Image Processing Reference
In-Depth Information
9
Fuzzy Math
ematical Morphology
9.1 Introduction
Mathematical morphology was born in 1964 from the collaborative work of
Georges Matheron and Jean Serra at the École des Mines de Paris, France. It
is a theory to analyse and process geometrical structures based on set theory,
lattice theory, topology and random functions. It is a tool for extracting dif-
ferent image components that are useful in the representation and descrip-
tion of image regions, boundaries, shapes or skeletons. Initially between the
1960s and 1970s, mathematical morphology dealt with binary images and
many binary operators were introduced such as erosion, dilation, opening,
closing, skeletonization, hit or miss transform. Later on in the mid-1970s and
mid-1980s, it was generalized to greyscale images that require more sophis-
ticated mathematical operations. Simultaneously the operators are extended
to new operators. Consequently, mathematical morphology gained much
recognition and is used widely in the image processing application.
Mathematical morphology is described almost entirely by set operation
such as union, intersection, difference and complement. Set is a collection
of pixels in an image. Binary morphology depends only on set membership
and does not take into account the grey value or colour of the image pixel.
9.2 Preliminaries on Morphology
As the chapter is related to fuzzy morphology, some basics of morphology
are discussed before detailing fuzzy morphology. Dilation and erosion are
the basic morphological processing operations. Both dilation and erosion are
produced by the interaction of a set called structuring element that has shape
and origin and can be described by many ways such as circular, pyramid,
linear and square. Dilation is a dual operation of erosion. Dilation dilates
the objects and closes holes and gaps of certain shapes and sizes, given by a
structuring element. It is an operation that combines two sets using vector
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