Inverse Problems in Topological Persistence

Oudot, Steve; Solomon, Elchanan

doi:10.1007/978-3-030-43408-3_16

Cited by 20 publications

(18 citation statements)

References 21 publications

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“…We use cones because vectors of Euler characteristics, taken along directions close together, express comparable information. That similiarity lets us leverage findings between them to increase our power of detecting truly associated shape vertices and regions-as opposed to antipodal directions where the lack of shared information may do harm when determining reconstructed manifolds (Supplementary Material, Section 1.4, ) (Curry, Mukherjee and Turner (2019), Fasy et al (2018), Oudot and Solomon (2018)).…”

Section: Statistical Model For Shape Classificationmentioning

confidence: 99%

A statistical pipeline for identifying physical features that differentiate classes of 3D shapes

et al. 2021

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The recent curation of large-scale databases with 3D surface scans of shapes has motivated the development of tools that better detect global patterns in morphological variation. Studies, which focus on identifying differences between shapes, have been limited to simple pairwise comparisons and rely on prespecified landmarks (that are often known). We present SINA-TRA, the first statistical pipeline for analyzing collections of shapes without requiring any correspondences. Our novel algorithm takes in two classes of shapes and highlights the physical features that best describe the variation between them. We use a rigorous simulation framework to assess our approach. Lastly, as a case study we use SINATRA to analyze mandibular molars from four different suborders of primates and demonstrate its ability recover known morphometric variation across phylogenies.

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Section: Statistical Model For Shape Classificationmentioning

confidence: 99%

A statistical pipeline for identifying physical features that differentiate classes of 3D shapes

et al. 2021

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“…This line of research goes back to the work of Schapira on when transforms of constructible functions are invertible [58]. We refer the interested reader to the original paper as well as [7,14,50] (as a non-comprehensive list).Here we present the basic form and refer the reader to [15] for more general and in-depth presentation.…”

Section: Simplicial Complexes 2 Cubical Complexes/imagesmentioning

confidence: 99%

Euler characteristic surfaces

Beltramo¹,

Škraba²,

Andreeva³

et al. 2022

FoDS

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<p style='text-indent:20px;'>In this paper, we investigate the use of the Euler characteristic for the topological data analysis, particularly over higher dimensional parameter spaces. The Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, primarily in the context of random fields. The goal of this paper, is to present the extension of using the Euler characteristic in higher dimensional parameter spaces. The topological data analysis of higher dimensional parameter spaces using stronger invariants such as homology, has and continues to be the subject of intense research. However, as important theoretical and computational obstacles remain, the use of the Euler characteristic represents an important intermediary step toward multi-parameter topological data analysis. We show the usefulness of the techniques using generated examples as well as a real world dataset of detecting diabetic retinopathy in retinal images.</p>

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“…where σ k are the singular values of F and v k are the associated vectors in R m from the singular value decomposition of matrix F in Equation (11). Then the parameterization…”

Section: The Choice Of Wavelet Basismentioning

confidence: 99%

“…The first two examples of persistence optimization are the computation of Fréchet mean of barcodes using gradients on Alexandrov spaces [9], and that of point cloud inference [10], where a point cloud is optimized so that its barcode fits a target fixed barcode. The latter is an instance of topological inverse problems (see Oudot and Solomon [11] for a recent overview of such). Another inverse problem is that of surface reconstruction [12].…”

Section: Introductionmentioning

confidence: 99%

Optimization of Spectral Wavelets for Persistence-Based Graph Classification

Yim

Leygonie

2021

Front. Appl. Math. Stat.

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A graph's spectral wavelet signature determines a filtration, and consequently an associated set of extended persistence diagrams. We propose a framework that optimizes the choice of wavelet for a dataset of graphs, such that their associated persistence diagrams capture features of the graphs that are best suited to a given data science problem. Since the spectral wavelet signature of a graph is derived from its Laplacian, our framework encodes geometric properties of graphs in their associated persistence diagrams and can be applied to graphs without a priori node attributes. We apply our framework to graph classification problems and obtain performances competitive with other persistence-based architectures. To provide the underlying theoretical foundations, we extend the differentiability result for ordinary persistent homology to extended persistent homology.

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Inverse Problems in Topological Persistence

Cited by 20 publications

References 21 publications

A statistical pipeline for identifying physical features that differentiate classes of 3D shapes

A statistical pipeline for identifying physical features that differentiate classes of 3D shapes

Euler characteristic surfaces

Optimization of Spectral Wavelets for Persistence-Based Graph Classification

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