gray morphology

molassesitalianAI and Robotics

Nov 6, 2013 (3 years and 5 months ago)

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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