gray morphology

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