Abstract
We first give a definition of simple sets of 1’s in 4D binary images that is consistent with “(8,80)-adjacency”—i.e., the use of 8-adjacency to define connectedness of sets of 1’s and 80-adjacency to define connectedness of sets of 0’s. Using this definition, it is shown that in any 4D binary image every minimal non-simple set of 1’s must be isometric to one of eight sets, the largest of which has just four elements. Our result provides the basis for a fairly general method of verifying that proposed 4D parallel thinning algorithms preserve topology in our “(8,80)” sense. This work complements the authors’ earlier work on 4D minimal non-simple sets, which essentially used “(80,8)-adjacency”—80-adjacency on 1’s and 8-adjacency on 0’s.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bertrand, G.: On P-simple points. C. R. Acad. Sci. Paris, Série I 321, 1077–1084 (1995)
Bertrand, G.: A Boolean characterization of three-dimensional simple points. Pattern Recogn. Lett. 17, 115–124 (1996)
Gau, C.J., Kong, T.Y.: Minimal nonsimple sets of voxels in binary images on a face-centered cubic grid. Int. J. Pattern Recogn. Artif. Intell. 13, 485–502 (1999)
Gau, C.J., Kong, T.Y.: Minimal nonsimple sets in 4D binary images. Graph. Models 65, 112–130 (2003)
Hall, R.W.: Tests for connectivity preservation for parallel reduction operators. Topology and Its Applications 46, 199–217 (1992)
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/North-Holland (1996)
Kinsey, L.C.: Topology of Surfaces. Springer, Heidelberg (1993)
Klette, R., Zamperoni, P.: Handbook of Image Processing Operators. Wiley, Chichester (1996)
Kong, T.Y.: On the problem of determining whether a parallel reduction operator for n-dimensional binary images always preserves topology. In: Melter, R.A., Wu, A.Y. (eds.) Vision Geometry II, Proceedings, Boston, September 1993. Proc. SPIE, vol. 2060, pp. 69–77 (2060)
Kong, T.Y.: On topology preservation in 2D and 3D thinning. Int. J. Pattern Recogn. Artif. Intell. 9, 813–844 (1995)
Kong, T.Y.: Topology preserving deletion of 1’s from 2-, 3- and 4-dimensional binary images. In: Ahronovitz, E. (ed.) DGCI 1997. LNCS, vol. 1347, pp. 3–18. Springer, Heidelberg (1997)
Kong, T.Y., Rosenfeld, A.: Digital topology: Introduction and survey. Computer Vision, Graphics, and Image Processing 48, 357–393 (1989)
Ma, C.M.: On topology preservation in 3D thinning. CVGIP: Image Understanding 59, 328–339 (1994)
Ronse, C.: Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images. Discrete Appl. Math. 21, 67–79 (1988)
Rosenfeld, A.: A characterization of parallel thinning algorithms. Information and Control 29, 286–291 (1975)
Spanier, E.H.: Algebraic Topology. Springer, Heidelberg (1989)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kong, T.Y., Gau, CJ. (2004). Minimal Non-simple Sets in 4-Dimensional Binary Images with (8,80)-Adjacency. In: Klette, R., Žunić, J. (eds) Combinatorial Image Analysis. IWCIA 2004. Lecture Notes in Computer Science, vol 3322. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30503-3_24
Download citation
DOI: https://doi.org/10.1007/978-3-540-30503-3_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23942-0
Online ISBN: 978-3-540-30503-3
eBook Packages: Computer ScienceComputer Science (R0)