Barbara Di Fabio

One-dimensional reduction of multidimensional persistent homology

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.

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

In algebraic topology it is well known that, using the Mayer–Vietoris sequence, the homology of a space X can be studied by splitting X into subspaces A and B and computing the homology of A, B, and A∩B. A natural question is: To what extent does persistent homology benefit from a similar property? In this paper we show that persistent homology has a Mayer–Vietoris sequence that is generally not exact but only of order 2. 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 persistent homological features of A and B can be found either as persistent homological features of X or of 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.

Persistent homology and partial similarity of shapes

Persistent homology provides shapes descriptors called persistence diagrams. We use persistence diagrams to address the problem of shape comparison based on partial similarity. We show that two shapes having a common sub-part in general present a common persistence sub-diagram. Hence, the partial Hausdorff distance between persistence diagrams measures partial similarity between shapes. The approach is supported by experiments on 2D and 3D data sets.

Comparing persistence diagrams through complex vectors

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.

Recognition of Occluded Shapes Using Size Functions

The robustness against occlusions and the ability to perform not only global matching, but also partial matching are investigated in computer vision in order to evaluate the performance of shape descriptors. In this paper we consider the size function shape descriptor, and we illustrate some results about size functions of occluded shapes. Theoretical results indicate that size functions are able to detect a partial matching between shapes by showing a common subset of cornerpoints. Experiments are presented which outline the potential of the proposed approach in recognition tasks in the presence of occlusions.

The Edit Distance for Reeb Graphs of Surfaces

Reeb graphs are structural descriptors that capture shape properties of a topological space from the perspective of a chosen function. In this work, we define a combinatorial distance for Reeb graphs of orientable surfaces in terms of the cost necessary to transform one graph into another by edit operations. The main contributions of this paper are the stability property and the optimality of this edit distance. More precisely, the stability result states that changes in the Reeb graphs, measured by the edit distance, are as small as changes in the functions, measured by the maximum norm. The optimality result states that the edit distance discriminates Reeb graphs better than any other distance for Reeb graphs of surfaces satisfying the stability property.

Comparing shapes through multi-scale approximations of the matching distance

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.

Multi-scale approximation of the matching distance for shape retrieval

This paper deals with the concepts of persistence diagrams and matching distance. They are two of the main ingredients of Topological Persistence, which has proven to be a promising framework for shape comparison. Persistence diagrams are descriptors providing a signature of the shapes under study, while the matching distance is a metric to compare them. One drawback in the application of these tools is the computational costs for the evaluation of the matching distance. The aim of the present paper is to introduce a new framework for the approximation of the matching distance, which does not affect the reliability of the entire approach in comparing shapes, and extremely reduces computational costs. This is shown through experiments on 3D-models.

Towards a Topological Fingerprint of Music

Can music be represented as a meaningful geometric and topological object? In this paper, we propose a strategy to describe some music features as a polyhedral surface obtained by a simplicial interpretation of the \textit{Tonnetz}. The \textit{Tonnetz} is a graph largely used in computational musicology to describe the harmonic relationships of notes in equal tuning. In particular, we use persistent homology in order to describe the \textit{persistent} properties of music encoded in the aforementioned model. Both the relevance and the characteristics of this approach are discussed by analyzing some paradigmatic compositional styles. Eventually, the task of automatic music style classification is addressed by computing the hierarchical clustering of the topological fingerprints associated with some collections of compositions.

Stability of Reeb Graphs of Closed Curves

Reeb graphs are very popular shape descriptors in computational frameworks as they capture both geometrical properties of the shape, and its topological features. Some different methodologies have been proposed in the literature to estimate the similarity of shapes through the comparison of the associated Reeb graphs. In this context, one of the most important open questions is whether Reeb graphs are robust against function perturbations. In fact, it is clear that any data acquisition is subject to perturbations, noise and approximation errors and, if Reeb graphs were not stable, then distinct computational investigations of the same object could produce completely different results. In this paper we present an initial contribution to establishing stability properties for Reeb graphs. More precisely, focusing our attention on 1-dimensional manifolds, we define an editing distance between Reeb graphs, in terms of the cost necessary to transform one graph into another. Our main result is that changes in Morse functions imply smaller changes in the editing distance between Reeb graphs.