A decomposition theorem for homogeneous sets with respect to diamond probes
Identifieur interne : 001732 ( Main/Exploration ); précédent : 001731; suivant : 001733A decomposition theorem for homogeneous sets with respect to diamond probes
Auteurs : D. Battaglino [Italie] ; A. Frosini [Italie] ; S. Rinaldi [Italie]Source :
- Computer vision and image understanding : (Print) [ 1077-3142 ] ; 2013.
Descripteurs français
- Pascal (Inist)
- Wicri :
- topic : Verre.
English descriptors
- KwdEn :
Abstract
An unknown planar discrete set of points A can be inspected by means of a probe P of generic shape that moves around it, and reveals, for each position, the number of its elements as a magnifying glass. All the data collected during this process can be naturally arranged in an integer matrix that we call the scan of the starting set A w.r.t. the probe P. In [10], Nivat conjectured that a discrete set whose scan w.r.t. an exact probe is k-homogeneous, shows a strong periodical behavior, and it can be decomposed into smaller 1-homogeneous subsets. In this paper, we prove this conjecture to be true when the probe is a diamond, and then we extend this result to exact polyominoes that can regarded as balls in a generalized L, norm of 2. Then we provide experimental evidence that the conjecture holds for each exact polyomino of small dimension, using the mathematical software Sage [13]. Finally, we give some hints to solve the related reconstruction problem.
Affiliations:
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Le document en format XML
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<front><div type="abstract" xml:lang="en">An unknown planar discrete set of points A can be inspected by means of a probe P of generic shape that moves around it, and reveals, for each position, the number of its elements as a magnifying glass. All the data collected during this process can be naturally arranged in an integer matrix that we call the scan of the starting set A w.r.t. the probe P. In [10], Nivat conjectured that a discrete set whose scan w.r.t. an exact probe is k-homogeneous, shows a strong periodical behavior, and it can be decomposed into smaller 1-homogeneous subsets. In this paper, we prove this conjecture to be true when the probe is a diamond, and then we extend this result to exact polyominoes that can regarded as balls in a generalized L, norm of <sup>2</sup>
. Then we provide experimental evidence that the conjecture holds for each exact polyomino of small dimension, using the mathematical software Sage [13]. Finally, we give some hints to solve the related reconstruction problem.</div>
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