Abstract
Morphological operators provide very efficient algorithms for signal (image) processing. The efficiency of morphological operators has been captured by using them as approximations of nonlinear operators in numerous applications (e.g., image restoration). Our approach to the approximation of nonlinear operators is the construction of morphological bounds on them. We present a general theory on the morphological bounds on nonlinear operators, propose conditions for the existence of these bounds, and derive several fundamental morphological bounds.
We also derive morphological bounds on iterations of nonlinear operators, which are superior to the original nonlinear operator in some applications. Because obtaining the results of the convergence of iterations of a nonlinear operator is often particularly desirable, we provide morphological bounds on the convergence of such iterations, and propose conditions for their convergence based on morphological properties. Finally, we propose several criteria for the morphological characterization of roots of nonlinear operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Matheron,Random Sets and Integral Geometry, Wiley: New York, 1975.
J. Serra,Image Analysis and Mathematical Morphology, vol. 1, Academic Press: San Diego, CL, 1982.
J. Serra (ed.),Image Analysis and Mathematical Morphology, vol. 2, Academic Press: New York, 1988.
J. Serra, “Introduction to mathematical morphology,”Comput. Vis. Graph. Image Process., vol. 35, pp. 283–305, 1986.
R.M. Haralick, S.R. Sternberg, and X. Zhuang, “Image analysis using mathematical morphology,”IEEE Trans. Patt. Anal. Mach. Intell., vol. PAMI-9, pp. 532–550, 1987
C.R. Giardina and E.R. Dougherty,Morphological Methods in Image and Signal Processing, Prentice-Hall: Englewood Cliffs, NJ, 1988.
J. Serra and L. Vincent,Lecture Notes on Morphological Filtering, Centre de Morphology Mathematique, Ecole Nationale Superieure des Mines de Paris: Fontainebleau Cedex, France, 1989.
P.A. Maragos, “A unified theory of translation-invariant systems with applications to morphological analysis and coding of images,” Ph.D. dissertation, School of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA, 1985.
P. Marogos and R.W. Schafer, “Morphological filters, part I: Their set-theoretic analysis and relations to linear shiftinvariant filters,”IEEE Trans. Acoust., Speech, Signal Process. vol. ASSP-35, pp. 1153–1169, 1987.
P. Maragos and R.W. 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.
P. Maragos, “A representation theory for morphological image and signal processing,”IEEE Trans. Patt. Anal. Mach. Intell. vol. PAMI-11, pp. 701–716. 1989.
M. Charif-Chefchaouni and D. Schonfeld, “Morphological bounds on order-statistics filtres,” inProc. Soc. Photo-Opt. Instrum. Eng., vol. 2030, 1993, pp. 24–32.
T.S. Huang (ed.),Two Dimensional Digital Signal Processing II: Transforms and Median Filters Springer-Verlag: New York, 1981.
N.C. Gallagher and G.L. Wise, “A theoretical analysis of the properties of median filters,”IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-29, pp. 1136–1141, 1981.
T.A. Nodes and N.C. Gallagher, “Median filters: some modifications and their properties,”IEEE Trans. Acoust., Speech, Signal Processing vol. ASSP-30, pp. 739–746, 1982.
T.A. Nodes and N.C. Gallagher, “Two-dimensional root structure and convergence properties of the separable median filter,”IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-31, pp. 1350–1365, 1983.
J.P. Fitch, E.J. Coyle, and N.C. Gallagher, “Root Properties and convergence rates of median filters,”IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-33, pp. 230–240, 1985.
H.D. Tagare and R.J.P. De Figueredo, “Order filters,”Proc. IEEE, vol. 73, pp. 163–165, 1985.
H.P. Kramer and J.B. Brucknert, “Iterations of a nonlinear transformation for enhancement of digital pictures,”Patt. Recog., vol. 7, pp. 53–58, 1975.
Y.H. Lee, S.J. Ko, and A.T. Fam, “Efficient impulsive noise suppression via nonlinear recursive filtering,”IEEE Trans. Acoust., Speech, Signal Process. vol. ASSP-37, pp. 303–306, 1989.
D. Schonfeld, “Optimal morphological representation and restoration of binary images: theory and applications,” Ph.D. dissertation, Department of Electrical and Computer Engineering, John Hopkins University, Baltimore, MD, 1990.
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, 1991.
G. Birkhoff,Lattice Theory, 3rd ed., American Mathematical Society Colloquium Publications, vol. 25, American Mathematical Society: Providence, RI, 1984.
H.J.A.M. Heijmans, “Iterations of morphological transformations,”CWI Quart., vol. 2, pp. 19–36, 1989.
H.J.A.M. Heijmans and C. Ronse, “The algebraic basis of mathematical morphology, I: dilations and erosions,”Comput. Vis., Graph. Image Process., vol. 50, pp. 245–295, 1990.
C. Ronse and H.J.A.M. Heijmans, “The algebraic basis of mathematical morphology, II: openings and closings,”Comput. Vis., Graph., Image Process. vol. 54, pp. 74–97, 1992.
Author information
Authors and Affiliations
Additional information
This work was supported by U.S. Office of Naval Research award N00014-91-J-1725.
Rights and permissions
About this article
Cite this article
Charif-Chefchaouni, M., Schonfeld, D. Morphological representation of nonlinear filters. J Math Imaging Vis 4, 215–232 (1994). https://doi.org/10.1007/BF01254100
Issue Date:
DOI: https://doi.org/10.1007/BF01254100