An Introduction to the Notion of Natural Pseudo-distance in Topological Data Analysis

The natural pseudo-distance dG associated with a group G of self-homeomorphisms of a topological space X is a pseudo-metric developed to compare real-valued functions defined on X, when the equivalence between functions is expressed by the group G. In this paper, we illustrate dG, its role in topological data analysis, its main properties and its link with persistent homology

On Some Geometric Aspects of the Class of hv-Convex Switching Components

In the usual aim of discrete tomography, the reconstruction of an unknown discrete set is considered, by means of projection data collected along a set U of discrete directions. Possible ambiguous reconstructions can arise if and only if switching components occur, namely, if and only if non-empty images exist having null projections along all the directions in U. In order to lower the number of allowed reconstructions, one tries to incorporate possible extra geometric constraints in the tomographic problem. In particular, the class P of horizontally and vertically convex connected sets (briefly, hv-convex polyominoes) has been largely considered. In this paper we introduce the class of hv-convex switching components, and prove some preliminary results on their geometric structure. The class includes all switching components arising when the tomographic problem is considered in P, which highly motivates the investigation of such configurations. It turns out that the considered class can be partitioned in two disjointed subclasses of closed patterns, called windows and curls, respectively. It follows that all windows have a unique representation, while curls consist of interlaced sequences of sub-patterns, called Z-paths, which leads to the problem of understanding the combinatorial structure of such sequences. We provide explicit constructions of families of curls associated to some special sequences, and also give additional details on further allowed or forbidden configurations by means of a number of illustrative examples

A Brief Introduction to Multidimensional Persistent Betti Numbers

In this paper, we propose a brief overview about multidimensional persistent Betti numbers (PBNs) and the metric that is usually used to compare them, i.e., the multidimensional matching distance. We recall the main definitions and results, mainly focusing on the 2-dimensional case. An algorithm to approximate n-dimensional PBNs with arbitrary precision is described

Characterization of hv-Convex Sequences

Reconstructing a discrete object by means of X-rays along a finite set U of (discrete) directions represents one of the main task in discrete tomography. Indeed, it is an ill-posed inverse problem, since different structures exist having the same projections along all lines whose directions range in U. Characteristic of ambiguous reconstructions are special configurations, called switching components, whose understanding represents a main issue in discrete tomography, and an independent interesting geometric problem as well. The investigation of switching component usually bases on some kind of prior knowledge that is incorporated in the tomographic problem. In this paper, we focus on switching components under the constraint of convexity along the horizontal and the vertical directions imposed to the unknown object. Moving from their geometric characterization in windows and curls, we provide a numerical description, by encoding them as special sequences of integers. A detailed study of these sequences leads to the complete understanding of their combinatorial structure, and to a polynomial-time algorithm that explicitly reconstructs any of them from a pair of integers arbitrarily given

Biomechanical analysis of pedalling for rehabilition purposes: experimental results on two pathological subjects and comparison with non-pathological findings

In this paper the experimental results obtained by means of a prototype measuring device dedicated to the evaluation of the rehabilitation level of the lower limb are presented. The analysis of the experimental data collected on non-pathological subjects allows the identification of the characteristic meaning of the most significant parameters typical of healthy subjects. These data have been employed for a systematic comparison with the same parameters measured on two pathological subjects, in order to define quantitative indicators of the rehabilitation degree of the lower limbs and indicators of the “quality” of the movement

Persistence modules, shape description, and completeness

Persistence modules are algebraic constructs that can be used to describe the shape of an object starting from a geometric representation of it. As shape descriptors, persistence modules are not complete, that is they may not distinguish non-equivalent shapes. In this paper we show that one reason for this is that homomorphisms between persistence modules forget the geometric nature of the problem. Therefore we introduce geometric homomorphisms between persistence modules, and show that in some cases they perform better. A combinatorial structure, the $H_0$-tree, is shown to be an invariant for geometric isomorphism classes in the case of persistence modules obtained through the 0th persistent homology functor

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.Comment: 11 pages, 4 figures, 2 table

ReLock: a resilient two-phase locking RESTful transaction model

Service composition and supporting transactions across composed services are among the major challenges characterizing service-oriented computing. REpresentational State Transfer (REST) is one of the approaches used for implementing Web services that is gaining momentum thanks to its features making it suitable for cloud computing and microservices-based contexts. This paper introduces ReLock, a resilient RESTful transaction model introducing general purpose transactions on RESTful services by a layered approach and a two-phase locking mechanism not requesting any change to the RESTful services involved in a transaction