Barbara Fabio

Università di Bologna, Matematica, Post-Doc

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Claudia Landi

Christoph Schweigert

Jie Gao

SUNY: Stony Brook University

Bauhaus-University-Weimar

Dr. J. M. Ashfaque (MInstP)

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Andrew Ranicki

University of Edinburgh

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Papers by Barbara Fabio

Shape from Functions:Enhancing Geometrical-Topological Descriptors

Parole chiave: Shape comparison, size function, natural pseudo-distance, persistent homology modu... more

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The Edit Distance for Reeb Graphs of Surfaces

Discrete & Computational Geometry, 2016

Reeb graphs of curves are stable under function perturbations

by Barbara Fabio and Claudia Landi

Persistent Homology and Partial Similarity of Shapes

The natural pseudo-distance as a quotient pseudo-metric, and applications

by Barbara Fabio and Claudia Landi

The natural pseudo-distance is a similarity measure conceived for the purpose of comparing shapes... more The natural pseudo-distance is a similarity measure conceived for the purpose of comparing shapes. In this paper we revisit this pseudo-distance from the point of view of quotients. In particular, we show that the natural pseudo-distance coincides with the quotient pseudo-metric on the space of continuous functions on a compact manifold, endowed with the uniform convergence metric, modulo self-homeomorphisms of the manifold. As applications of this result, the natural pseudo-distance is shown to be actually a metric on a number of function subspaces such as the space of topological embeddings, of isometries, and of simple Morse functions on surfaces.

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Multi-scale approaches for the matching distance estimation

by Barbara Fabio and Grzegorz Jabłoński

Comparing Persistence Diagrams Through Complex Vectors

Lecture Notes in Computer Science, 2015

The natural pseudo-distance of spaces endowed with filtering functions is precious for shape clas... more The natural pseudo-distance of spaces endowed with filtering functions is precious for shape classification and retrieval; its optimal estimate coming from persistence diagrams is the bottleneck distance, which unfortunately suffers from combinatorial explosion. A possible algebraic representation of persistence diagrams is offered by complex polynomials; since far polynomials represent far persistence diagrams, a fast comparison of the coefficient vectors can reduce the size of the database to be classified by the bottleneck distance. This article explores experimentally three transformations from diagrams to polynomials and three distances between the complex vectors of coefficients.

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Stable Shape Comparison of Surfaces via Reeb Graphs

by Barbara Fabio and Claudia Landi

Lecture Notes in Computer Science, 2014

Computational Topology in Image Context

Pattern Recognition Letters, 2012

Livre: Computational topology in image context FERRI Massimo, FROSINI Patrizio, LANDI Claudia, CE... more

Persistent homology and partial similarity of shapes

by Barbara Fabio and Claudia Landi

Pattern Recognition Letters, 2012

A Mayer–Vietoris Formula for Persistent Homology with an Application to Shape Recognition in the Presence of Occlusions

Foundations of Computational Mathematics, 2011

In algebraic topology it is well-known that, using the Mayer-Vietoris sequence, the homology of a... more In algebraic topology it is well-known that, using the Mayer-Vietoris sequence, the homology of a space X can be studied splitting X into subspaces A and B and computing the homology of A, B, A ∩ B. A natural question is to which an extent persistent homology benefits of a similar property. In this paper we show that persistent homology has a Mayer-Vietoris sequence that in general is not exact but only of order two. However, we obtain a Mayer-Vietoris formula involving the ranks of the persistent homology groups of X, A, B and A ∩ B plus three extra terms. This implies that topological features of A and B either survive as topological features of X or are hidden in A ∩ B. As an application of this result, we show that persistence diagrams are able to recognize an occluded shape by showing a common subset of points.

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The natural pseudo-distance as a quotient pseudo-metric, and applications

by Barbara Fabio and Claudia Landi

Forum Mathematicum, 2000

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Stability of Reeb Graphs of Closed Curves

by Barbara Fabio and Claudia Landi

Electronic Notes in Theoretical Computer Science, 2012

Calcolo e visualizzazione delle funzioni di taglia di curve spline

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Filtrations induced by continuous functions

Topology and its Applications, 2013

In Persistent Homology and Topology, filtrations are usually given by introducing an ordered coll... more In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to R n . A natural question arises, whether these approaches are equivalent or not. In this paper we study this problem and prove that, while the answer to the previous question is negative in the general case, the approach by continuous functions is not restrictive with respect to the other, provided that some natural stability and completeness assumptions are made. In particular, we show that every compact and stable 1-dimensional filtration of a compact metric space is induced by a continuous function. Moreover, we extend the previous result to the case of multi-dimensional filtrations, requiring that our filtration is also complete. Three examples show that we cannot drop the assumptions about stability and completeness.

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Multi-scale approximation of the matching distance for shape retrieval

by Filippo Medri and Barbara Fabio

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2012

This paper deals with the concepts of persistence diagrams and matching distance. They are two of... more This paper deals with the concepts of persistence diagrams and matching distance. They are two of the main ingredients of Topo- logical Persistence, which has proven to be a promising framework for shape comparison. Persistence diagrams are descriptors providing a sig- nature of the shapes under study, while the matching distance is a metric to compare them. One drawback in

Comparing shapes through multi-scale approximations of the matching distance

by Filippo Medri, Grzegorz Jabłoński, and Barbara Fabio

Computer Vision and Image Understanding, 2014

Two of the main ingredients of topological persistence for shape comparison are persistence diagr... more Two of the main ingredients of topological persistence for shape comparison are persistence diagrams and the matching distance. Persistence diagrams are signatures capturing meaningful properties of shapes, while the matching distance can be used to stably compare them. From the application viewpoint, one drawback of these tools is the computational cost for evaluating the matching distance. In this paper we introduce a new framework for the matching distance estimation: It preserves the reliability of the entire approach in comparing shapes, extremely reducing the computational cost. Theoretical results are supported by experiments on 3D-models.

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One-dimensional reduction of multidimensional persistent homology

Proceedings of The American Mathematical Society, 2010

A recent result on size functions is extended to higher homology modules: the persistent homology... more A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a stable distance for multidimensional persistent homology. Some reflections on i-essentiality of homological critical values conclude the paper.

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Optimal homeomorphisms between closed curves

The concept of natural pseudo-distance has proven to be a powerful tool for measuring the dissimi... more The concept of natural pseudo-distance has proven to be a powerful tool for measuring the dissimilarity between topological spaces endowed with continuous real-valued functions. Roughly speaking, the natural pseudo-distance is defined as the infimum of the change of the functions' values, when moving from one space to the other through homeomorphisms, if possible. In this paper, we prove the first available result about the existence of optimal homeomorphisms between closed curves, i.e. inducing a change of the function that equals the natural pseudo-distance.

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Multidimensional persistent homology is stable

Multidimensional persistence studies topological features of shapes by analyzing the lower level ... more Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants such that small changes of the function imply only small changes of the rank invariant. This result can be obtained by assuming the function to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence.

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Shape from Functions:Enhancing Geometrical-Topological Descriptors

Parole chiave: Shape comparison, size function, natural pseudo-distance, persistent homology modu... more

Download

The Edit Distance for Reeb Graphs of Surfaces

Discrete & Computational Geometry, 2016

Reeb graphs of curves are stable under function perturbations

by Barbara Fabio and Claudia Landi

Persistent Homology and Partial Similarity of Shapes

The natural pseudo-distance as a quotient pseudo-metric, and applications

by Barbara Fabio and Claudia Landi

Download