The Euclidean plane can be partitioned into three kinds of

regular polygons, which results in triangular, square and hexagonal grids.

While the topology of the square grid is well-established, less emphasis

is put on the remaining two regular sampling schemes. In this paper we

summarize the results of our research that aimed to give some general

characterizations of simple pixels and sufficient conditions for topology-

preserving operators in the mentioned grids.

Thinning is a frequently applied technique for extracting centerlines from 2D binary objects. Parallel thinning algorithms can remove a set of object points simultaneously, while sequential algorithms traverse the boundary of objects, and consider the actually visited single point for possible removal. Two thinning algorithms are called equivalent if they produce the same result for each input picture. This paper presents the very first pair of equivalent 2D sequential and parallel subiteration-based thinning algorithms. These algorithms can be implemented directly on a conventional sequential computer or on a parallel computing device. Both of them preserve topology for (8, 4) pictures sampled on the square grid.

1 aPalágyi, Kálmán1 aNémeth, Gábor1 aKardos, Péter1 aLoncaric, S1 aLerski, D1 aEskola, H1 aBregovic, R uhttps://www.inf.u-szeged.hu/en/publication/topology-preserving-equivalent-parallel-and-sequential-4-subiteration-2d-thinning-algorithms01180nas a2200145 4500008004100000245006100041210006000102260004000162300001200202520066300214100002000877700001900897700002300916856009500939 2015 hun d00aVékonyítás a végpont-megőrzés felülvizsgálatáva0 aVékonyítás a végpontmegőrzés felülvizsgálatáva aKecskemét, MagyarországcJan 2015 a578-5873 aA vékonyítás mint iteratív objektum redukció gyakran alkalmazott

vázkijelölo módszer. A legtöbb létezo vékonyító algoritmus végpontok - vagyis releváns geometriai információt hordozó objektumpontok - megorzésével biztosítja azt, hogy ne törlodjenek az objektumok alakját reprezentáló fontos részletek. Ennek a megközelítésnek hátránya, hogy számos nemkívánatos vázágat eredményezhet. Ebben a cikkben egy olyan módszert mutatunk be, amellyel jelentosen csökkentheto a hamis vázágak száma. Ráadásul az itt bemutatott megközelítés tetszoleges végpont-megorzo 2D vékonyító algoritmusban alkalmazható.

An important requirement for various applications of binary image processing is to preserve topology. This issue has been earlier studied for two special types of image operators, namely, reductions and additions, and there have been some sufficient conditions proposed for them. In this paper, as an extension of those earlier results, we give novel sufficient criteria for general operators working on 2D pictures.

1 aKardos, Péter1 aPalágyi, Kálmán1 aBarneva, Reneta, P1 aBrimkov, Valentin E1 aŠlapal, Josef uhttp://dx.doi.org/10.1007/978-3-319-07148-0_1001138nas a2200157 4500008004100000020002300041245004500064210004500109260002900154300001400183520063300197100001900830700002300849700002000872856008800892 2013 eng d a978-1-4799-1543-9 00aParallel Thinning on the Triangular Grid0 aParallel Thinning on the Triangular Grid aBudapestbIEEEcDec 2013 a277 - 2823 a

One of the fundamental issues of human and computational cognitive psychology is pattern or shape recognition. Various applications in image processing and computer vision rely on skeleton-like shape features A possible technique for extracting these feautures is thinning. Although the majority of 2D thinning algorithms work on digital pictures sampled onthe conventional square grid, the role of some non-conventional grids, like the hexagonal and triangular grid, are of increasing importance as well. In this paper we propose numerous topolgy preserving parallel thinning algorithms that work on the triangular grid.

1 aKardos, Péter1 aPalágyi, Kálmán1 aBaranyi, Péter uhttps://www.inf.u-szeged.hu/en/publication/parallel-thinning-on-the-triangular-grid01600nas a2200145 4500008004100000245008200041210006900123260004300192300001400235520102400249100001901273700002301292700001401315856012501329 2013 eng d00aSufficient Conditions for Topology Preserving Additions and General Operators0 aSufficient Conditions for Topology Preserving Additions and Gene aCalgarybIASTED - Acta PresscFeb 2013 a107 - 1143 aTopology preservation is a crucial issue of digital topology. Various applications of binary image processing rest on topology preserving operators. Earlier studies in this topic mainly concerned with reductions (i.e., operators that only delete some object points from binary images), as they form the basis for thinning algorithms. However, additions (i.e., operators that never change object points) also play important role for the purpose of generating discrete Voronoi diagrams or skeletons by influence zones (SKIZ). Furthermore, the use of general operators that may both add and delete some points to and from objects in pictures are suitable for contour smoothing. Therefore, in this paper we present some new sufficient conditions for topology preserving reductions, additions, and general operators. Two additions for 2D and 3D contour smoothing are also reported.

1 aKardos, Péter1 aPalágyi, Kálmán1 aLinsen, L uhttps://www.inf.u-szeged.hu/en/publication/sufficient-conditions-for-topology-preserving-additions-and-general-operators01104nas a2200181 4500008004100000245007200041210006700113260002800180300001400208520046000222100001900682700002300701700002200724700002100746700002000767700002200787856011300809 2013 eng d00aOn Topology Preservation in Triangular, Square, and Hexagonal Grids0 aTopology Preservation in Triangular Square and Hexagonal Grids aTriestebIEEEcSep 2013 a782 - 7873 a

There are three possible partitionings of the continuous plane into regular polygons that leads to triangular, square, and hexagonal grids. The topology of the square grid is fairly well-understood, but it cannot be said of the remaining two regular sampling schemes. This paper presents a general characterization of simple pixels and some simplified sufficient conditions for topology-preserving operators in all the three types of regular grids.

1 aKardos, Péter1 aPalágyi, Kálmán1 aRamponi, Giovanni1 aLončarić, Sven1 aCarini, Alberto1 aEgiazarian, Karen uhttps://www.inf.u-szeged.hu/en/publication/on-topology-preservation-in-triangular-square-and-hexagonal-grids00516nas a2200133 4500008004100000245006100041210006100102260003800163300001400201100001900215700002300234700002100257856010400278 2013 eng d00aTopology preserving parallel thinning on hexagonal grids0 aTopology preserving parallel thinning on hexagonal grids aVeszprémbNJSZT-KÉPAFcJan 2013 a250 - 2641 aKardos, Péter1 aPalágyi, Kálmán1 aCzúni, László uhttps://www.inf.u-szeged.hu/en/publication/topology-preserving-parallel-thinning-on-hexagonal-grids01108nas a2200157 4500008004100000020001400041245004300055210004200098260002700140300001600167490000700183520064200190100001900832700002300851856007600874 2013 eng d a0020-716000aTopology-preserving hexagonal thinning0 aTopologypreserving hexagonal thinning bTaylor & Francisc2013 a1607 - 16170 v903 aThinning is a well-known technique for producing skeleton-like shape features from digital binary objects in a topology-preserving way. Most of the existing thinning algorithms work on input images that are sampled on orthogonal grids; however, it is also possible to perform thinning on hexagonal grids (or triangular lattices). In this paper, we point out to the main similarities and differences between the topological properties of these two types of sampling schemes. We give various characterizations of simple points and present some new sufficient conditions for topology-preserving reductions working on hexagonal grids.

1 aKardos, Péter1 aPalágyi, Kálmán uhttp://www.tandfonline.com/doi/abs/10.1080/00207160.2012.724198#preview01376nas a2200193 4500008004100000020002200041245010000063210006900163260005500232300001200287520059000299100001900889700002300908700002800931700002400959700002600983700003001009856014301039 2012 eng d a978-0-415-62134-200aHexagonal parallel thinning algorithms based on sufficient conditions for topology preservation0 aHexagonal parallel thinning algorithms based on sufficient condi aLondonbCRC Press - Taylor and Frances Groupc2012 a63 - 683 aThinning is a well-known technique for producing skeleton-like shape features from digital

binary objects in a topology preserving way. Most of the existing thinning algorithms presuppose that the input

images are sampled on orthogonal grids.This paper presents new sufficient conditions for topology preserving

reductions working on hexagonal grids (or triangular lattices) and eight new 2D hexagonal parallel thinning

algorithms that are based on our conditions.The proposed algorithms are capable of producing both medial lines

and topological kernels as well.

Thinning as a layer-by-layer reduction is a frequently used technique for skeletonization. Sequential thinning algorithms usually suffer from the drawback of being order-dependent, i.e., their results depend on the visiting order of object points. Earlier order-independent sequential methods are based on the conventional thinning schemes that preserve endpoints to provide relevant geometric information of objects. These algorithms can generate centerlines in 2D and medial surfaces in 3D. This paper presents an alternative strategy for order-independent thinning which follows an approach, proposed by Bertrand and Couprie, which accumulates so-called isthmus points. The main advantage of this order-independent strategy over the earlier ones is that it makes also possible to produce centerlines of 3D objects.

1 aKardos, Péter1 aPalágyi, Kálmán1 aPetrou, M1 aSappa, A, D1 aTriantafyllidis, A G uhttp://www.actapress.com/Content_of_Proceeding.aspx?proceedingID=73601390nas a2200157 4500008004100000020002300041245004700064210004400111260003400155300001400189520091000203100001901113700002301132700000501155856007201160 2012 eng d a978-1-4673-5187-4 00aOn Order–Independent Sequential Thinning0 aOrder–Independent Sequential Thinning aKosice, Slovakia bIEEEc2012 a149 - 1543 aThe visual world composed by the human and computational cognitive systems strongly relies on shapes of objects. Skeleton is a widely applied shape feature that plays an important role in many fields of image processing, pattern recognition, and computer vision. Thinning is a frequently used, iterative object reduction strategy for skeletonization. Sequential thinning algorithms, which are based on contour tracking, delete just one border point at a time. Most of them have the disadvantage of order-dependence, i.e., for dissimilar visiting orders of object points, they may generate different skeletons. In this work, we give a survey of our results on order-independent thinning: we introduce some sequential algorithms that produce identical skeletons for any visiting orders, and we also present some sufficient conditions for the order-independence of templatebased sequential algorithms.

1 aKardos, Péter1 aPalágyi, Kálmán1 a uhttp://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=641330501137nas a2200181 4500008004100000020002200041245006400063210006100127260004700188300001400235520048800249100001900737700002300756700002300779700002400802700002200826856010700848 2012 eng d a978-3-642-34731-300aOn topology preservation for triangular thinning algorithms0 atopology preservation for triangular thinning algorithms aAustin, TX, USAbSpringer VerlagcNov 2012 a128 - 1423 aThinning is a frequently used strategy to produce skeleton-like shape features of binary objects. One of the main problems of parallel thinning is to ensure topology preservation. Solutions to this problem have been already given for the case of orthogonal and hexagonal grids. This work introduces some characterizations of simple pixels and some sufficient conditions for parallel thinning algorithms working on triangular grids (or hexagonal lattices) to preserve topology.

1 aKardos, Péter1 aPalágyi, Kálmán1 aBarneva, Reneta, P1 aBrimkov, Valentin E1 aAggarwal, Jake, K uhttps://www.inf.u-szeged.hu/en/publication/on-topology-preservation-for-triangular-thinning-algorithms01194nas a2200181 4500008004100000020002200041245005600063210005600119260002600175300001400201520058900215100002300804700002000827700001900847700002400866700002300890856009900913 2012 eng d a978-94-007-4173-700aTopology Preserving Parallel 3D Thinning Algorithms0 aTopology Preserving Parallel 3D Thinning Algorithms bSpringer-Verlagc2012 a165 - 1883 aA widely used technique to obtain skeletons of binary objects is thinning, which is an iterative layer-by-layer erosion in a topology preserving way. Thinning in 3D is capable of extracting various skeleton-like shape descriptors (i.e., centerlines, medial surfaces, and topological kernels). This chapter describes a family of new parallel 3D thinning algorithms for (26, 6) binary pictures. The reported algorithms are derived from some sufficient conditions for topology preserving parallel reduction operations, hence their topological correctness is guaranteed. ` `

Thinning and shrinking algorithms, respectively, are capable of extracting medial lines and topological kernels from digital binary objects in a topology preserving way. These topological algorithms are composed of reduction operations: object points that satisfy some topological and geometrical constraints are removed until stability is reached. In this work we present some new sufficient conditions for topology preserving parallel reductions and fiftyfour new 2D parallel thinning and shrinking algorithms that are based on our conditions. The proposed thinning algorithms use five characterizations of endpoints.

1 aNémeth, Gábor1 aKardos, Péter1 aPalágyi, Kálmán uhttps://www.inf.u-szeged.hu/en/publication/2d-parallel-thinning-and-shrinking-based-on-sufficient-conditions-for-topology-preservation01821nas a2200217 4500008004100000020002200041245008300063210006900146260004500215300001200260520102200272100002001294700001901314700002301333700002201356700002301378700002401401700002801425700002401453856012601477 2011 eng d a978-3-642-21072-300aA family of topology-preserving 3d parallel 6-subiteration thinning algorithms0 afamily of topologypreserving 3d parallel 6subiteration thinning aMadrid, SpainbSpringer VerlagcMay 2011 a17 - 303 aThinning is an iterative layer-by-layer erosion until only the skeleton-like shape features of the objects are left. This paper presents a family of new 3D parallel thinning algorithms that are based on our new sufficient conditions for 3D parallel reduction operators to preserve topology. The strategy which is used is called subiteration-based: each iteration step is composed of six parallel reduction operators according to the six main directions in 3D. The major contributions of this paper are: 1) Some new sufficient conditions for topology preserving parallel reductions are introduced. 2) A new 6-subiteration thinning scheme is proposed. Its topological correctness is guaranteed, since its deletion rules are derived from our sufficient conditions for topology preservation. 3) The proposed thinning scheme with different characterizations of endpoints yields various new algorithms for extracting centerlines and medial surfaces from 3D binary pictures. © 2011 Springer-Verlag Berlin Heidelberg.

1 aNémeth, Gábor1 aKardos, Péter1 aPalágyi, Kálmán1 aAggarwal, Jake, K1 aBarneva, Reneta, P1 aBrimkov, Valentin E1 aKoroutchev, Kostadin, N1 aKorutcheva, Elka, R uhttps://www.inf.u-szeged.hu/en/publication/a-family-of-topology-preserving-3d-parallel-6-subiteration-thinning-algorithms00520nas a2200157 4500008004100000245006000041210006000101260002800161300001400189100001900203700002000222700002300242700001700265700002300282856005700305 2011 eng d00aIterációnkénti simítással kombinált vékonyítás0 aIterációnkénti simítással kombinált vékonyítás aSzegedbNJSZTcJan 2011 a174 - 1891 aKardos, Péter1 aNémeth, Gábor1 aPalágyi, Kálmán1 aKato, Zoltan1 aPalágyi, Kálmán uhttp://www.inf.u-szeged.hu/kepaf2011/pdfs/S05_01.pdf01093nas a2200157 4500008004100000245006600041210006500107260004900172300001400221520051000235100001900745700002300764700002300787700001600810856010900826 2011 eng d00aOrder-independent sequential thinning in arbitrary dimensions0 aOrderindependent sequential thinning in arbitrary dimensions aCrete, GreekbIASTED - Acta PresscJune 2011 a129 - 1343 aSkeletons are region based shape descriptors that play important role in shape representation. This paper introduces a novel sequential thinning approach for n-dimensional binary objects (*n* =1,2,3, ...). Its main strength lies in its order--independency, i.e., it can produce the same skeletons for any visiting orders of border points. Furthermore, this is the first scheme in this field that is also applicable for higher dimensions.

The main issue of this paper is to introduce some conditions for template-based sequential thinning that are capable of producing the same skeleton for a given binary image, independent of the visiting order of object points. As an example, we introduce two order-independent thinning algorithms for 2D binary images that satisfy these conditions. ` `

In this work we present a new thinning scheme for reducing the noise sensitivity of 3D thinning algorithms. It uses iteration-by-iteration smoothing that removes some border points that are considered as extremities. The proposed smoothing algorithm is composed of two parallel topology preserving reduction operators. An efficient implementation of our algorithm is sketched and its topological correctness for (26, 6) pictures is proved. © 2011 Elsevier Inc. All rights reserved.

1 aNémeth, Gábor1 aKardos, Péter1 aPalágyi, Kálmán uhttps://www.inf.u-szeged.hu/en/publication/thinning-combined-with-iteration-by-iteration-smoothing-for-3d-binary-images00578nas a2200157 4500008004100000245010100041210007700142260002800219300001400247100002000261700001900281700002300300700001700323700002300340856005700363 2011 hun d00aA topológia-megőrzés elegendő feltételein alapuló 3D párhuzamos vékonyító algoritmusok0 atopológiamegőrzés elegendő feltételein alapuló 3D párhuzamos vék aSzegedbNJSZTcJan 2011 a190 - 2051 aNémeth, Gábor1 aKardos, Péter1 aPalágyi, Kálmán1 aKato, Zoltan1 aPalágyi, Kálmán uhttp://www.inf.u-szeged.hu/kepaf2011/pdfs/S05_02.pdf01289nas a2200205 4500008004100000020002200041245007200063210006900135260004500204300001200249520054400261100001900805700002300824700002200847700002300869700002400892700002800916700002400944856011500968 2011 eng d a978-3-642-21072-300aOn topology preservation for hexagonal parallel thinning algorithms0 atopology preservation for hexagonal parallel thinning algorithms aMadrid, SpainbSpringer VerlagcMay 2011 a31 - 423 aTopology preservation is the key concept in parallel thinning algorithms on any sampling schemes. This paper establishes some sufficient conditions for parallel thinning algorithms working on hexagonal grids (or triangular lattices) to preserve topology. By these results, various thinning (and shrinking to a residue) algorithms can be verified. To illustrate the usefulness of our sufficient conditions, we propose a new parallel thinning algorithm and prove its topological correctness. © 2011 Springer-Verlag Berlin Heidelberg.

1 aKardos, Péter1 aPalágyi, Kálmán1 aAggarwal, Jake, K1 aBarneva, Reneta, P1 aBrimkov, Valentin E1 aKoroutchev, Kostadin, N1 aKorutcheva, Elka, R uhttps://www.inf.u-szeged.hu/en/publication/on-topology-preservation-for-hexagonal-parallel-thinning-algorithms00498nas a2200157 4500008004100000020001400041245005300055210005300108260000900161300001200170490000700182100001900189700002000208700002300228856008900251 2010 eng d a0133-339900aBejárásfüggetlen szekvenciális vékonyítás0 aBejárásfüggetlen szekvenciális vékonyítás c2010 a17 - 400 v271 aKardos, Péter1 aNémeth, Gábor1 aPalágyi, Kálmán uhttps://www.inf.u-szeged.hu/en/publication/bejarasfuggetlen-szekvencialis-vekonyitas01183nas a2200181 4500008004100000245005800041210005700099260005200156300001400208520057500222100002000797700001900817700002300836700001300859700001500872700001300887856010100900 2010 eng d00aTopology preserving 2-subfield 3D thinning algorithms0 aTopology preserving 2subfield 3D thinning algorithms aInnsbruck, AustriabIASTED ACTA PresscFeb 2010 a310 - 3163 aThis paper presents a new family of 3D thinning algorithms for extracting skeleton-like shape features (i.e, centerline, medial surface, and topological kernel) from volumetric images. A 2-subfield strategy is applied: all points in a 3D picture are partitioned into two subsets which are alternatively activated. At each iteration, a parallel operator is applied for deleting some border points in the active subfield. The proposed algorithms are derived from Ma's sufficient conditions for topology preservation, and they use various endpoint characterizations.

1 aNémeth, Gábor1 aKardos, Péter1 aPalágyi, Kálmán1 aZagar, B1 aKuijper, A1 aSahbi, H uhttps://www.inf.u-szeged.hu/en/publication/topology-preserving-2-subfield-3d-thinning-algorithms01489nas a2200181 4500008004100000245007800041210006900119260005900188300001400247490000900261520081200270100002001082700001901102700002301121700002301144700001901167856012101186 2010 eng d00aTopology Preserving 3D Thinning Algorithms using Four and Eight Subfields0 aTopology Preserving 3D Thinning Algorithms using Four and Eight aPóvoa de Varzim, PortugalbSpringer VerlagcJune 2010 a316 - 3250 v61113 aThinning is a frequently applied technique for extracting skeleton-like shape features (i.e., centerline, medial surface, and topological kernel) from volumetric binary images. Subfield-based thinning algorithms partition the image into some subsets which are alternatively activated, and some points in the active subfield are deleted. This paper presents a set of new 3D parallel subfield-based thinning algorithms that use four and eight subfields. The three major contributions of this paper are: 1) The deletion rules of the presented algorithms are derived from some sufficient conditions for topology preservation. 2) A novel thinning scheme is proposed that uses iteration-level endpoint checking. 3) Various characterizations of endpoints yield different algorithms. © 2010 Springer-Verlag.

1 aNémeth, Gábor1 aKardos, Péter1 aPalágyi, Kálmán1 aCampilho, Aurélio1 aKamel, Mohamed uhttps://www.inf.u-szeged.hu/en/publication/topology-preserving-3d-thinning-algorithms-using-four-and-eight-subfields01336nas a2200217 4500008004100000245006400041210006400105260004400169300001400213490000900227520058400236100002000820700001900840700002300859700002300882700002400905700002500929700002600954700003100980856010701011 2010 eng d00aTopology Preserving Parallel Smoothing for 3D Binary Images0 aTopology Preserving Parallel Smoothing for 3D Binary Images aBuffalo, USAbSpringer VerlagcMay 2010 a287 - 2980 v60263 aThis paper presents a new algorithm for smoothing 3D binary images in a topology preserving way. Our algorithm is a reduction operator: some border points that are considered as extremities are removed. The proposed method is composed of two parallel reduction operators. We are to apply our smoothing algorithm as an iteration-by-iteration pruning for reducing the noise sensitivity of 3D parallel surface-thinning algorithms. An efficient implementation of our algorithm is sketched and its topological correctness for (26,6) pictures is proved. © 2010 Springer-Verlag.

1 aNémeth, Gábor1 aKardos, Péter1 aPalágyi, Kálmán1 aBarneva, Reneta, P1 aBrimkov, Valentin E1 aHauptman, Herbert, A1 aJorge, Renato M Natal1 aTavares, João, Manuel R S uhttps://www.inf.u-szeged.hu/en/publication/topology-preserving-parallel-smoothing-for-3d-binary-images00396nas a2200097 4500008004100000245009200041210007800133260003300211300001000244856004400254 2009 hun d00aKritikus párokat vizsgáló bejárásfüggetlen szekvenciális vékonyító algoritmus0 aKritikus párokat vizsgáló bejárásfüggetlen szekvenciális vékonyí aBudapestbAkaprintcJan 2009 a1 - 8 uhttps://www.inf.u-szeged.hu/en/node/97301114nas a2200181 4500008004100000020002200041245005500063210005100118260005600169300001400225520052000239100001900759700002000778700002300798700002200821700002300843856006600866 2009 eng d a978-3-642-10208-000aAn order-independent sequential thinning algorithm0 aorderindependent sequential thinning algorithm aPlaya del Carmen, MexicobSpringer VerlagcNov 2009 a162 - 1753 aThinning is a widely used approach for skeletonization. Sequential thinning algorithms use contour tracking: they scan border points and remove the actual one if it is not designated a skeletal point. They may produce various skeletons for different visiting orders. In this paper, we present a new 2-dimensional sequential thinning algorithm, which produces the same result for arbitrary visiting orders and it is capable of extracting maximally thinned skeletons. © Springer-Verlag Berlin Heidelberg 2009.

1 aKardos, Péter1 aNémeth, Gábor1 aPalágyi, Kálmán1 aWiederhold, Petra1 aBarneva, Reneta, P uhttp://link.springer.com/chapter/10.1007/978-3-642-10210-3_13