Gray

Scale
Morphological Filtering
•
Generalization from binary to gray level
•
Use f(x,y) and b(x,y) to denote an image and a
structuring element
•
Gray

scale dilation of
f
by
b
–
(f
⊕b)(s,t)=max{f(s

x,t

y)+b(x,y)(s

x), (t

y)
D
f
and (x,y)
D
b
}
–
D
f
and D
b
are the domains of f and b respectively
–
(f
⊕b) chooses the maximum value of (f+ ) in the interval defined by ,
where is structuring element after rotation by 180 degree ( )
•
Similar to the definition of convolution with
–
The max operation replacing the summation and
–
Addition replacing the product
•
b(x,y) functions as the mask in convolution
–
It needs to be rotated by 180 degree first
b
ˆ
b
ˆ
b
ˆ
)
,
(
ˆ
y
x
b
b
Gray

Scale Dilation
•
Illustrated in 1D
–
(f
⊕b)(s)=max{f(s

x)+b(x)(s

x)
D
f
and x
D
b
}
f(x)
with slope 1
x
x
A
a
b
(x)
s
{f(s
1

x)+b(x) )(s

x)
D
f
and x
[

a/2,a/2]
}
A
s
1
max{f(s
1

x)+b(x) )(s

x)
D
f
and x
[

a/
2
,a/
2
]
}
=f(s
1
+a/
2
)+b(

a/
2
)= f(s
1
+a/
2
)+A=f(s
1
)+a/
2
+A
s
A
f
⊕
b
A+a/2
Flat Gray Scale Dilation
•
In practice, gray

scale dilation is performed
using
flat
structuring element
–
b(x,y)=0 if (x,y)
D
b
; otherwise, b(x,y) is not defined
–
In this case, D
b
needs to be specified as a binary
matrix with 1s being its domain
–
(f
⊕b)(x,y)=max{f(x

x’,y

y’),
(x’,y’)
D
b
}
–
It is the same as the “max” filter in order statistic
filtering with arbitrarily shaped domain
–
D
b
can be obtained using strel function as in binary
dilation case
Flat Gray

Scale Dilation
(f
⊕b)(s)=max{f(s

x)
x
D
b
}
f(x)
with slope 1
x
x
A
a
D
b
s
f
⊕
b
1
Effects of Gray

Scale Dilation
•
Depending on the structuring element
adopted
•
If all the values are non

negative
(including flat gray scale dilation), the
resulting image tends to be brighter
•
Dark details are either reduced or
eliminated
–
Wrinkle removal
Gray

Scale Erosion
•
Gray

scale erosion of
f
by
b
–
(f
⊖b)(s,t)=
min
{f(s
+
x,t
+
y)

b(x,y)(s
+
x), (t
+
y)
D
f
and (x,y)
D
b
}
–
D
f
and D
b
are the domains of
f
and
b
respectively
–
(f
⊖b) chooses the minimum value of (f

b) in the domain defined
by the structuring element
Gray

Scale Erosion
(f
⊖b)(s)=min{f(s+x)

b(x)(s+x)
D
f
and x
D
b
}
f(x)
with slope
1
x
x
A
a
b
(x)
s
{f(s
1
+x)

b(x) )(s+x)
D
f
and x
[

a/2,a/2]
}
A
s
1
min{f(s
1
+x)

b(x) )(s+x)
D
f
and x
[

a/2,a/2]
}
=f(s
1

a/2)

b(

a/2)= f(s
1

a/2)

A= f(s
1
)

a/2

A
s
A+a/
2
f
⊖
b
Flat Gray Scale Erosion
•
In practice, gray

scale erosion is performed
using
flat
structuring element
–
b(x,y)=0 if (x,y)
D
b
; otherwise, b(x,y) is not defined
–
In this case, D
b
needs to be specified as a binary
matrix with 1s being its domain
–
(f
⊖b)(x,y)=min{f(x+x’,y+y’),
(x’,y’)
D
b
}
–
It is the same as the “min” filter in order statistic
filtering with arbitrarily shaped domain
–
D
b
can be obtained using strel function as in binary
case
Effects of Gray

Scale Erosion
•
Depending on the structuring element
adopted
•
If all the values are non

negative
(including flat gray scale dilation), the
resulting image tends to be darker
•
Bright details are either reduced or
eliminated
Examples
Reduced
Eliminated
Reduced
Eliminated
Dual Operations
•
Gray

scale dilation and erosion are duals
with respect to function complementation
and reflection
–
⊖
–
It means dilation of a bright object is equal to
erosion of its dark background
)
,
(
ˆ
and
)
,
(
where
)
,
(
)
(
)
,
)(
ˆ
(
y
x
b
b
y
x
f
f
t
s
b
f
t
s
b
f
c
c
c
Gray

Scale Opening and Closing
•
The definitions of gray

scale opening and
closing are similar to that of binary case
–
Both are defined in terms of dilation and erosion
•
Opening (erosion followed by dilation)
–
A
◦
b=(A
⊖
b)
⊕
b
•
Closing (dilation followed by erosion)
–
A
•b=(A
⊕
b)
⊖
b
•
Again, opening and closing are dual to each
other with respect to complementation and
reflection
–
)
ˆ
(
or
)
ˆ
(
b
f
b
f
b
f
b
f
c
c
Geometric Interpretation
Properties of Gray

Scale
Opening and Closing
•
Opening
1.
(
f
◦
b
)
f
2.
If
f
1
f
2
, then (
f
1
◦
b
)
(
f
2
◦
b
)
3.
(
f
◦
b
)
◦
b
=
f
◦
b
•
Closing
1.
f
(
f
•
b
)
2.
If
f
1
f
2
, then (
f
1
•
b
)
(
f
2
•
b
)
3.
(
f
•
b
•
b
) =
f
•
b
•
The notation e
r
is used to indicate that the domain
of e is a subset of r and e(x,y)
r(x,y)
•
The above properties can be justified using the the
geometric interpretation of opening and closing shown
previously
Example
9.31
(a) may
be not correct
Applications of Gray

Scale
Morphology
•
Morphological smoothing
–
Opening (reduce bright details) followed by closing (reduce dark details)
–
Alternating sequential filtering
•
Repeat opening followed by closing with structuring elements of increasing
sizes
A
◦
b
5
A
◦
b
5
•b
5
A
•b
2
◦
b
2
•b
3
◦
b
3
•b
4
◦
b
4
•b
5
◦
b
5
Applications of Gray

Scale
Morphology
•
Morphological gradient
–
Effects of dilation (brighter) and erosion (darker) are
manifested on edges of an image
–
g = (f
⊕b)

(f⊖b) can be used to bring out edges of
an image
Applications of Gray

Scale
Morphology
•
Top

hat transform
–
Defined as h = f
–
(f
◦
b)
–
Useful for enhancing details in the presence
of shading
Another Application of Top

Hat
Transform
•
Compensation for nonuniform background
illumination
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