Mathematical Morphology
L.J. van Vliet TNW-IST Quantitative Imaging
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The basic operations are for gray-value images are, f(x) a) Complement = gray-scale inversion b) Translation: c) Offset = gray addition: d) Multiplication = gray scaling: f(x+v) f(x) + t a f(x) f1(x) À f2(x)
L.J. van Vliet TNW-IST Quantitative Imaging
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f
f ⊗ g(σ)
dytB f
tetB f
Segmentation: Thresholding
L.J. van Vliet TNW-IST Quantitative Imaging
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Divide the image into objects and background
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Local MIN filter
[ εB f ]( x ) = min f ( x + β )
β ∈B
a
f(x)
minf(a,5)
g(x)
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minf(a,9)
Opening & Top Hat
L.J. van Vliet TNW-IST Quantitative Imaging
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Opening (or lower-envelope): min-filter followed by max-filter.
Subtract the background
I ( x, y )
I ( x, y ) − Iˆwhite ( x, y )
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Duality opening and closing
L.J. van Vliet TNW-IST Quantitative Imaging
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(Re-)define erosion and dilation
[ f B ]( x ) [ f • B ]( x )
[ ( f B ) ⊕ B ]( x ) [ ( f ⊕ B ) B ]( x )
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Erosion of a function f(x) by a set B a function g(x)
[ εB f ]( x ) = ∧ f−β = min f ( x + β )
β ∈B β ∈B
[ εg f ]( x ) = min { f ( x + β ) − g ( β )}
β ∈D{ g }
f c (x)
−f ( x )
ˆ = { a a = −b, for b ∈ B } B
Opening
L.J. van Vliet TNW-IST Quantitative Imaging
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Erosion followed by dilation
[ γg f ]( x ) = [ δg εg f ]( x )
f(x)
f(x)
g(x)
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g(x)
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Closing
L.J. van Vliet TNW-IST Quantitative Imaging
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Dilation followed by erosion.
[ φg f ]( x ) = [ εg δg f ]( x )
f(x)
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(Re)define erosion and dilation
f B f ⊕B
[ εB f ]( x ) [ δB f ]( x )
and the duality relation becomes
⎡ ( f B )c ⎤ ( x ) = ⎢⎣ ⎥⎦
with
ˆ ⎤ (x) ⎡ f c ⊕B ⎢⎣ ⎥⎦
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Texture smoothing = texture threshold
1 φ +γ [ tet B f ]( x ) = ⎡⎣ 2 ( B B )⎤ ⎦f (x)
Not idempotent for all f(x) Self-dual
f
tetB f
Smoothing: morphology vs linear
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Local MAX filter
[ δB f ]( x ) = max f ( x − β )
β ∈B
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f(x)
maxf(a,5)
g(x)
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maxf(a,9)
Erosion: Local minimum filter
L.J. van Vliet TNW-IST Quantitative Imaging
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Байду номын сангаас
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I ( x, y ) Iˆwhite ( x, y )
Application: Shading correction
1 δ +ε [ dyt B f ]( x ) = ⎡⎣ 2 ( B B )⎤ ⎦f (x)
Not idempotent Self-dual: dytB –f = – dytB f
f
dytB f
Application: Smoothing 2
L.J. van Vliet TNW-IST Quantitative Imaging
e) Intersection = minimum operator: f1(x) ¿ f2(x) f) Union = maximum operator:
a)
b)
c)
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e)
f)
Dilation: Local maximum filter
L.J. van Vliet TNW-IST Quantitative Imaging
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max distance
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minf ( maxf ( f , size ), size )
f(x)
size
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bot_hat(f , size ) = f − upp(f , size )
Dilation
L.J. van Vliet TNW-IST Quantitative Imaging
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Dilation of a function f(x) by a set B a function g(x)
L.J. van Vliet TNW-IST Quantitative Imaging
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f ( x, y )
⎡ δB f ⎤ ( x ) ⎣ size ⎦
⎡ εB δB f ⎤ ( x ) ⎣ size size ⎦
size = 3 size = 7 size = 13 size = 27
size = 3 size = 7 size = 13 size = 27
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dilation
closing
erosion
opening
Ramp + Texture
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Morphological filters can unravel an image into ramps and textures Textures cannot be distinguished from noise
Mathematical Morphology
Introduction to functions
Lucas J. van Vliet
http://homepage.tudelft.nl/e3q6n/
1 Quantitative Imaging Group Department Imaging Science & Technology Faculty of Applied Sciences
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