3. Soft mathematical morphology (Koskinen, et al.,
1991, Kuosmanen and
Astola, 1995)
In soft mathematical
morphology the structuring element is divided into two parts: the core and the soft boundary. The basic concept is that the min/max operators
which are used in standard morphology are substituted by other order
statistics. The pixels of the core have weight k greater than or equal to 1 and the pixels of the soft boundary
have weight equal to 1. This means that the results (differences or sums)
involving the pixels of the core are repeated k times. The parameter k
is called repetition parameter. It has
been proven that soft morphological operations are advantageous regarding
additive noise elimination and sensitivity to small variations in object shape,
when compared to standard morphological transforms. In the literature the
definitions of grey-scale soft morphology were reported first. Thus, we present
them first here.
3.1 Grey-scale soft
morphology
Let 
denote the k-times repetition of x. The core and the soft boundary of the
structuring element g (z) are denoted by á(zá) and â(zâ),
respectively, where
and \ denotes the set difference.
Soft Erosion
The soft erosion
of f by a structuring element g (with core á and soft boundary b) at
a point x is defined as follows:
where min(k) is the kth smallest of the multi-set in the parentheses. A
multi-set is a collection of objects, where the repetition of objects is
allowed.
This means that
in order to perform the soft erosion of f
by g, we translate spatially g by x
(so that its origin is located at the point x)
and then:
(i) we find all
differences of values of f with the
corresponding values of the translated core á,
.
(ii) we find all
differences of values of f with the
corresponding values of the translated soft boundary â,
(iii) we
classify all differences in ascending order (the differences involving the
values of the core repeated k times).
The kth smallest of these
differences constitutes 
.
Figure 9 depicts the soft erosion of grey-scale image f by a grey-scale structuring element g.
Figure 9. f(1,1)=7, x=(1,1) and (f È[â,á,2])(1,1)=2
Indeed, according to eqn (34): 
Soft Dilation
The soft
dilation of f by a structuring element g (with core á and soft boundary â) at
a point x is defined as follows:
(35)
where: min(k) is the kth smallest of the multi-set in the parentheses.
This means that in order to perform the soft dilation of f by g at point x, we translate spatially the reflection of g, g (-z), so that its origin is located at point x and then we follow the same steps of the soft erosion case. The difference is that we calculate sums instead of differences and that we finally seek for the kth largest of these sums. Figure 10 depicts the soft dilation of grey-scale image f by a grey-scale structuring element g.
In the case of k=1 or D[á]=D[g], it is obvious
from definitions (34) and (35) that soft erosion and soft dilation are identical
to standard erosion and standard dilation respectively.
Figure 10. f(1,2)=7,
x=(1,2) and 
Indeed, according to eqn (35): 
Soft erosion and
dilation possesses all basic properties mentioned in section 2.1. Thus, soft
erosion is anti-extensive and soft dilation is extensive, provided that the
structuring element includes the origin. In particular the smaller the
repetition parameter k is the more
the input image shrinks or expands. Soft erosion is dual to soft dilation and
vice versa. Furthermore, both soft erosion and soft dilation are translation
invariant and monotonically increasing.
Soft Opening and Soft Closing
Soft opening and
soft closing are defined as follows:
(36)
(37)
Soft opening is
dual to soft closing and vice versa, but unlike standard opening and closing,
soft opening and closing are neither extensive nor anti-extensive. In general,
both soft opening and soft closing are not idempotent. It has been proven that
soft opening and closing are idempotent only under specific conditions (e.g. in
case that g=gS and the core includes only one pixel, which is
located at the origin).
3.2 Binary soft morphology (Pu and Shih,
1995)
In this section
the core and the soft boundary of the structuring element B are denoted by B1
and B2, respectively 
.
Binary soft erosion
and dilation are defined as follows:
(38)
(39)
Card[X] denotes the cardinality (the number of
elements) of set X.
Definition (38) implies
that in order to form soft erosion of a binary input image A by a binary
structuring element B we translate B by x (so that its origin is located at
point x). Next we find the cardinality of the intersection of A with the
translated core B1, we multiply it with k and we finally add it to the cardinality of the intersection of A
with the translated soft boundary B2. If the result is greater than
or equal to 
, the eroded image has the value 1 at point x. Definition
(39) has a similar meaning. Figures 11 and 12 illustrate examples of binary
soft erosion and dilation, respectively.
Figure 11. k=2, Card[B1]=2, Card[B2]=2 and k / Card[B1]+Card[B2]-k+1 = 2 / 2 + 2 - 2 + 1 = 5
For x=(1,0): 
and, therefore, 
.
For x=(1,1): 
and, therefore, 
. In a similar way we proceed with any x.
Figure 12. k=2
For x=(1,0): 
and, therefore, 
.
For x=(1,-1): 
and, therefore, 
.
For x=(5,3): 
and, therefore, 
. In a similar way we proceed with any x.
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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.