Abstract
The connection between morphological and stack filters is used in the analysis of the statistical properties of morphological filters. Closed-form expressions for the output distributions of morphological filters are given, and their statistical symmetry properties are analyzed. Asytotically tight bounds on the expectations of two-dimensional morphological filters, and asymptotic formulas for the variances of one-dimensional morphological filters are derived. These results form the basis for analyzing general asymptotic properties of morphological filters.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Stevenson and G. Arce, “Morphological filters: statistics and further syntactic properties, IEEE Trans. Circuits and Systems, vol. ASSP-34, pp. 1292–1305, 1987.
D. Schonfeld and J. Goutsias, “Optimal morphological pattern restoration from noisy binary images,” IEEE Trans. Patt. Anal. Mach. Intell., vol. PAMI-13, pp. 14–29, 1990.
P. Maragos and R. Schafer, “Morphological filters—part II: their relations to median, order-statistics, and stack filters,” IEEE Trans. Acoust., Speech, Signal Process. vol. ASSP-35, pp. 1170–1184, 1987.
L. Koskinen, J. Astola, and Y. Neuvo, “Morphological filtering of noisy images,” in Visual Communications and Image Processing'90: Fifth in a Series, M. Kunt, ed., Proc. Soc. Photo-Opt. Instrum. Eng., vol. 1360, pp. 155–165, 1990.
L. Koskinen, J. Astola, and Y. Neuvo, “Analysis of noise attenuation in morphological image processing,” in Nonlinear Image Processing II, G.R. Arce, C.G. Boncelet, and E.R. Dougherty, eds., Proc. Soc. Photo-Opt. Instrum. Eng., vol. 1451, pp. 102–113, 1991.
P. Rustanius, L. Koskinen, and J. Astola, “Theoretical and experimental analysis of the effects of noise in morphological image processing,” in Proc. SPIE Symp. on Image Algebra and Morphological Image Processing III, San Diego, CA, July 1992.
E. Dougherty and C. Ciardiana, Morphological Methods in Image and Signal Processing, Prentice-Hall: Englewood Cliffs, NJ, 1988.
S. Serra, Image Analysis and Mathematical Morphology, Academic Press: London, 1988.
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 Process. vol. ASSP-35, pp. 1153–1169, 1987.
G. Matheron, Random Sets and Integral Geometry, John Wiley: New York, 1975.
C. Chu and E. Delp, “Impulsive noise suppression and background normalization of electrocardiogram signals using morphological operators,” IEEE Trans. Biomed. Eng., vol. 36, pp. 226–273, 1989.
P. Wendt, E. Coyle, and N. Callager, “Stack filters,” IEEE Trans. Acoust., Speech, Signal Process. vol. ASSP-34, pp. 898–911, 1986.
O. Yli-Harja, J. Astola, and Y. Neuvo, “Analysis of the properties of median and weighted median filters using threshold logic and stack filter representation,” IEEE Trans. Signal Process., vol. 39, pp. 395–410, 1991.
L. Koskinen, J. Astola, and Y. Neuvo, “Statistical properties of discrete morphological filters,” in Proc. IEEE Int. Symp. on Circuits and Systems, New Orleans, LA, May 1990, pp. 1219–1222.
E. Castillo, Extreme Value Theory in Engineering, Academic Press: London, 1988.
J. Galambos, The Asymptotic Theory of Order Statistics, John Wiley: New York, 1978.
L. Koskinen and J. Astola, “Statistical properties of soft morphological filters,” in Proc. SPIE Symp. on Nonlinear Image Processing III, San Jose, CA, February 1992.
J. Astola and Y. Neuvo, “An efficient tool for analyzing weighted median and stack filters,” submitted to IEEE Trans. Circuits and Systems.
B. Justusson, “Median filtering: statistical properties,” in Topics in Applied Physics, Two Dimensional Digital Signal Processing II, T.S. Huang, ed., Springer-Verlag: Berlin, 1981.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Koskinen, L., Astola, J. Asymptotic behavior of morphological filters. J Math Imaging Vis 2, 117–135 (1992). https://doi.org/10.1007/BF00118585
Issue Date:
DOI: https://doi.org/10.1007/BF00118585