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Detection of the Discrete Convexity of Polyominoes [chapter]

Isabelle Debled-Rennesson, Rémy Jean-Luc, Jocelyne Rouyer-Degli
2000 Lecture Notes in Computer Science  
The convexity of a discrete region is a property used in numerous domains of computational imagery. We study its detection in the particular case of polyominoes.  ...  Correlatively, we obtain a characterisation of lower and upper convex hulls of a discrete line segment. Finally, we evoke some applications of these results to the problem of discrete tomography.  ...  Acknowledgements We thank the members of the PolKA research group of LORIA for their review and constructive remarks.  ... 
doi:10.1007/3-540-44438-6_40 fatcat:eria52ncyzh27j74wjvgoafdxi

Detection of the discrete convexity of polyominoes

Isabelle Debled-Rennesson, Jean-Luc Rémy, Jocelyne Rouyer-Degli
2003 Discrete Applied Mathematics  
The convexity of a discrete region is a property used in numerous domains of computational imagery. We study its detection in the particular case of polyominoes.  ...  Correlatively, we obtain a characterisation of lower and upper frontiers of the convex hull of a discrete line segment.  ...  Acknowledgements We thank the members of the PolKA research group of LORIA for their review and constructive remarks.  ... 
doi:10.1016/s0166-218x(02)00227-5 fatcat:nuoo44apb5amzicyem4qt2mgei

Further steps on the reconstruction of convex polyominoes from orthogonal projections

Paolo Dulio, Andrea Frosini, Simone Rinaldi, Lama Tarsissi, Laurent Vuillon
2021 Journal of combinatorial optimization  
AbstractA remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean  ...  In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of  ...  A possible approach to the reconstruction of convex polyominoes consists in modifying the above algorithm as follows: Stage 1 is enriched with a further operation that produces the convex hull of the detected  ... 
doi:10.1007/s10878-021-00751-z fatcat:c4djpxeoyvc6xcvx23gsabk2a4

How to Decompose a Binary Matrix into Three hv-convex Polyominoes [chapter]

Andrea Frosini, Christophe Picouleau
2013 Lecture Notes in Computer Science  
Given a binary matrix, deciding wether it can be decomposed into three hv-convex matrices is an N P-complete problem, whereas its decomposition into two hv-convex matrices or two hv-polyominoes can be  ...  These problems are motivated by the Intensity Modulated Radiation Therapy (IMRT).  ...  of the theoretical discrete tomography.  ... 
doi:10.1007/978-3-642-37067-0_27 fatcat:pfuozoy5wfbulgc7ngnjkizq5i

First Steps in the Algorithmic Reconstruction of Digital Convex Sets [chapter]

Paolo Dulio, Andrea Frosini, Simone Rinaldi, Lama Tarsissi, Laurent Vuillon
2017 Lecture Notes in Computer Science  
Digital convex (DC) sets plays a prominent role in the framework of digital geometry providing a natural generalization to the concept of Euclidean convexity when we are dealing with polyominoes, i.e.,  ...  The intent of this paper is to provide some local properties that a boundary words has to fulfill in order to allow a single point modifications that preserves the convexity of the polyomino.  ...  Debled-Rennesson, Rémy and Rouyer-Degli have worked on the arithmetical properties of discrete segments in order to detect the convexity of polyominoes (see [8] ).  ... 
doi:10.1007/978-3-319-66396-8_16 fatcat:keo6ividf5fxdin65hgyq6dfei

Reconstructing hv-convex multi-coloured polyominoes

Adam Bains, Therese Biedl
2010 Theoretical Computer Science  
We focus on the case where there are multiple disjoint polyominoes (of different colours) that are hv-convex, i.e., any intersection with a horizontal or vertical line is contiguous.  ...  In this paper, we consider the problem of reconstructing polyominoes from information about the thickness in vertical and horizontal directions.  ...  Introduction The field of discrete tomography concerns reconstruction of objects given information about the thickness of the object in various projections.  ... 
doi:10.1016/j.tcs.2010.04.041 fatcat:lgm5xwakqnbcddw7xnnixfrb6y

Region Detection in Markov Random Fields: Gaussian Case [article]

Ilya Soloveychik, Vahid Tarokh
2018 arXiv   pre-print
When the number of samples is less than this amount, reliable detection of all edges is impossible.  ...  The benchmark information-theoretic results in the case of d-regular graphs require the number of samples to be at least proportional to the logarithm of the number of vertices to allow consistent graph  ...  APPENDIX C ENUMERATION OF CONVEX POLYOMINOES AND POLYGONS As mentioned in Section III-C, another possible way to discretize the plane consists in using convex lattice polygons with similar restrictions  ... 
arXiv:1802.03848v8 fatcat:v6i4yspchjbz7km5vioaj2k4wu

Interactions between Digital Geometry and Combinatorics on Words

Srečko Brlek
2011 Electronic Proceedings in Theoretical Computer Science  
Discrete figures are identified with polyominoes encoded by words. The combinatorial tools lead to elegant descriptions of geometrical features and efficient algorithms.  ...  Motivated on the one hand by the well-known theory of Sturmian words which model conveniently discrete lines in the plane, and on the other hand by the development of digital geometry, this study reveals  ...  detecting if a discrete region of the plane is convex requires a deeper analysis.  ... 
doi:10.4204/eptcs.63.1 fatcat:fpk5rmh7r5dvbp6st2jjxh4bju

On the shape of permutomino tiles

A. Blondin Massé, A. Frosini, S. Rinaldi, L. Vuillon
2013 Discrete Applied Mathematics  
In this paper we relate the two concepts by considering the pseudo-square polyominoes which are also convex permutominoes.  ...  On the other hand, pseudo-square polyominoes are a class of polyominoes tiling the the plane by translation.  ...  plane by translation; in [8] the authors present a linear time algorithm for detecting pseudo-square polyominoes.  ... 
doi:10.1016/j.dam.2012.08.034 fatcat:xuy7z2bu45abjpv7r56fvu277i

Lyndon Christoffel digitally convex

S. Brlek, J.-O. Lachaud, X. Provençal, C. Reutenauer
2009 Pattern Recognition  
The notion of convexity does not translate trivially, and detecting if a discrete region of the plane is convex requires a deeper analysis.  ...  In this paper we provide first a fast optimal algorithm checking digital convexity of polyominoes coded by their contour word.  ...  polyomino and appears as a discrete version of it.  ... 
doi:10.1016/j.patcog.2008.11.010 fatcat:wuqja7h5d5hjtloqsm5a3dpija

Page 3318 of Mathematical Reviews Vol. , Issue 2004d [page]

2004 Mathematical Reviews  
(F-INRIA3-LRI; Vandoeuvre-les-Nancy) ; Rouyer-Degli, Jocelyne (F-NANC-LRI; Vandoeuvre-les-Nancy) Detection of the discrete convexity of polyominoes.  ...  The algorithm first tries to verify the hv-convexity (cells of each row and column are consecutive) of the polyomino and then a procedure for discrete line segments recog- nition [cf. 1.  ... 

Large Deviations of Convex Polyominoes [article]

Ilya Soloveychik, Vahid Tarokh
2018 arXiv   pre-print
In this work, we develop a large deviation principle for convex polyominoes under different restrictions, such as fixed area and/or perimeter.  ...  Enumeration of various types of lattice polygons and in particular polyominoes is of primary importance in many machine learning, pattern recognition, and geometric analysis problems.  ...  Figure 1 shows an example of a convex polyomino on a square lattice and its circumscribed rectangle. In the discrete scenario we have the following analog of the isoperimetric inequality.  ... 
arXiv:1802.03849v5 fatcat:xkf2uit3jffoneud7xbasxeft4

An Optimal Algorithm for Detecting Pseudo-squares [chapter]

Srečko Brlek, Xavier Provençal
2006 Lecture Notes in Computer Science  
We consider the problem of determining if a given word, which encodes the boundary of a discrete figure, tiles the plane by translation.  ...  These words have been characterized by the Beauquier-Nivat condition, for which we provide a linear time algorithm in the case of pseudo-square polyominoes, improving the previous quadratic algorithm of  ...  The authors are grateful to the anonymous referees for their careful reading and valuable comments.  ... 
doi:10.1007/11907350_34 fatcat:gb4ep4xdfzgahkhn4mvffvo76i

Reconstruction of 8-connected but not 4-connected hv-convex discrete sets

Péter Balázs, Emese Balogh, Attila Kuba
2005 Discrete Applied Mathematics  
The reconstruction of 8-connected but not 4-connected hv-convex discrete sets from few projections is considered.  ...  Finally, we consider the possible generalizations of our results to solve the problem in more general classes.  ...  An algorithm for reconstructing hv-convex polyominoes was presented in [3, 4] . Then, the method was improved to reconstruct discrete sets of S 8 , too [7] .  ... 
doi:10.1016/j.dam.2004.09.009 fatcat:rwvajzlmbjhgvnyqion4v7qpwi

Page 584 of Mathematical Reviews Vol. , Issue 2003A [page]

2003 Mathematical Reviews  
The sets considered are hv-convex polyominoes and hv-convex 8-connected sets.  ...  (H-SZEG-C; Szeged) ; Kuba, Attila (H-SZEG-C; Szeged) ; Dévényi, Csaba (H-SZEG-C; Szeged) ; Dei Lungo, Alberto (I-SIN; Siena) Comparison of algorithms for reconstructing hv-convex discrete sets.  ... 
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