318 research outputs found
Invariance properties of the multidimensional matching distance in Persistent Topology and Homology
Persistent Topology studies topological features of shapes by analyzing the
lower level sets of suitable functions, called filtering functions, and
encoding the arising information in a parameterized version of the Betti
numbers, i.e. the ranks of persistent homology groups. Initially introduced by
considering real-valued filtering functions, Persistent Topology has been
subsequently generalized to a multidimensional setting, i.e. to the case of
-valued filtering functions, leading to studying the ranks of
multidimensional homology groups. In particular, a multidimensional matching
distance has been defined, in order to compare these ranks. The definition of
the multidimensional matching distance is based on foliating the domain of the
ranks of multidimensional homology groups by a collection of half-planes, and
hence it formally depends on a subset of inducing a
parameterization of these half-planes. It happens that it is possible to choose
this subset in an infinite number of different ways. In this paper we show that
the multidimensional matching distance is actually invariant with respect to
such a choice.Comment: 14 pages, 2 figure
A closed formula for the number of convex permutominoes
In this paper we determine a closed formula for the number of convex
permutominoes of size n. We reach this goal by providing a recursive generation
of all convex permutominoes of size n+1 from the objects of size n, according
to the ECO method, and then translating this construction into a system of
functional equations satisfied by the generating function of convex
permutominoes. As a consequence we easily obtain also the enumeration of some
classes of convex polyominoes, including stack and directed convex
permutominoes
A new approximation Algorithm for the Matching Distance in Multidimensional Persistence
Topological Persistence has proven to be a promising framework for dealing with problems concerning shape analysis and comparison. In this contexts, it was originally introduced by taking into account 1-dimensional properties of shapes, modeled by real-valued functions. More recently, Topological Persistence has been generalized to consider multidimensional properties of shapes, coded by vector-valued functions. This extension has led to introduce suitable shape descriptors, named the multidimensional persistence Betti numbers functions, and a distance to compare them, the so-called multidimensional matching distance. In this paper we propose a new computational framework to deal with the multidimensional matching distance. We start by proving some new theoretical results, and then we use them to formulate an algorithm for computing such a distance up to an arbitrary threshold error
Necessary Conditions for Discontinuities of Multidimensional Size Functions
Some new results about multidimensional Topological Persistence are
presented, proving that the discontinuity points of a k-dimensional size
function are necessarily related to the pseudocritical or special values of the
associated measuring function.Comment: 23 pages, 4 figure
Multidimensional persistent homology is stable
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.Comment: 14 pages, 3 figure
On the geometrical properties of the coherent matching distance in 2D persistent homology
In this paper we study a new metric for comparing Betti numbers functions in
bidimensional persistent homology, based on coherent matchings, i.e. families
of matchings that vary in a continuous way. We prove some new results about
this metric, including its stability. In particular, we show that the
computation of this distance is strongly related to suitable filtering
functions associated with lines of slope 1, so underlining the key role of
these lines in the study of bidimensional persistence. In order to prove these
results, we introduce and study the concepts of extended Pareto grid for a
normal filtering function as well as of transport of a matching. As a
by-product, we obtain a theoretical framework for managing the phenomenon of
monodromy in 2D persistent homology.Comment: 39 pages, 15 figures. Corrected the definition of multiplicity of
points in the extended Pareto grid and the definition of normal function.
Removed Rem. 3.3. Added Ex. 3.9, Fig. 11, Fig. 12, Rem. 5.3 and Fig. 15.
Changed Rem. 4.9 into regular text. Reformulated statements of Theorems 5.1,
5.2, 5.4. Some changes in their proofs. Added references. Some small changes
in the text and in the figure
Keypics: free–hand drawn iconic keywords
We propose an iconic indexing of images to be exposed on the Web. This should be accomplished by “Keypics”, i.e. auxiliary, simplified pictures referring to the geometrical and/or the semantic content of the indexed image. Keypics should not be rigidly standardized; they should be left free to evolve, to express nuances and to stress details. A mathematical tool for dealing with such freedom, in the retrieval task, already exists: Size Functions. An experiment on 494 Keypics with Size Functions based on three measuring functions (distances, projections and jumps) and their combination is presented
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