1,721,067 research outputs found
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
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
Range size functions
No full textA 2-parameter family of size functions is introduced, which allows the recognition of planar finite sets up to congruence. Some experiments on digital images are shown
Lower bounds for natural pseudodistances via size functions
No full textLet us consider two closed homeomorphic manifolds
, and two functions
,
, called measuring
functions. The natural pseudodistance between the pairs
, is defined as
the infimum of , as varies in the set of
all homeomorphisms from onto . In this
paper we show that size functions allow us to get a lower bound for
. Furthermore, we prove that this lower bound can be assumed
equal either to or to , where
, are two suitable critical values of the measuring
functions
Range size functions
No full textA 2-parameter family of size functions is introduced, which allows the recognition of planar finite sets up to congruence. Some experiments on digital images are shown
Connections between size functions and morphological transformations
No full textInequalities involving size functions of a subset of the Euclidean plane, its dilation and its skeleton are given, which lead to new techniques of computation of size functions. Some experiments on digital images are shown
Connections between size functions and morphological transformations
No full textInequalities involving size functions of a subset of the Euclidean plane, its dilation and its skeleton are given, which lead to new techniques of computation of size functions. Some experiments on digital images are shown
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