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.

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}, 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 {1153, title = {Binary image reconstruction from two projections and skeletal information}, booktitle = {Combinatorial Image Analysis}, series = {Lecture Notes in Computer Science}, number = {7655}, year = {2012}, note = {ScopusID: 84869986820doi: 10.1007/978-3-642-34732-0_20}, month = {Nov 2012}, pages = {263 - 273}, publisher = {Springer Verlag}, organization = {Springer Verlag}, type = {Conference paper}, address = {Berlin; Heidelberg; New York; London; Paris; Tokyo}, abstract = {In binary tomography, the goal is to reconstruct binary images from a small set of their projections. However, especially when only two projections are used, the task can be extremely underdetermined. In this paper, we show how to reduce ambiguity by using the morphological skeleton of the image as a priori. Three different variants of our method based on Simulated Annealing are tested using artificial binary images, and compared by reconstruction time and error. {\textcopyright} 2012 Springer-Verlag.

}, doi = {10.1007/978-3-642-34732-0_20}, author = {Norbert Hantos and P{\'e}ter Bal{\'a}zs and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Reneta P Barneva and Valentin E Brimkov and Jake K Aggarwal} } @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 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} } @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} } @inbook {1110, title = {Direction-dependency of a binary tomographic reconstruction algorithm}, booktitle = {Computational Modeling of Objects Represented in Images}, series = {Lecture Notes in Computer Science}, number = {6026}, year = {2010}, note = {UT: 000279020400022ScopusID: 77952365308doi: 10.1007/978-3-642-12712-0_22}, month = {May 2010}, pages = {242 - 253}, publisher = {Springer Verlag}, organization = {Springer Verlag}, type = {Conference paper}, address = {Buffalo, NY, USA}, abstract = {We study how the quality of an image reconstructed by a binary tomographic algorithm depends on the direction of the observed object in the scanner, if only a few projections are available. To do so we conduct experiments on a set of software phantoms by reconstructing them form different projection sets using an algorithm based on D.C. programming (a method for minimizing the difference of convex functions), and compare the accuracy of the corresponding reconstructions by two suitable approaches. Based on the experiments, we discuss consequences on applications arising from the field of non-destructive testing, as well.

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}, isbn = {978-3-642-12711-3}, doi = {10.1007/978-3-642-12712-0_22}, author = {L{\'a}szl{\'o} G{\'a}bor Varga and P{\'e}ter Bal{\'a}zs and Antal Nagy}, 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} } @inbook {959, title = {Parallel Thinning Algorithms Based on Ronse{\textquoteright}s Sufficient Conditions for Topology Preservation}, booktitle = {Progress in Combinatorial Image Analysis}, year = {2010}, month = {May 2010}, pages = {183 - 194}, publisher = {Scientific Research Publishing Inc.}, organization = {Scientific Research Publishing Inc.}, type = {Book chapter}, address = {Singapore}, url = {http://rpsonline.com.sg/rpsweb/iwcia09.html}, author = {G{\'a}bor N{\'e}meth and K{\'a}lm{\'a}n Pal{\'a}gyi}, editor = {Petra Wiederhold and Reneta P Barneva} } @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} } @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} } @inbook {1134, title = {Reconstruction of canonical hv-convex discrete sets from horizontal and vertical projections}, booktitle = {Combinatorial Image Analysis}, number = {5852}, year = {2009}, note = {UT: 000279344100022ScopusID: 78650444641doi: 10.1007/978-3-642-10210-3_22}, month = {Nov 2009}, pages = {280 - 288}, publisher = {Springer Verlag}, organization = {Springer Verlag}, type = {Conference paper}, address = {Berlin; Heidelberg; New York; London; Paris; Tokyo}, abstract = {The problem of reconstructing some special hv-convex discretesets from their two orthogonal projections is considered. In general, the problem is known to be NP-hard, but it is solvable in polynomial time if the discrete set to be reconstructed is also 8-connected. In this paper, we define an intermediate class - the class of hv-convex canonical discrete sets - and give a constructive proof that the above problem remains computationally tractable for this class, too. We also discuss some further theoretical consequences and present experimental results as well. {\textcopyright} Springer-Verlag Berlin Heidelberg 2009.

}, isbn = {978-3-642-10208-0}, doi = {10.1007/978-3-642-10210-3_22}, author = {P{\'e}ter Bal{\'a}zs}, editor = {Petra Wiederhold and Reneta P Barneva} } @inbook {1138, title = {On the number of hv-convex discrete sets}, booktitle = {Combinatorial Image Analysis}, series = {Lecture Notes in Computer Science}, number = {4958}, year = {2008}, note = {UT: 000254600100010ScopusID: 70249110264doi: 10.1007/978-3-540-78275-9_10}, month = {Apr 2008}, pages = {112 - 123}, publisher = {Springer Verlag}, organization = {Springer Verlag}, type = {Conference paper}, address = {Buffalo, NY, USA}, abstract = {One of the basic problems in discrete tomography is thereconstruction of discrete sets from few projections. Assuming that the set to be reconstructed fulfills some geometrical properties is a commonly used technique to reduce the number of possibly many different solutions of the same reconstruction problem. The class of hv-convex discrete sets and its subclasses have a well-developed theory. Several reconstruction algorithms as well as some complexity results are known for those classes. The key to achieve polynomial-time reconstruction of an hv- convex discrete set is to have the additional assumption that the set is connected as well. This paper collects several statistics on hv-convex discrete sets, which are of great importance in the analysis of algorithms for reconstructing such kind of discrete sets. {\textcopyright} 2008 Springer-Verlag Berlin Heidelberg.

}, isbn = {978-3-540-78274-2}, doi = {10.1007/978-3-540-78275-9_10}, author = {P{\'e}ter Bal{\'a}zs}, editor = {Valentin E Brimkov and Reneta P Barneva and Herbert A Hauptman} }