Random Sets pp 73-95 | Cite as

# Logical Granulometric Filtering in the Signal—Union—Clutter Model

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## Abstract

A basic problem of binary morphological image filtering is to remove background clutter (noise) in order to reveal a desired target (signal). The present paper discusses the manner in which filtering can be achieved using morphological granulometric filters. Logical granulometries are unions of intersections of reconstructive openings and these use shape elements to identify image components to be passed (in full), whereas others are deleted. Assuming opening structuring elements are parameterized, the task is to find parameters that result in optimal filtering. Optimization is achieved via the notion of granulometric sizing. For situations where optimization is impractical or intractable, filter design can be achieved via adaptation. Based upon correct or incorrect decisions as to whether or not to pass a component, the filter parameter vector is adapted during training in accordance with a protocol that adapts towards correct decisions. The adaptation scheme yields a Markov chain in which the parameter space is the state space of the chain. Convergence of the adaptation procedure is characterized by the stationary distribution of the parameter vector. State-probability equations are derived via the Chapman-Kolmogorov equations and these are used to describe the steady-state distribution.

## Key words

Mathematical Morphology Logical Granulometries Size Distribution Optimal Morphological Filtering Adaptive Morphological Filtering Markov Chains## Preview

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## References

- [1]G. Matheron
*Random Sets and Integral Geometry*John Wiley, New York, 1975.zbMATHGoogle Scholar - [2]E.R. Dougherty, R.M. Haralick, Y. Chen, C. Agerskov, U. Jacobi, and P.H. Sloth
*Estimation of optimal T-opening parameters based on independent observation of signal and noise pattern spectra*Signal Processing, 29 (1992) pp. 265–281.zbMATHCrossRefGoogle Scholar - [3]E.R. Dougherty and C. Cuciurean-Zapan
*Optimal reconstructive T-openings for disjoint and statistically modeled nondisjoint grains*Signal Processing, 56 (1997), pp. 45–58.zbMATHCrossRefGoogle Scholar - [4]Y. Chen and E.R. Dougherty
*Adaptive reconstructive T-openings: Convergence and the steady-state distribution*Electronic Imaging, 5 (1996), pp. 266–282.CrossRefGoogle Scholar - [5]Y. Chen and E.R. Dougherty
*Markovian analysis of adaptive reconstructive multiparameter r-openings*Journal of Mathematical Imaging and Vision (submitted).Google Scholar - [6]E.R. Dougherty, J.T. Newell, and J.B. Pelz
*Morphological texture-based maximum-likelihood pixel classification based on local granulometric moments*Pattern Recognition, 25 (1992), pp. 1181–1198.CrossRefGoogle Scholar - [7]E.R. Dougherty and F. Sand
*Representation of linear granulometric moments for deterministic and random binary Euclidean images*Journal of Visual Communication and Image Representation, 6 (1995), pp. 69–79.CrossRefGoogle Scholar - [8]S. Batman and E.R. Dougherty
*Size distributions for multivariate morphological granulometries: Texture classification and statistical properties*Optical Engineering, 36 (1997), pp. 1518–1529.CrossRefGoogle Scholar