological invariants of data set in each scale summarizing them in different representations, persistence barcode (PB) [11,12], persistence diagram (PD) [13,14], persistence landscape (PL) [15], persistence image (PI) [16], persistence surface (PS) and β-curve, which reveal topological information of data set. Being robust to noise, PH is able to show us the essential features of the systems with high internal degrees of freedom and is capable to classify underlying data sets [17,18].…”

Section: Introductionmentioning

confidence: 99%

Persistent Homology of Weighted Visibility Graph from Fractional Gaussian Noise

Masoomy,

Askari,

Najafi

et al. 2021

Preprint

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In this paper, we utilize persistent homology technique to examine the topological properties of the visibility graph constructed from fractional Gaussian noise (fGn). We develop the weighted natural visibility graph algorithm and the standard network in addition to the global properties in the context of topology, will be examined. Our results demonstrate that the distribution of eigenvector and betweenness centralities behave as power-law decay. The scaling exponent of eigenvector centrality and the moment of eigenvalue distribution, Mn, for n ≥ 1 reveal the dependency on the Hurst exponent, H, containing the sample size effect. We also focus on persistent homology of kdimensional topological holes incorporating the filtration of simplicial complexes of associated graph. The dimension of homology group represented by Betti numbers demonstrates a strong dependency on the Hurst exponent. More precisely, the scaling exponent of the number of k-dimensional topological holes appearing and disappearing at a given threshold, depends on H which is almost not affected by finite sample size. We show that the distribution function of lifetime for k-dimensional topological holes decay exponentially and corresponding slope is an increasing function versus H and more interestingly, the sample size effect is completely disappeared in this quantity. The persistence entropy logarithmically grows with the size of visibility graph of system with almost H-dependent prefactors.

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Reconstructing embedded graphs from persistence diagrams

Cited by 13 publications

References 8 publications

The Fiber of Persistent Homology for simplicial complexes

The Fiber of Persistent Homology for simplicial complexes

Measuring hidden phenotype: Quantifying the shape of barley seeds using the Euler Characteristic Transform

Persistent Homology of Weighted Visibility Graph from Fractional Gaussian Noise

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