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The number of convex polyominoes reconstructible from their orthogonal projections

Alberto Del Lungo, Maurice Nivat, Renzo Pinzani
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
ambiguity, thus making the set's reconstruction very difficult [l] , In this paper, we study the ambiguity problem with respect to a certain class of sets, called convex polyominoes, on which some connection  ...  (P%,Y) we indicate the class of convex polyominoes (directed convex and stack).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(96)83007-x">doi:10.1016/s0012-365x(96)83007-x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hqmyuzv27zd65gw6pe4uhuzwtm">fatcat:hqmyuzv27zd65gw6pe4uhuzwtm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190417184656/https://core.ac.uk/download/pdf/82777839.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e7/30/e7308bfa076b0c2e4f187341e98b025256cf8e75.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(96)83007-x"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Two subclasses of 2-convex polyominoes: properties for reconstruction

Tawbe K.
<span title="">2010</span> <i title="National Library of Serbia"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/smpfgyjbcnc4bpmdvwtg66ueyq" style="color: black;">Applicable Analysis and Discrete Mathematics</a> </i> &nbsp;
This paper studies the tomographical aspects of two subclasses of 2-convex polyominoes called η2L and η 2L .  ...  A polyomino P is called 2-convex if for every two cells there exists a monotone path included in P with at most 2 changes of direction.  ...  In [6] it is given an algorithm that reconstructs an L-convex polyomino from the set of its maximal L-polyominoes.  ... 
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L-Convex Polyominoes: Discrete Tomographical Aspects

Khalil Tawbe, Salwa Mansour
<span title="">2018</span> <i title="Scientific Research Publishing, Inc,"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/t3hxsybnxjg55o6z4c2pbf2sge" style="color: black;">Open Journal of Discrete Mathematics</a> </i> &nbsp;
This paper uses the geometrical properties of L-convex polyominoes in order to reconstruct these polyominoes.  ...  In [5] it is given an algorithm that reconstructs an L-convex polyomino from the set of its maximal L-polyominoes.  ...  Conflicts of Interest The authors declare no conflicts of interest regarding the publication of this paper.  ... 
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Polyominoes defined by their vertical and horizontal projections

Elena Barcucci, Alberto Del Lungo, Renzo Pinzani, Renzo Sprugnoli
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
In this paper, we prove that there are not two directed column-or row-convex polyominoes having the same vertical and horizontal projections (V,Z-Z).  ...  Therefore, every subclass of these polyominoes is defined by their orthogonal projections. We consider the enumeration of these subclasses according to the number of their columns and rows.  ...  (l-kt)=~~~(l-it)~~~(l-(j-l)t)(l-(j+l)t)...(l_~)' The number Vn,k of directed column-convex polyominoes having n columns and k rows is j=l i=l (k -j)! (j -i)! ' 4.2.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0304-3975(95)00205-7">doi:10.1016/0304-3975(95)00205-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wdaujxpdyfeudlwwz5ckygypi4">fatcat:wdaujxpdyfeudlwwz5ckygypi4</a> </span>
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A benchmark set for the reconstruction of hv-convex discrete sets

Péter Balázs
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
By using this benchmark set the paper also collects several statistics on hv-convex discrete sets, which are of great importance in the analysis of algorithms for reconstructing such kinds of discrete  ...  of hv-convex discrete sets and its subclasses.  ...  Preliminary version of the paper was presented at the 12th International Workshop on Combinatorial Image Analysis, Buffalo, NY, USA, April 2008 [1] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2009.02.019">doi:10.1016/j.dam.2009.02.019</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/455iya3j2vaozjnootgdxgl3qm">fatcat:455iya3j2vaozjnootgdxgl3qm</a> </span>
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The Number of Line-Convex Directed Polyominoes Having the Same Orthogonal Projections [chapter]

Péter Balázs
<span title="">2006</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
The number of line-convex directed polyominoes with given horizontal and vertical projections is studied.  ...  It is proven that diagonally convex directed polyominoes are uniquely determined by their orthogonal projections. The proof of this result is algorithmical.  ...  . , l} be a finite set of directions and P be a polyomino line-convex along every directions of D.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11907350_7">doi:10.1007/11907350_7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jibjzbz5snallh65sd4opkkej4">fatcat:jibjzbz5snallh65sd4opkkej4</a> </span>
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Reconstruction and Enumeration of hv-Convex Polyominoes with Given Horizontal Projection [chapter]

Norbert Hantos, Péter Balázs
<span title="">2013</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In this paper, we provide a linear-time algorithm for reconstructing hv-convex polyominoes with minimal number of columns satisfying a given horizontal projection.  ...  Enumeration and reconstruction of certain types of polyominoes, according to several parameters, are frequently studied problems in combinatorial image processing.  ...  In Section 3, we provide a linear-time algorithm for reconstructing hv-convex polyominoes from the horizontal projection.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-41822-8_13">doi:10.1007/978-3-642-41822-8_13</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kyxkcsp5sjcebhfu52nwvkxudi">fatcat:kyxkcsp5sjcebhfu52nwvkxudi</a> </span>
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L-Convex Polyominoes: Geometrical Aspects

Khalil Tawbe, S. Mansour
<span title="">2019</span> <i title="Scientific Research Publishing, Inc,"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/t35nyaf3lzb6lei7onyd4hnf4e" style="color: black;">Applied Mathematics</a> </i> &nbsp;
This paper is a theoretical step for the reconstruction of all L-convex polyominoes by using the geometrical paths.  ...  Finally, we study the subclasses of directed L-convex polyominoes and we give necessary and sufficient conditions for polyominoes to be L-convex.  ...  In [2] it is given an algorithm that reconstructs an L-convex polyomino from the set of its maximal L-polyominoes.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4236/am.2019.108046">doi:10.4236/am.2019.108046</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hl35w7htkjh35geoi46q3mmyf4">fatcat:hl35w7htkjh35geoi46q3mmyf4</a> </span>
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Reconstruction of 2-Convex Polyominoes with Non-Empty Corners

Khalil Tawbe, Salwa Mansour
<span title="">2019</span> <i title="Scientific Research Publishing, Inc,"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/t3hxsybnxjg55o6z4c2pbf2sge" style="color: black;">Open Journal of Discrete Mathematics</a> </i> &nbsp;
Some modifications are made in the reconstruction algorithm of Chrobak and Dürr for HV-convex polyominoes in order to impose these geometries.  ...  This paper uses the theoretical material developed in a previous study by the authors in order to reconstruct a subclass of 2-convex polyominoes called 1,1 2L α where the upper left corner and the lower  ...  Conflicts of Interest The authors declare no conflicts of interest regarding the publication of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4236/ojdm.2019.94009">doi:10.4236/ojdm.2019.94009</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jnncsdeprrfvfgjhq2ocugmaka">fatcat:jnncsdeprrfvfgjhq2ocugmaka</a> </span>
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Reconstruction in Different Classes of 2D Discrete Sets [chapter]

Attila Kuba
<span title="">1999</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
The reconstruction algorithms and complexity results are summarized in the case of hv-convex sets, hv-convex polyominoes, hv-convex 8-connected sets, and directed h-convex sets.  ...  We show that the reconstruction algorithms used in the class of hv-convex 4-connected sets (polyominoes) can be used, with small modifications, for reconstructing hv-convex 8connected sets.  ...  Acknowledgments This work was supported by FKFP 0908/1997 Grant of the Hungarian Ministry of Education.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-49126-0_12">doi:10.1007/3-540-49126-0_12</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ahjo6tqblfd33kjgyfrh6v3iai">fatcat:ahjo6tqblfd33kjgyfrh6v3iai</a> </span>
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Reconstructing digital sets from X-rays [chapter]

E. Barcucci, A. Del Lungol, M. Nivat, R. Pinzani, A. Zurli
<span title="">1997</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In order to give experimental evidence of this conjecture, we present a polynomial algorithm that reconstructs convex polyominoes from their X-rays.  ...  We make the conjecture that there is a certain set U of four directions that characterizes the class of convex polyominoes (sets which are convex with respect to only two directions: horizontal and vertical  ...  One of these classes is the set of column-convex polyominoes.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-63507-6_198">doi:10.1007/3-540-63507-6_198</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3uy2hunsxrhxfaiuui7nbcdhli">fatcat:3uy2hunsxrhxfaiuui7nbcdhli</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180726114040/https://link.springer.com/content/pdf/10.1007%2F3-540-63507-6_198.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/16/2e/162e88adea140609193d1c02b42acfbd0d2a0553.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-63507-6_198"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

A framework for generating some discrete sets with disjoint components by using uniform distributions

Péter Balázs
<span title="">2008</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
Especially, the uniform random generation of hv-convex discrete sets and Q-convex discrete sets according to the size of the minimal bounding rectangle are discussed.  ...  of the same reconstruction problem.  ...  Besides, for some subclasses of the hv-convex class (hv-convex polyominoes and hv-convex 8-connected sets) different polynomial-time reconstruction algorithms have been developed.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.tcs.2008.06.010">doi:10.1016/j.tcs.2008.06.010</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/t6nhxoxbbbawrmrmgy7uerxjhu">fatcat:t6nhxoxbbbawrmrmgy7uerxjhu</a> </span>
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Reconstruction of convex polyominoes from orthogonal projections of their contours

C. Picouleau
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
We prove the NP-hardness of this problem for several classes of polyominoes: general, h-convex, v-convex. For hv-convex polyominoes we give a polynomial time algorithm for the reconstruction problem.  ...  The problem of reconstructing a convex polyominoes from its horizontal and vertical projections when the projections are defined as the number of cells of the polyomino in the different lines and columns  ...  Here, we will study the reconstruction of convex polyominoes when the orthogonal projections represent the length of their contour lines.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.tcs.2005.08.023">doi:10.1016/j.tcs.2005.08.023</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/stpodkwdk5bq7htqjbuxm4asta">fatcat:stpodkwdk5bq7htqjbuxm4asta</a> </span>
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Reconstructing hv-Convex Polyominoes from Orthogonal Projections [article]

Christoph Durr, Marek Chrobak
<span title="1999-06-22">1999</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In addition, the problem of reconstructing a special class of centered hv-convex polyominoes is addressed.  ...  A simple, polynomial algorithm for reconstructing hv-convex polyominoes is provided, which is several orders of magnitudes faster than the best previously known algorithm from Barcucci et al.  ...  F k,l (r, c) is satisfiable iff there is an hv-convex polyomino realization T of (r, c) that is anchored at (k, l).  ... 
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A reconstruction algorithm for L-convex polyominoes

Giusi Castiglione, Antonio Restivo, Roberto Vaglica
<span title="">2006</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
We give an algorithm that uniquely reconstruct an L-convex polyomino from the size of some special paths, called bordered L-paths.  ...  Acknowledgments We would like to thank the anonymous reviewers for the detailed referee reports and for the constructive suggestions which helped us to improve the presentation of the paper.  ...  Then, we prove that the pair of sets (SE, EN ) uniquely determines an L-convex polyomino by describing an algorithm of reconstruction.  ... 
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