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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)Df and (x,y)Db}– Df and Db 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
b̂ b̂b̂ ),(ˆ yxbb
Gray-Scale Dilation• Illustrated in 1D
– (f⊕b)(s)=max{f(s-x)+b(x)|(s-x)Df and xDb}
f(x) with slope 1
x xA
a
b(x)
s
{f(s1-x)+b(x)| )|(s-x)Df and x[-a/2,a/2]}
A
s1
max{f(s1-x)+b(x)| )|(s-x)Df and x[-a/2,a/2]}
=f(s1+a/2)+b(-a/2)= f(s1+a/2)+A=f(s1)+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)Db; otherwise, b(x,y) is not defined
– In this case, Db 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’)Db}
– It is the same as the “max” filter in order statistic filtering with arbitrarily shaped domain
– Db can be obtained using strel function as in binary dilation case
Flat Gray-Scale Dilation
(f⊕b)(s)=max{f(s-x)| xDb}
f(x) with slope 1
x xA
a
Db
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)Df and
(x,y)Db}
– Df and Db 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)Df and xDb}
f(x) with slope 1
x xA
a
b(x)
s
{f(s1+x)-b(x)| )|(s+x)Df and x[-a/2,a/2]}A
s1
min{f(s1+x)-b(x)| )|(s+x)Df and x[-a/2,a/2]}
=f(s1-a/2)-b(-a/2)= f(s1-a/2)-A= f(s1)-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)Db; otherwise, b(x,y) is not defined
– In this case, Db 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’)Db}
– It is the same as the “min” filter in order statistic filtering with arbitrarily shaped domain
– Db 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(
yxbbyxff
tsbftsbfc
cc
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( bfbfbfbf cc
Geometric Interpretation
Properties of Gray-Scale Opening and Closing
• Opening1. ( f ◦ b ) f2. If f1 f2, then (f 1◦ b) (f 2◦ b )3. (f ◦ b )◦ b =f ◦ b
• Closing– f ( f • b ) – If f1 f2, then (f 1 • b) (f 2 • b )– (f • b • b) =f • b
1. 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)
2. The above properties can be justified using the the geometric interpretation of opening and closing shown previously
Example9.31 (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◦b5
A◦b5•b5 A•b2◦b2•b3◦b3•b4◦b4•b5◦b5
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