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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

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**Gray-Scale Dilation (f⊕b)(s)=max{f(s-x)+b(x)|(s-x)Df and xDb}**

Illustrated in 1D (f⊕b)(s)=max{f(s-x)+b(x)|(s-x)Df and xDb} f(x) with slope 1 b(x) A x x a 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 {f(s1-x)+b(x)| )|(s-x)Df and x[-a/2,a/2]} f ⊕ b A A A+a/2 s s s1

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**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

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**Flat Gray-Scale Dilation**

(f⊕b)(s)=max{f(s-x)| xDb} f(x) with slope 1 Db 1 A x x a f ⊕ b s

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**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

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**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

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**Gray-Scale Erosion (f⊖b)(s)=min{f(s+x)-b(x)|(s+x)Df and xDb}**

f(x) with slope 1 b(x) A x x a 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 f ⊖ b A+a/2 A {f(s1+x)-b(x)| )|(s+x)Df and x[-a/2,a/2]} s s s1

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**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

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**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

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Examples Reduced Eliminated Eliminated Reduced

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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

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**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

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**Geometric Interpretation**

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**Properties of Gray-Scale Opening and Closing**

( f ◦ b ) f If f1 f2, then (f 1◦ b) (f 2◦ b ) (f ◦ b )◦ b =f ◦ b Closing f ( f • b ) If f1 f2, then (f 1 • b) (f 2 • b ) (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

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Example 9.31 (a) may be not correct

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**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

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**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

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**Applications of Gray-Scale Morphology**

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

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**Another Application of Top-Hat Transform**

Compensation for nonuniform background illumination

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