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.

\

}, isbn = {978-3-319-07147-3}, issn = {978-3-319-07147-3}, doi = {10.1007/978-3-319-07148-0_10}, url = {http://dx.doi.org/10.1007/978-3-319-07148-0_10}, author = {P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Reneta P Barneva and Valentin E Brimkov and Josef {\v S}lapal} } @inbook {983, title = {Parallel Thinning on the Triangular Grid}, booktitle = {International Conference on Cognitive Infocommunications (CogInfoCom)}, year = {2013}, month = {Dec 2013}, pages = {277 - 282}, publisher = {IEEE}, organization = {IEEE}, type = {Conference paper}, address = {Budapest}, abstract = {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.

}, isbn = {978-1-4799-1543-9 }, doi = {10.1109/CogInfoCom.2013.6719256}, author = {P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {P{\'e}ter Baranyi} } @inbook {976, title = {Sufficient Conditions for Topology Preserving Additions and General Operators}, booktitle = {Proceedings of the IASTED International Conference on Computer Graphics and Imaging (CGIM)}, year = {2013}, month = {Feb 2013}, pages = {107 - 114}, publisher = {IASTED - Acta Press}, organization = {IASTED - Acta Press}, type = {Conference paper}, address = {Calgary}, abstract = {Topology 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.

\

}, doi = {10.2316/P.2013.797-040}, author = {P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {L Linsen} } @inbook {980, title = {On Topology Preservation in Triangular, Square, and Hexagonal Grids}, booktitle = {Proceedings of International Symposium on Image and Signal Processing and Analysis (ISPA)}, year = {2013}, month = {Sep 2013}, pages = {782 - 787}, publisher = {IEEE}, organization = {IEEE}, type = {Conference paper}, address = {Trieste}, abstract = {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.

}, doi = {10.1109/ISPA.2013.6703844}, author = {P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Giovanni Ramponi and Sven Lon{\v c}ari{\'c} and Alberto Carini and Karen Egiazarian} } @conference {974, title = {Topology preserving parallel thinning on hexagonal grids}, booktitle = {A K{\'e}pfeldolgoz{\'o}k {\'e}s Alakfelismer{\H o}k T{\'a}rsas{\'a}g{\'a}nak konferenci{\'a}ja - K{\'E}PAF 2013}, year = {2013}, month = {Jan 2013}, pages = {250 - 264}, publisher = {NJSZT-K{\'E}PAF}, organization = {NJSZT-K{\'E}PAF}, type = {Conference paper}, address = {Veszpr{\'e}m}, author = {P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {L{\'a}szl{\'o} Cz{\'u}ni} } @article {975, title = {Topology-preserving hexagonal thinning}, journal = {INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS}, volume = {90}, year = {2013}, note = {doi: 10.1080/00207160.2012.724198}, month = {2013}, pages = {1607 - 1617}, publisher = {Taylor \& Francis}, type = {Journal article}, abstract = {Thinning 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.

}, isbn = {0020-7160}, doi = {10.1080/00207160.2012.724198}, url = {http://www.tandfonline.com/doi/abs/10.1080/00207160.2012.724198$\#$preview}, author = {P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi} } @inbook {942, title = {Hexagonal parallel thinning algorithms based on sufficient conditions for topology preservation}, booktitle = {Computational Modelling of Objects Represented in Images: Fundamentals, Methods and Applications III}, year = {2012}, month = {2012}, pages = {63 - 68}, publisher = {CRC Press - Taylor and Frances Group}, organization = {CRC Press - Taylor and Frances Group}, type = {Conference paper}, address = {London}, abstract = {Thinning 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.

}, doi = {10.2316/P.2012.778-025}, url = {http://www.actapress.com/Content_of_Proceeding.aspx?proceedingID=736}, author = {P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {M Petrou and A D Sappa and A G Triantafyllidis} } @conference {944, title = {On Order{\textendash}Independent Sequential Thinning}, booktitle = {IEEE International Conference on Cognitive Infocommunications (CogInfoCom)}, year = {2012}, month = {2012}, pages = {149 - 154}, publisher = {IEEE}, organization = {IEEE}, type = {Conference paper}, address = {Kosice, Slovakia }, abstract = {The 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.

}, isbn = {978-1-4673-5187-4 }, doi = {10.1109/CogInfoCom.2012.6421971}, url = {http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=6413305}, author = {P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {IEEE} } @inbook {943, title = {On topology preservation for triangular thinning algorithms}, booktitle = {Combinatorial Image Analysis (IWCIA)}, series = {Lecture Notes in Computer Science}, number = {7655}, year = {2012}, note = {doi: 10.1007/978-3-642-34732-0_10Lecture Notes in Computer Science, Volume 7655}, month = {Nov 2012}, pages = {128 - 142}, publisher = {Springer Verlag}, organization = {Springer Verlag}, type = {Conference paper}, address = {Austin, TX, USA}, abstract = {Thinning 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.

}, isbn = {978-3-642-34731-3}, doi = {10.1007/978-3-642-34732-0_10}, author = {P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Reneta P Barneva and Valentin E Brimkov and Jake K Aggarwal} } @inbook {875, title = {Topology Preserving Parallel 3D Thinning Algorithms}, booktitle = {Digital Geometry Algorithms}, series = {Lecture Notes in Computational Vision and Biomechanics}, number = {2}, year = {2012}, note = {doi: 10.1007/978-94-007-4174-4_6}, month = {2012}, pages = {165 - 188}, publisher = {Springer-Verlag}, organization = {Springer-Verlag}, type = {Book chapter}, chapter = {6}, abstract = {A 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.

}, isbn = {0324-721X}, author = {G{\'a}bor N{\'e}meth and P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi} } @inbook {864, title = {A family of topology-preserving 3d parallel 6-subiteration thinning algorithms}, booktitle = {Combinatorial Image Analysis (IWCIA)}, series = {Lecture Notes in Computer Science}, number = {6636}, year = {2011}, note = {ScopusID: 79957651399doi: 10.1007/978-3-642-21073-0_5}, month = {May 2011}, pages = {17 - 30}, publisher = {Springer Verlag}, organization = {Springer Verlag}, type = {Conference paper}, address = {Madrid, Spain}, abstract = {Thinning 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. {\textcopyright} 2011 Springer-Verlag Berlin Heidelberg.

}, isbn = {978-3-642-21072-3}, doi = {10.1007/978-3-642-21073-0_5}, author = {G{\'a}bor N{\'e}meth and P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Jake K Aggarwal and Reneta P Barneva and Valentin E Brimkov and Kostadin N Koroutchev and Elka R Korutcheva} } @conference {876, title = {Iter{\'a}ci{\'o}nk{\'e}nti sim{\'\i}t{\'a}ssal kombin{\'a}lt v{\'e}kony{\'\i}t{\'a}s}, booktitle = {A K{\'e}pfeldolgoz{\'o}k {\'e}s Alakfelismer{\H o}k T{\'a}rsas{\'a}g{\'a}nak konferenci{\'a}ja - K{\'E}PAF 2011}, year = {2011}, month = {Jan 2011}, pages = {174 - 189}, publisher = {NJSZT}, organization = {NJSZT}, type = {Conference paper}, address = {Szeged}, url = {http://www.inf.u-szeged.hu/kepaf2011/pdfs/S05_01.pdf}, author = {P{\'e}ter Kardos and G{\'a}bor N{\'e}meth and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Zoltan Kato and K{\'a}lm{\'a}n Pal{\'a}gyi} } @conference {933, title = {Order-independent sequential thinning in arbitrary dimensions}, booktitle = {Signal and Image Processing and Applications (SIPA)}, year = {2011}, note = {doi: 10.2316/P.2011.738-021}, month = {June 2011}, pages = {129 - 134}, publisher = {IASTED - Acta Press}, organization = {IASTED - Acta Press}, type = {Conference paper}, address = {Crete, Greek}, abstract = {Skeletons 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. {\textcopyright} 2011 Elsevier Inc. All rights reserved.

}, isbn = {1524-0703}, doi = {10.1016/j.gmod.2011.02.001}, author = {G{\'a}bor N{\'e}meth and P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi} } @conference {877, title = {A topol{\'o}gia-meg{\H o}rz{\'e}s elegend{\H o} felt{\'e}telein alapul{\'o} 3D p{\'a}rhuzamos v{\'e}kony{\'\i}t{\'o} algoritmusok}, booktitle = {A K{\'e}pfeldolgoz{\'o}k {\'e}s Alakfelismer{\H o}k T{\'a}rsas{\'a}g{\'a}nak konferenci{\'a}ja - K{\'E}PAF 2011}, year = {2011}, month = {Jan 2011}, pages = {190 - 205}, publisher = {NJSZT}, organization = {NJSZT}, type = {Conference paper}, address = {Szeged}, url = {http://www.inf.u-szeged.hu/kepaf2011/pdfs/S05_02.pdf}, author = {G{\'a}bor N{\'e}meth and P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Zoltan Kato and K{\'a}lm{\'a}n Pal{\'a}gyi} } @inbook {935, title = {On topology preservation for hexagonal parallel thinning algorithms}, booktitle = {Combinatorial Image Analysis (IWCIA)}, series = {Lecture Notes in Computer Science}, number = {6636}, year = {2011}, note = {ScopusID: 79957628214doi: 10.1007/978-3-642-21073-0_6}, month = {May 2011}, pages = {31 - 42}, publisher = {Springer Verlag}, organization = {Springer Verlag}, type = {Conference paper}, address = {Madrid, Spain}, abstract = {Topology 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. {\textcopyright} 2011 Springer-Verlag Berlin Heidelberg.

}, isbn = {978-3-642-21072-3}, doi = {10.1007/978-3-642-21073-0_6}, author = {P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Jake K Aggarwal and Reneta P Barneva and Valentin E Brimkov and Kostadin N Koroutchev and Elka R Korutcheva} } @article {862, title = {Bej{\'a}r{\'a}sf{\"u}ggetlen szekvenci{\'a}lis v{\'e}kony{\'\i}t{\'a}s}, journal = {ALKALMAZOTT MATEMATIKAI LAPOK}, volume = {27}, year = {2010}, month = {2010}, pages = {17 - 40}, type = {Journal article}, isbn = {0133-3399}, author = {P{\'e}ter Kardos and G{\'a}bor N{\'e}meth and K{\'a}lm{\'a}n Pal{\'a}gyi} } @conference {872, title = {Topology preserving 2-subfield 3D thinning algorithms}, booktitle = {Proceedings of the International Conference on Signal Processing, Pattern Recognition and Applications (SPPRA)}, year = {2010}, note = {ScopusID: 77954590365}, month = {Feb 2010}, pages = {310 - 316}, publisher = {IASTED ACTA Press}, organization = {IASTED ACTA Press}, type = {Conference paper}, address = {Innsbruck, Austria}, abstract = {This 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{\textquoteright}s sufficient conditions for topology preservation, and they use various endpoint characterizations.

}, author = {G{\'a}bor N{\'e}meth and P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {B Zagar and A Kuijper and H Sahbi} } @inbook {865, title = {Topology Preserving 3D Thinning Algorithms using Four and Eight Subfields}, booktitle = {Proceedings of the International Conference on Image Analysis and Recognition (ICIAR)}, series = {Lecture Notes in Computer Science}, volume = {6111}, year = {2010}, note = {ScopusID: 77955432947doi: 10.1007/978-3-642-13772-3_32}, month = {June 2010}, pages = {316 - 325}, publisher = {Springer Verlag}, organization = {Springer Verlag}, type = {Conference paper}, address = {P{\'o}voa de Varzim, Portugal}, abstract = {Thinning 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. {\textcopyright} 2010 Springer-Verlag.

}, doi = {10.1007/978-3-642-13772-3_32}, author = {G{\'a}bor N{\'e}meth and P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Aur{\'e}lio Campilho and Mohamed Kamel} } @inbook {866, title = {Topology Preserving Parallel Smoothing for 3D Binary Images}, booktitle = {Proceedings of the Computational Modeling of Objects Represented in Images (CMORI)}, volume = {6026}, year = {2010}, note = {ScopusID: 77952401887doi: 10.1007/978-3-642-12712-0_26}, month = {May 2010}, pages = {287 - 298}, publisher = {Springer Verlag}, organization = {Springer Verlag}, type = {Conference paper}, address = {Buffalo, USA}, abstract = {This 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. {\textcopyright} 2010 Springer-Verlag.

}, doi = {10.1007/978-3-642-12712-0_26}, author = {G{\'a}bor N{\'e}meth and P{\'e}ter Kardos and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Reneta P Barneva and Valentin E Brimkov and Herbert A Hauptman and Renato M Natal Jorge and Jo{\~a}o Manuel R S Tavares} } @conference {973, title = {Kritikus p{\'a}rokat vizsg{\'a}l{\'o} bej{\'a}r{\'a}sf{\"u}ggetlen szekvenci{\'a}lis v{\'e}kony{\'\i}t{\'o} algoritmus}, booktitle = {A K{\'e}pfeldolgoz{\'o}k {\'e}s Alakfelismer{\H o}k T{\'a}rsas{\'a}g{\'a}nak konferenci{\'a}ja - K{\'E}PAF 2009}, year = {2009}, month = {Jan 2009}, pages = {1 - 8}, publisher = {Akaprint}, organization = {Akaprint}, type = {Conference paper}, address = {Budapest}, author = {P{\'e}ter Kardos and G{\'a}bor N{\'e}meth and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Dmitrij Chetverikov and Tamas Sziranyi} } @inbook {868, title = {An order-independent sequential thinning algorithm}, booktitle = {Proceedings of the International Workshop on Combinatorial Image Analysis (IWCIA)}, number = {5852}, year = {2009}, note = {ScopusID: 78650496028doi: 10.1007/978-3-642-10210-3_13}, month = {Nov 2009}, pages = {162 - 175}, publisher = {Springer Verlag}, organization = {Springer Verlag}, type = {Conference paper}, address = {Playa del Carmen, Mexico}, abstract = {Thinning 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. {\textcopyright} Springer-Verlag Berlin Heidelberg 2009.

}, isbn = {978-3-642-10208-0}, doi = {10.1007/978-3-642-10210-3_13}, url = {http://link.springer.com/chapter/10.1007/978-3-642-10210-3_13}, author = {P{\'e}ter Kardos and G{\'a}bor N{\'e}meth and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Petra Wiederhold and Reneta P Barneva} }