1,721,027 research outputs found
Uniqueness of models in persistent homology: the case of curves
No full textWe consider generic curves in R^2, i.e. generic C^1 functions f:S^1->R^2. We analyze these curves through the persistent homology groups of a filtration induced on S^1 by f. In particular, we consider the question whether these persistent homology groups uniquely characterize f, at least up to re-parameterizations of S^1. We give a partially positive answer to this question. More precisely, we prove that f=goh, where h:S^1->S^1$ is a C^1-diffeomorphism, if and only if the persistent homology groups of sof and sog coincide, for every s belonging to the group Sigma_2 generated by reflections in the coordinate axes. Moreover, for a smaller set of generic functions, we show that f and g are close to each other in the max-norm (up to re-parameterizations) if and only if, for every s in Sigma_2, the persistent Betti numbers functions of sof and sog are close to each other, with respect to a suitable distance
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
New pseudo-distances for the size function space
No full textA method to construct new pseudo-distances for the size function space based on the formal series representation of size functions is introduced. These new pseudo-distances allow to measure quantitatively the differences in shapes by comparing size functions. Some experiments on digital images are shown
Size theory as a topological tool for computer vision
No full textIn this paper we give an outline of Size Theory and its mainresults. The usefulness of such a theory in comparing shapes is high-lighted by showing some examples. The robustness of Size Theory withrespect to noise and occlusions is pointed out. In addition, an algebraicapproach to the theory is presented
Intrinsic harmonicity of Morse functions
No full textConsider a real valued Morse function f on a C-2 closed connected n-dimensional manifold M. It is proved that a suitable Riemannian metric exists on M, such that f is harmonic outside the set of critical points off of index 0 and n. The proof is based on a result of Calabi [1], providing a criterion for a closed one-form on a closed connected manifold to be harmonic with respect to some Riemannian metric
Size functions and morphological transformations
No full textIn this paper changes of size functions under morphological transformations are studied. Some inequalities concerning size functions of a subset of the Euclidean plane, its dilation by an open disk and its skeleton are proved. Such inequalities prove the stability of size functions with respect to these morphological transformations and give a new approach for computation in Size Theory
Size functions and formal series
No full textIn this paper we consider a mathematical tool for shape description called size function. We prove that every size function can be represented as a set of points and lines in the real plane, with multiplicities. This allows for an algebraic approach to size functions and the construction of new pseudo-distances between size functions for comparing shapes
Size functions as complete invariants for image recognition
No full textThe problem of completeness for invariant size functions isstudied. Families of size functions are introduced, which allowrecognition of some classes of plane curves up to transformations of increasing generality
Etudes hydrologiques italiennes. Le régime du Tibre (d'après les travaux du Professeur P. Frosini)
No full textPardé Maurice. Etudes hydrologiques italiennes. Le régime du Tibre (d'après les travaux du Professeur P. Frosini). In: Revue de géographie alpine, tome 21, n°2, 1933. pp. 289-335
Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval
No full textIn content-based image retrieval a major problem is the presence of noisy shapes. Noise
can present itself not only in the form of continuous deformations, but also as topological
changes. It is well known that persistent Betti numbers are a shape descriptor that admits
dissimilarity distances stable under continuous shape deformations. In this paper we focus
on the problem of dealing with noise that alters the topology of the studied objects. We
present a general method to turn persistent Betti numbers into stable descriptors also in
the presence of topological changes. Retrieval tests on the Kimia-99 database show the
effectiveness of the method
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