Abstract
As known from the works of Serra, Ronse, and Haralick and Shapiro, the connectivity relations are found to be useful in filtering binary images. But it can be used also to find roadmaps in robot motion planning, i.e. to build discrete networks of simple paths connecting points in the robot’s configuration space capturing the connectivity of this space. This paper generalises and puts together the notion of a connectivity class and the notion of a separation relation. This gives an opportunity to introduce approximate epsilon-connectivity, and thus we show the relation between our approach and the Epsilon Geometry introduced by Guibas, Salesin and Stolfi. Ronse and Serra have defined connectivity analogues on complete lattices with certain properties. As a particular case of their work we consider the connectivity of fuzzy compact sets, which is a natural way to study the connectivity of greyscale images. This idea can be transferred also in planning robot trajectories in the presence of uncertainties. Since based on fuzzy sets theory, our approach is intuitively closer to the classical set oriented approach, used for binary images and robot path planning in known environment with obstacles.
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References
I. Bloch and H. Maître, Fuzzy mathematical morphologies: A comparative study, Pattern Recognition, Vol. 28(9), pp. 1341–1387, 1995.
I. Bloch, Fuzzy connectivity and mathematical morphology, Pattern Recognition Letters, Vol.14, 483–488, 1993.
L. Guibas, D. Salesin, J. Stolfi, Epsilon geometry: building robust algorithms from imprecise computations. Proc. 5th Annual ACM Symposium on Computational Geometry, pp. 100–111, 1983.
L. Guibas, D. Salesin, J. Stolfi, Constructing strongly convex approximate hulls with inaccurate primitives. Algorithmica, Vol. 9, pp. 534–560, 1993.
P. Diamond, P. Kloeden, Metric spaces of fuzzy sets, Fuzzy sets and systems, Vol. 35, pp. 241–249, 1990.
Haralick, R.M. and L. G. Shapiro, Computer and robot vision, volume 1. Addison-Wesley, 1992.
H. J. A. M. Heijmans. Morphological image operators, Academic Press, Boston 1994.
J.-C. Latombe, Robot motion planning. Kluwer Academic Publishers, Boston, 1991.
G. Matheron. Random sets and integral geometry, Wiley, New York 1975.
A. T. Popov, Morphological operations on fuzzy sets, Proc. 5th Int. Conf. on Image. Processing and its Applications, Edinburgh, pp. 837–840, 1995.
A. T. Popov, Convexity indicators based on fuzzy morphology, Pattern Recognition. Letters, 18(3), pp. 259–267, 1997.
C. Ronse, Openings: main properties, and how to construct them. Unpublished, 1990.
C. Ronse, Set theoretical algebraic approach to connectivity in continuous or digital spaces. Journal of Mathematical Imaging and Vision, 8, pp. 41–58, 1998.
A. Rosenfeld, The fuzzy geometry of image subsets. Pattern Recognition Letters, 2, pp. 311–317, 1984.
J. Serra. Image analysis and mathematical morphology, Academic Press, London 1982.
J. Serra. Mathematical morphology for complete lattices. In J. Serra, editor, Image. analysis and mathematical morphology, vol. 2, Academic Press, London 1988.
J. Serra, Connectivity on complete lattices. Journal of Mathematical Imaging and Vision, 9, pp. 231–251, 1998.
D. Sinha and E. R. Dougherty, Fuzzification of Set Inclusion, Theory and Applications. Fuzzy Sets and Systems, 55, pp. 15–42, 1993.
Werman M. and Peleg S., Min — max operators in texture analysis, IEEE Trans. on Pattern Analysis and Machine Intelligence, 7(2), pp. 730–733, 1985.
L. Zadeh, Fuzzy sets and their applications to cognitive processes. Academic Press, London, 1975.
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Popov, A.T. (2002). Approximate Connectivity and Mathematical Morphology. In: Goutsias, J., Vincent, L., Bloomberg, D.S. (eds) Mathematical Morphology and its Applications to Image and Signal Processing. Computational Imaging and Vision, vol 18. Springer, Boston, MA. https://doi.org/10.1007/0-306-47025-X_17
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DOI: https://doi.org/10.1007/0-306-47025-X_17
Publisher Name: Springer, Boston, MA
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