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.
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- Pascal (Inist)
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- topic : Verre.
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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.
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Rinaldi</name><affiliation wicri:level="1"><inist:fA14 i1="01"><s1>Università di Siena, Dipartimento di Matematica e Informatica, Pian dei Mantellini 44</s1><s2>Siena</s2><s3>ITA</s3><sZ>1 aut.</sZ><sZ>3 aut.</sZ></inist:fA14><country>Italie</country><wicri:noRegion>Siena</wicri:noRegion></affiliation></author></analytic><series><title level="j" type="main">Computer vision and image understanding : (Print)</title><title level="j" type="abbreviated">Comput. vis. image underst. : (Print)</title><idno type="ISSN">1077-3142</idno><imprint><date when="2013">2013</date></imprint></series></biblStruct></sourceDesc><seriesStmt><title level="j" type="main">Computer vision and image understanding : (Print)</title><title level="j" type="abbreviated">Comput. vis. image underst. : (Print)</title><idno type="ISSN">1077-3142</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Computational geometry</term><term>Computer vision</term><term>Discrete geometry</term><term>Glass</term><term>Integer lattice</term><term>Integer matrix</term><term>Small dimension</term><term>Tiling</term><term>Tomography</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>Vision ordinateur</term><term>Géométrie algorithmique</term><term>Tomographie</term><term>Verre</term><term>Géométrie discrète</term><term>Petite dimension</term><term>Réseau arithmétique</term><term>Matrice nombre entier</term><term>Pavage</term><term>.</term></keywords><keywords scheme="Wicri" type="topic" xml:lang="fr"><term>Verre</term></keywords></textClass></profileDesc></teiHeader><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></front></TEI><affiliations><list><country><li>Italie</li></country></list><tree><country name="Italie"><noRegion><name sortKey="Battaglino, D" sort="Battaglino, D" uniqKey="Battaglino D" first="D." last="Battaglino">D. Battaglino</name></noRegion><name sortKey="Frosini, A" sort="Frosini, A" uniqKey="Frosini A" first="A." last="Frosini">A. Frosini</name><name sortKey="Rinaldi, S" sort="Rinaldi, S" uniqKey="Rinaldi S" first="S." last="Rinaldi">S. Rinaldi</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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