1.
Introduction
As a discipline mathematical
morphology has its roots in the pioneering work of G. Matheron (1975) and J. Serra (1982). It is a powerful tool for solving problems ranging
over the entire imaging spectrum, including character recognition, medical
imaging, microscopy, inspection, metallurgy and robot vision (Matheron,
1975, Serra, 1982, Dougherty
and Astola, 1994, Gonzalez and Woods,
1992, Haralick and Shapiro,
1992, Pitas
and Venetsanopoulos, 1990, Serra, 1989,
Serra and Soille, 1994, Maragos, et
al., 1996, Heijmans and Roerdink,
1998). Morphology is now a necessary tool for engineers
involved with imaging applications. Morphological operations have been viewed
as filters the properties of which have been well studied (Heijmans,
1994). Another well-known class of non-linear filters is
the class of rank order filters (Pitas
and Venetsanopoulos, 1990). Soft morphological filters are a combination of
morphological and weighted rank order filters (Koskinen,
et al., 1991, Kuosmanen and Astola,
1995). They have been introduced to improve the behaviour
of traditional morphological filters in noisy environments. The idea was to
slightly relax the typical morphological definitions in such a way that a
degree of robustness is achieved, while most of the desirable properties of
typical morphological operations are maintained. Soft morphological filters are
less sensitive to additive noise and to small variations in object shape than
typical morphological filters. They can remove positive and negative impulse
noise, preserving at the same time small details in images.
The basic definitions and properties of standard and soft morphological
operations for both binary and grey-scale images are presented in this paper.
Several graphical illustrative examples are also included. The rest of the
paper is organised as follows. In section 2 the fundamental operations of binary
and grey-scale morphology as well as their basic properties are discussed. In
section 3 the soft morphological operations along with their basic properties
are presented. Concluding remarks are made in section
4.
G. Matheron (1975): Random
Sets and Integral Geometry, Wiley, New York.
J.
Serra (1982): Image Analysis and
Mathematical Morphology, Academic Press, London.
R.
C. Gonzalez and R. E. Woods (1992): Digital
Image Processing, Addison-Wesley, New York.
R.
M. Haralick and L. G. Shapiro (1992): Computer
and Robot Vision, Addison-Wesley, New York.
H. J.
A. M. Heijmans (1994): Morphological
Image Operators, Academic Press, New York.