221 research outputs found
G-invariant Persistent Homology
No full textClassical persistent homology is not tailored to study the action of transformation groups different from the group Homeo(X) of all self-homeomorphisms of a topological space X. In order to obtain better lower bounds for the natural pseudo-distance d_G associated with a subgroup G of Homeo(X), we need to adapt persistent homology and consider G-invariant
persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of chains that is invariant under the action of G. In this paper we formalize this idea, and prove the stability of G-invariant persistent homology with respect to the natural pseudo-distance d_G. We also show how G-invariant persistent homology could be used in applications concerning shape comparison
Stable comparison of multidimensional persistent homology groups with torsion
No full textThe present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance that represents a possible solution to this problem. Indeed, is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the new pseudo-distance with respect to the change of the filtering function, expressed both with respect to the max-norm
and to the natural pseudo-distance between topological spaces endowed with -valued filtering functions. Furthermore, we prove a result showing the relationship between and the matching distance in the 1-dimensional case, when the homology coefficients are taken in a field and hence the comparison can be made
A note on the linearity of real-valued functions with respect to suitable metrics
No full textIn this paper we prove that for every real-valued Morse function
on a smooth closed manifold and every
neighborhood of its critical points a suitable Riemannian metric
exists such that is linear outside
Measuring shapes by size functions
No full textWe define the concept of size functions. They are functions from the real plane to the natural numbers which describe the `shape of the objects' (seen as submanifolds of a Euclidean space). We give two different techniques of computation of size functions and some actual examples of computation. Moreover, we present the concept of deformation distance between manifolds (i.e., curves, surfaces, etc.). It is a distance which measures the `difference in shape' of two manifolds. Finally we point out the link between deformation distances and size functions
Measuring shapes by size functions
No full textWe define the concept of size functions. They are functions from the real plane to the natural numbers which describe the `shape of the objects' (seen as submanifolds of a Euclidean space). We give two different techniques of computation of size functions and some actual examples of computation. Moreover, we present the concept of deformation distance between manifolds (i.e., curves, surfaces, etc.). It is a distance which measures the `difference in shape' of two manifolds. Finally we point out the link between deformation distances and size functions
Does intelligence imply contradiction
No full textContradiction is often seen as a defect of intelligent systems and a dangerous limi-tation on efficiency. In this paper we raise the question of whether, on the contrary, it could be considered a key tool in increasing intelligence in biological structures. A possible way of answering this question in a mathematical context is shown, for-mulating a proposition that suggests a link between intelligence and contradiction. A concrete approach is presented in the well-defined setting of cellular automata. Here we define the models of “observer”, “entity”, “environment”, “intelligence” and “contradiction”. These definitions, which roughly correspond to the common meaning of these words, allow us to deduce a simple but strong result about these concepts in an unbiased, mathematical manner. Evidence for a real-world counterpart to the demonstrated formal link between intelligence and contradiction is provided by three computational experiments. The structure of this pape
No embedding of the automorphisms of a topological space into a compact metric space endows them with a composition that passes to the limit
No full textThe Hausdorff distance, the Gromov-Hausdorff, the Fréchet and the natural pseudo-distances are instances of dissimilarity measures widely used in
shape comparison. We show that they share the property of being defined as where is a suitable functional and varies in a set of correspondences containing the set of homeomorphisms. Our main result states that the set of homeomorphisms cannot be enlarged to a metric space , in such a way that the composition in (extending the composition of homeomorphisms) passes to the limit and, at the same time, is compact
Corrigendum to “Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval”
No full textThis paper is a corrigendum to the paper “Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval”, Pattern Recognition Letters, vol. 34 (2013), 863-872. DOI: 10.1016/j.patrec.2012.10.015. Table 1 and Figure 3 have been corrected
Towards an observer-oriented theory of shape comparison
No full textIn this position paper we suggest a possible metric approach to shape comparison that is based on a mathematical formalization of the concept of observer, seen as a collection of suitable operators acting on a metric space of functions. These functions represent the set of data that are accessible to the observer, while the operators describe the way the observer elaborates the data and enclose the invariance that he/she associates with them. We expose this model and illustrate some theoretical reasons that justify its possible use for shape comparison
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