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Multidimensional Size Functions for Shape Comparison
TLDR
It is proved that in Size Theory the comparison of multidimensional size functions can be reduced to the 1-dimensional case by a suitable change of variables, and the definition of a new distance between multiddimensional size functions is defined, and to the proof of their stability with respect to that distance.
Betti numbers in multidimensional persistent homology are stable functions
Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector‐valued functions, called filtering functions. As is well known, in the case of
Natural Pseudo-Distance and Optimal Matching between Reduced Size Functions
TLDR
The matching distance is shown to be resistant to perturbations, implying that it is always smaller than the natural pseudo-distance, and it is proved that the lower bound so obtained is sharp and cannot be improved by any other distance between size functions.
The rank invariant stability via interleavings
  • C. Landi
  • Mathematics, Computer Science
    ArXiv
  • 10 December 2014
TLDR
A lower bound for the interleaving distance on persistence modules is given in terms of matching distance of rank invariants, and the internal stability of the rank invariant is proved in Terms of interleavings.
A Mayer–Vietoris Formula for Persistent Homology with an Application to Shape Recognition in the Presence of Occlusions
TLDR
It is shown that persistence diagrams are able to recognize an occluded shape by showing a common subset of points and a Mayer–Vietoris formula involving the ranks of the persistent homology groups of X, A, B, and A∩B plus three extra terms is obtained.
Describing shapes by geometrical-topological properties of real functions
TLDR
This survey is to provide a clear vision of what has been developed so far, focusing on methods that make use of theoretical frameworks that are developed for classes of real functions rather than for a single function, even if they are applied in a restricted manner.
Size theory as a topological tool for computer vision
TLDR
The usefulness of such a theory in comparing shapes is high lighted by showing some examples and the robustness of Size Theory with respect to noise and occlusions is pointed out.
Size Functions and Formal Series
  • P. Frosini, C. Landi
  • Mathematics
    Applicable Algebra in Engineering, Communication…
  • 1 August 2001
TLDR
It is proved that every size function can be represented as a set of points and lines in the real plane, with multiplicities, which allows for an algebraic approach to size functions and the construction of new pseudo-distances between size functions for comparing shapes.
The Edit Distance for Reeb Graphs of Surfaces
TLDR
A combinatorial distance for Reeb graphs of orientable surfaces in terms of the cost necessary to transform one graph into another by edit operations is defined in order to determine the stability property of these graphs.
Stability in multidimensional Size Theory
This paper proves that in Size Theory the comparison of multidimensional size functions can be reduced to the 1-dimensional case by a suitable change of variables. Indeed, we show that a foliation in
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