Abstract
The characterization of the root-signal set of a nonlinear operator has proved to be a crucial step in understanding the utility and usefulness of the operator. The set of root signals constitutes the passband of the nonlinear operator, and the complement of this set represents the stopband of the operator. Knowledge of these two sets for all operators determines which one must be used for any particular task. In this paper we investigate the root signals of the basic morphological filters, we study the properties of these signals, and we derive a system of equations to compute the number of binary-root signals for these morphological filters with structuring element of width k and signals of length n. The derivation is based on the state description for these root signals. Simple recursive equations are derived for counting the number of root signals of opening, closing, open-closing, and clos-opening. An application example in which these root signals are used in block truncation coding for image compression is discussed.
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References
E.J. Coyle and J.H. Lin, “Stack filters and the mean absolute error criterion,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-36, pp. 1244–1254, 1988.
E.J. Coyle, J.H. Lin, and M. Gabbouj, “Optimal stack filtering and the estimation and structural approaches to image processing,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-37, pp. 2037–2066, 1989.
L. Yin, J. Astola, and Y. Neuvo, “Optimal weighted order statistic filter under the mean absolute error criterion,” in Proc. ICASSP 91, Int. Conf. on Acoustics, Speech, and Signal Processing, Toronto, May 1991, pp. 2539–2532.
L. Yin, J. Astola, and Y. Neuvo, “Adaptive stack filtering with application to image processing,” IEEE Trans. Signal Process., to be published.
E.R. Dougherty, “Minimal search for the optimal meansquare digital gray-scale morphological filter,” Visual Communications and Image Processing '90: Fifth in a Series, M. Kunt, ed., Proc. Soc. Photo-Opt. Instrum. Eng., vol. 1360, pp. 214–226, 1990.
J. Nieweglowski, L. Yin, M. Gabbouj, and Y. Neuvo, “Optimal weighted order statistic filters under structural constraints,” in Proc. 1992 IEEE Int. Symp. on Circuits and Systems, San Diego, CA, May 1992, pp. 2621–2624.
Q. Wang and Y. Neuvo, “On two-dimensional root structures of separable and cross median filters,” in Proc. 1991 IEEE Int. Symp. on Circuits and Systems, Singapore, June 1991, pp. 104–107.
P.T. Yu and E.J. Coyle, “Convergence behavior and N-roots of stack filters,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-38, pp. 1529–1544, 1990.
M. Gabbouj and E.J. Coyle, “Minimum mean absolute error stack filtering with structural constraints and goals,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-38, pp. 995–968, 1990.
P.T. Yu and E.J. Coyle, “On the existence and design or the best stack filter based associative memory,” IEEE Trans. Circuits and Systems II, vol. 39, pp. 171–184, 1992.
M. Gabbouj, P.T. Yu, and E.J. Coyle, “Convergence behavior and root signal sets of stack filters,” Circuits, Systems, and Signal Processing, Special Issue on Median and Morphological filtering, 1991.
P.T. Yu, W.L. Wang, and S.S. Hung. “Root properties of median filters under three appending strategies,” in Proc. 1991 IEEE Int. Symp. on Circuits and Systems, Singapore, June 1991.
G. Matheron, Random Sets and Integral Geometry, Wiley: New York, 1975.
J. Serra, Image Analysis and Mathematical Morphology, Academic Press: New York, 1982.
P.A. Maragos and R.W. 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.
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.
P.A. Maragos and R.W. Schafer, “Morphological filters—part II: their relations to median, order-statistic, and stack filters,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-35, pp. 1170–1184, 1987.
S.G. Tyan, “Median filtering: deterministic properties,” in Topics in Applied Physics, Two-Dimensional Digital Signal Processing II, T.S. Huang, ed., Springer: Berlin, 1981, pp. 197–217.
J.P. Fitch, E.J. Coyle, and N.C. GallagherJr., “Root properties and convergence rates of median filters,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-33, pp. 230–240, 1985.
G.R. Arce and N.C. Gallagher, “State description for the root-signal set of median filters,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-30, pp. 894–902, 1982.
Q. Wang, M. Gabbouj, and Y. Neuvo, “State description for the root signal sets of morphological filters,” in Proc. 1992 IEEE Int. Symp. on Circuits and Systems, San Diego, CA, May 1992, pp. 113–116.
Q. Wang, M. Gabbouj, and Y. Neuvo, “Root signal sets of morphological filters,” Electron. Lett., vol. 28, pp. 952–953, 1992.
E.N. Gilbert, “Lattice-theoretic properties of frontal switching functions,” J. Math. Phys., vol. 33, April 1954.
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.
R.L. Stevenson and G.R. Arce, “Morphological filters: statistics and further syntactic properties,” IEEE trans. Circuits and Systems, vol. CAS-34, pp. 1292–1305, 1987.
Q. Wang, M. Gabbouj, and Y. Neuvo, “Root properties of morphological filters,” submitted to IEEE Trans. Signal Process.
E.J. Delp and O.R. Mitchell, “Image compression using block truncation coding,” IEEE Trans. Commun., vol. COM-27, pp. 1335–1342, 1979.
G.R. Arce and N.C. Gallagher, “BTC image coding using median filter roots,” IEEE Trans. Commun., vol. COM-31, pp. 784–793, 1983.
B. Zeng, Q. Wang, and Y. Neuvo, “BTC image coding using two-dimensional median filter roots,” in Proc. 1991 IEEE Int. Symp. on Circuits and Systems, Singapore, June 1991, pp. 400–403.
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Wang, Q., Gabbouj, M. & Neuvo, Y. Root-signal sets of morphological filters and their use in variable-length BTC image coding. J Math Imaging Vis 2, 155–171 (1992). https://doi.org/10.1007/BF00118587
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DOI: https://doi.org/10.1007/BF00118587