Shih_Image_Processing_and_Mathematical_Morpholo.pdf

20 Image Processing and Mathematical Morphology
corners rounded; however, when erosion is applied, the result is exactly a
shrunken version of the square.
Given a Cartesian grid, the two commonly used structuring elements are
the 4-connected (or cross-shaped) and 8-connected (or square-shaped) sets,
N 4 and N 8. They are given by
2.5 Opening and Closing
È0 1 0˘ È1 1 1˘
Í ˙ Í ˙
BN = 1 1 1 , B 1 1 1 .
4 Í ˙ N = 8 Í ˙
ÍÎ0 1 0˙˚ ÍÎ1 1 1˙˚
In practical applications, dilation and erosion pairs are combined in sequence:
either dilation of an image followed by erosion of the dilated result, or vice
versa. In either case, the result of iteratively applying dilations and erosions
is an elimination of specifi c image details whose sizes are smaller than the
structuring element without the global geometric distortion of unsuppressed
features. These properties were fi rst explored by Matheron [1975] and Serra
[1982]. Their defi nitions for both opening and closing are identical to the ones
given here, but their formulas appear different because they use the symbol
� to mean Minkowski subtraction rather than erosion. The algebraic basis of
morphological opening and closing has been further developed by Heijmans
and Ronse [1991].
Defi nition 2.4: The opening of an image A by a structuring element B,
denoted by A ° B, is defi ned as
A ° B = (A � B) � B. (2.17)
Defi nition 2.5: The closing of an image A by a structuring element B,
denoted by A • B, is defi ned as
A • B = (A � B) � B. (2.18)
Note that in opening and closing, the symbols � and � can respectively
represent either binary or grayscale dilation and erosion. Grayscale morphological
operations will be presented in the next chapter. Equivalently, opening
can be expressed as the union of all translations of B that are contained in A
and is formulated as
A�B = ∪ By.
ByÕA (2.19)