Abstract
This paper investigates two constraints for the connected operator class. For binary images, connected operators are those that treat grains and pores of the input in an all or nothing way, and therefore they do not introduce discontinuities. The first constraint, called connected-component (c.c.) locality, constrains the part of the input that can be used for computing the output of each grain and pore. The second, called adjacency stability, establishes an adjacency constraint between connected components of the input set and those of the output set. Among increasing operators, usual morphological filters can satisfy both requirements. On the other hand, some (non-idempotent) morphological operators such as the median cannot have the adjacency stability property. When these two requirements are applied to connected and idempotent morphological operators, we are lead to a new approach to the class of filters by reconstruction. The important case of translation invariant operators and the relationships between translation invariance and connectivity are studied in detail. Concepts are developed within the binary (or set) framework; however, conclusions apply as well to flat non-binary (gray-level) operators.
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References
G. Birkhoff, Lattice Theory, American Mathematical Society, Providence, 1984.
J. Crespo, “Morphological connected filters and intra-region smoothing for image segmentation,” PhD. Thesis, School of Electrical Engineering, Georgia Institute of Technology, Dec. 1993.
J. Crespo, “Space connectivity and translation-invariance,” in Workshop on Mathematical Morphology and Its Applications to Image Processing, Atlanta, May 1996.
J. Crespo, J. Serra, and R. Schafer, “Theoretical aspects of morphological filters by reconstruction,” Signal Processing, Vol. 47, No.2, pp. 201–225, Nov. 1995.
J. Crespo, J. Serra, and R. Schafer, “Image segmentation using connected filters,” in Workshop on Mathematical Morphology, Barcelona, pp. 52–57, May 1993.
C. Giardina and E. Dougherty, Morphological Methods in Image and Signal Processing, Prentice-Hall: Englewood Clliffs, 1988.
H. Heijmans, “Theoretical aspects of gray-level morphology,” IEEE Trans. Pattern Anal. Machine Intell., Vol. 13, pp. 568–582, 1991.
H. Heijmans, Morphological Image Operators (Advances in Electronics and Electron Physics, P. Hawkes (Ed.), Academic Press: Boston, 1994.
H. Heijmans and C. Ronse, “The algebraic basis of mathematical morphology, part I: Dilations and erosions,” Comput. Vision, Graphics and Image Processing, Vol. 50, pp. 245–295, 1990.
C. Lantuéjoul and S. Beucher, “On the use of the geodesic metric in image analysis,” J. Microsc., Vol. 121, pp. 39–49, 1981.
P. Maragos, “A representation theory for morphological image and signal processing,” IEEE Trans. Pattern Anal. Machine Intell., Vol. 11, pp. 586–599, June 1989.
P. Maragos and R. Schafer, “Morphological filters—Part I: Their set-theoretic analysis and relations to linear-shift-invariant filters,” IEEE Trans. Acoust. Speech Signal Processing, Vol. 35, pp. 1153–1169, Aug. 1987.
P. Maragos and R. Schafer, “Morphological filters—Part II: Their relations to median, order-statistic, and stack filters,” IEEE Trans. Acoust. Speech Signal Processing, Vol. 35, pp. 1170–1184, Aug. 1987.
P. Maragos and R. Schafer, “Morphological systems for multidimensional signal processing,” Proc. of the IEEE, Apr. 1990, Vol. 78, No.4, pp. 690–710.
G. Matheron, Éléments pour une Théorie des Milieux Poreux. Masson: Paris, 1965.
G. Matheron, Random Sets and Integral Geometry, Wiley: New York, 1975.
G. Matheron, Les Applications Idempotentes, Report Centre de Géostatistique et de Morphologie Mathématique, E.N.S. des Mines de Paris, 1982.
G. Matheron, Surpotentes et Sous-potentes, Report Centre de Géostatistique et de Morphologie Mathématique, E.N.S. des Mines de Paris, 1982.
G. Matheron, Filters and Lattices, Report Centre de Géostatistique et de Morphologie Mathématique, E.N.S. des Mines de Paris, 1983.
G. Matheron, “Filters and lattices,” in Mathematical Morphology Volume II: Theoretical Advances, J. Serra (Ed.), Academic Press: London, Chap. 6, pp. 115–140, 1988.
G. Matheron and J. Serra, “Strong filters and connectivity,” in: Mathematical Morphology Volume II: Theoretical Advances, J. Serra (Ed.), Academic Press: London, Chap. 7, pp. 141–157, 1988.
J. Serra, Mathematical Morphology, Academic Press: London, Vol. I, 1982.
J. Serra, Éléments de Théorie pour l’Optique Morphologique, Thèse d’État, Université de Paris VI, 1986.
J. Serra, Mathematical Morphology. Volume II: Theoretical Advances, Academic Press: London, 1988.
J. Serra, “Anamorphoses and function lattices,” in Mathematical Morphology in Image Processing, E. Dougherty (Ed.), Marcel Dekker: New York, Chap. 13, pp. 483–523, 1993.
J. Serra and P. Salembier, “Connected operators and pyramids,” in Proceedings of SPIE, Non-Linear Algebra and Morphological Image Processing, San Diego, July 1993, Vol. 2030, pp. 65–76.
J. Serra and P. Salembier, “Opérateurs connexes et pyramides,” in 9ème Congrès RFIA, AFCET/AFIA, Paris, Janvier, Vol. 1, pp. 243–254, 1994.
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Crespo, J., Schafer, R.W. Locality and Adjacency Stability Constraints for Morphological Connected Operators. Journal of Mathematical Imaging and Vision 7, 85–102 (1997). https://doi.org/10.1023/A:1008270125009
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DOI: https://doi.org/10.1023/A:1008270125009