Abstract
Parallel reductions transform binary pictures only by changing a set of black points to white ones simultaneously. Topology preservation is a major concern of some topological algorithms composed of parallel reductions. For 3D binary pictures sampled on the body-centered cubic (BCC) grid, we propose a new sufficient condition for topology-preserving parallel reductions. This condition takes some configurations of deleted points into consideration, and it provides a method of verifying that formerly constructed parallel reductions preserve the topology. We present two further sufficient conditions that investigate individual points, directly provide deletion rules of topology-preserving parallel reductions, and allow us to construct parallel thinning algorithms.
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References
Bertrand, G., Couprie, M.: On parallel thinning algorithms: minimal non-simple sets, P-simple points and critical kernels. J. Math. Imaging Vis. 35, 23–35 (2009). https://doi.org/10.1007/s10851-009-0152-3
Čomić, L., Nagy, B.: A combinatorial coordinate system for the body-centered cubic grid. Graph. Models 87, 11–22 (2016). https://doi.org/10.1016/j.gmod.2016.08.001
Csébfalvi, B.: Cosine-weighted B-spline interpolation: a fast and high-quality reconstruction scheme for the body-centered cubic lattice. IEEE Trans. Visual. Comput. Graph. 19, 1455–1466 (2013). https://doi.org/10.1109/TVCG.2013.7
Hall, R.W.: Parallel connectivity-preserving thinning algorithms. In: Kong, T.Y., Rosenfeld, A. (eds.) Topological Algorithms for Digital Image Processing, pp. 145–179, Elsevier Science (1996). https://doi.org/10.1016/S0923-0459(96)80014-0
Kardos, P.: Topology preservation on the BCC grid. J. Comb. Optim. 44, 1981–2995 (2021). https://doi.org/10.1007/s10878-021-00828-9
Kardos, P., Palágyi, K.: Topology-preserving hexagonal thinning. Int. J. Comput. Math. 90, 1607–1617 (2013). https://doi.org/10.1080/00207160.2012.724198
Kardos, P., Palágyi, K.: Topology preservation on the triangular grid. Ann. Math. Artif. Intell. 75, 53–68 (2015). https://doi.org/10.1007/s10472-014-9426-6
Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Visi. Graph. Image Process. 48, 357–393 (1989). https://doi.org/10.1016/0734-189X(89)90147-3
Kong, T.Y.: On topology preservation in 2-D and 3-D thinning. Int. J. Pattern Recognit Artif Intell. 9, 813–844 (1995). https://doi.org/10.1142/S0218001495000341
Ma, C.M.: On topology preservation in 3D thinning. CVGIP: Image Underst. 59, 328–339 (1994). https://doi.org/10.1006/ciun.1994.1023
Matej, S., Lewitt, R.M.: Efficient 3D grids for image reconstruction using spherically-symmetric volume elements. IEEE Trans. Nucl. Sci. 42, 1361–1370 (1995). https://doi.org/10.1109/23.467854
Németh, G., Palágyi, K.: Topology-preserving hexagonal thinning. Int. J. Comput. Math. 90, 1607–1617 (2013). https://doi.org/10.1007/s10472-014-9426-6
Palágyi, K., Németh, G., Kardos, P.: Topology preserving parallel 3D thinning algorithms. In: Brimkov, V.E., Barneva, R.P. (eds.) Digital Geometry Algorithms: Theoretical Foundations and Applications to Computational Imaging, pp. 165–188. Springer, Heidelberg (2012). https://doi.org/10.1007/978-94-007-4174-4_6
Ronse, C.: Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images. Discrete Appl. Math. 21, 67–79 (1988). https://doi.org/10.1016/0166-218X(88)90034-0
Strand, R.: Surface skeletons in grids with non-cubic voxels. In Proceedings of 17th International Conference on Pattern Recognition, ICPR 2004, pp. 548–551 (2004). https://doi.org/10.1109/ICPR.2004.1334195
Strand, R., Brunner, D.: Simple points on the body-centered cubic grid. Technical report 42, Centre for Image Analysis, Uppsala University, Uppsala, Sweden (2006)
Strand, R., Nagy, B.: Weighted neighbourhood sequences in non-standard three-dimensional grids - Metricity and algorithms. In Proceedings of 14th IAPR International Conference on Discrete Geometry for Computer Imagery, DGCI 2008, pp. 201–212 (2008). https://doi.org/10.1007/978-3-540-79126-3_19
Theussl, T., Möller, T., Grölle, M.E.: Optimal regular volume sampling. In Proceedings of IEEE Visualization, VIS 2001, pp. 91–98 (2001). https://doi.org/10.1109/VISUAL.2001.964498
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Project no. TKP2021-NVA-09 has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme.
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Palágyi, K., Karai, G., Kardos, P. (2023). Sufficient Conditions for Topology-Preserving Parallel Reductions on the BCC Grid. In: Barneva, R.P., Brimkov, V.E., Nordo, G. (eds) Combinatorial Image Analysis. IWCIA 2022. Lecture Notes in Computer Science, vol 13348. Springer, Cham. https://doi.org/10.1007/978-3-031-23612-9_5
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